diff --git a/ReadMe.md b/ReadMe.md
index 3e57fe6..2607ac8 100644
--- a/ReadMe.md
+++ b/ReadMe.md
@@ -104,7 +104,7 @@ github](https://github.com/gutow/Algebra_with_Sympy/issues).
 
 #### Change Log
 
-* 0.9.2rc0
+* 0.9.2
   * `=@` shorthand syntax for defining equations in IPython compatible 
     environments.
   * Fixed bug where `root()` override called `sqrt()` on bare expressions.
diff --git a/docs/algebra_with_sympy.html b/docs/algebra_with_sympy.html
index 3704d1c..216fe21 100644
--- a/docs/algebra_with_sympy.html
+++ b/docs/algebra_with_sympy.html
@@ -3,14 +3,14 @@
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@@ -47,13 +47,14 @@ <h2>Contents</h2>
 
         <h2>Submodules</h2>
         <ul>
-                <li><a href="algebra_with_sympy/algebraic_equation.html">algebra_with_sympy.algebraic_equation</a></li>
+                <li><a href="algebra_with_sympy/algebraic_equation.html">algebraic_equation</a></li>
+                <li><a href="algebra_with_sympy/preparser.html">preparser</a></li>
         </ul>
 
 
-            <footer>Algebra with Sympy v0.9.1</footer>
+            <footer>Algebra with Sympy v0.9.2</footer>
 
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+        <a class="attribution" title="pdoc: Python API documentation generator" href="https://pdoc.dev" target="_blank">
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@@ -61,7 +62,7 @@ <h2>Submodules</h2>
 </div>
     </nav>
     <main class="pdoc">
-            <section>
+            <section class="module-info">
                     <h1 class="modulename">
 algebra_with_sympy    </h1>
 
@@ -98,13 +99,26 @@ <h4 id="introduction">Introduction</h4>
 
 <p>A simple example as it would appear in a <a href="https://jupyter.org">Jupyter</a> 
 notebook is shown immediately below:
-<img src="./resources/simple_example.png" alt="screenshot of simple example" /></p>
+<img src="https://gutow.github.io/Algebra_with_Sympy/resources/simple_example.png" alt="screenshot of simple example" /></p>
 
 <p>Once the algebra is complete it is possible to substitute numbers with 
 units into the solved equation to calculate a numerical solution with 
 proper units.</p>
 
-<p><a href="Demonstration%20of%20equation%20class.html">More examples here</a>.</p>
+<p>In IPython environments (IPython and Jupyter) there is also a shorthand 
+syntax for entering equations provided through the IPython preparser. An 
+equation can be specified as <code>eq1 =@ a/b = c/d</code>. If no Python name is 
+specified for the equation (<code>eq1</code> left out in the previous example), the 
+equation will still 
+be defined, but will not be easily accessible for further computation. The 
+<code>=@</code> symbol combination was chosen to avoid conflicts with reserved python 
+symbols while minimizing impacts on syntax highlighting and autoformatting. A 
+screenshot of an example in Jupyter is shown immediately below:</p>
+
+<p><img src="https://gutow.github.io/Algebra_with_Sympy/resources/short_syntax.png" alt="screenshot of short syntax" /></p>
+
+<p><a href="https://gutow.github.io/Algebra_with_Sympy/Demonstration%20of%20equation%20class.html">More examples of the capabilities of Algebra with Sympy are 
+here</a>.</p>
 
 <p>Many math packages such as <a href="https://www.sagemath.org/">SageMath</a> 
 and <a href="http://maxima.sourceforge.net/">Maxima</a> have similar capabilities, 
@@ -116,7 +130,7 @@ <h4 id="controlling-the-format-of-interactive-outputs">Controlling the Format of
 <ul>
 <li><p><strong>In graphical environments (Jupyter)</strong> you will get rendered Latex such as 
 $\frac{a}{b} = \frac{c}{d}$. To also see the code representation (what can 
-be copy and pasted for 
+be copied and pasted for 
 additional computation) set <code>algwsym_config.output.show_code = True</code>. 
 This will print the code version (e.g. <code>Equation(a,b/c)</code>) of the equation as 
 well. This code version can be accessed directly by calling <code>repr()</code> on the 
@@ -163,6 +177,12 @@ <h4 id="issues-or-comments">Issues or Comments</h4>
 <h4 id="change-log">Change Log</h4>
 
 <ul>
+<li>0.9.2
+<ul>
+<li><code>=@</code> shorthand syntax for defining equations in IPython compatible 
+environments.</li>
+<li>Fixed bug where <code>root()</code> override called <code>sqrt()</code> on bare expressions.</li>
+</ul></li>
 <li>0.9.1
 <ul>
 <li>Equations labeled with their python name, if they have one.</li>
@@ -191,21 +211,23 @@ <h5 id="this-software-is-distributed-under-the-gnu-v3-licensehttpsgnuorglicenses
 <p>Copyright - Jonathan Gutow 2021, 2022</p>
 </div>
 
-                        <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">.. include:: ../docs/intro.md</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="n">__docformat__</span> <span class="o">=</span> <span class="s2">&quot;numpy&quot;</span>
-
-<span class="kn">from</span> <span class="nn">algebra_with_sympy.algebraic_equation</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="c1"># Set the output formatting defaults</span>
-<span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span> <span class="o">=</span> <span class="kc">False</span>
-<span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span> <span class="o">=</span> <span class="kc">False</span>
-<span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="kc">True</span>
-</pre></div>
-
-        </details>
+                        <input id="algebra_with_sympy-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+
+                        <label class="view-source-button" for="algebra_with_sympy-view-source"><span>View Source</span></label>
+
+                        <div class="pdoc-code codehilite"><pre><span></span><span id="L-1"><a href="#L-1"><span class="linenos"> 1</span></a><span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-2"><a href="#L-2"><span class="linenos"> 2</span></a><span class="sd">.. include:: ../docs/intro.md</span>
+</span><span id="L-3"><a href="#L-3"><span class="linenos"> 3</span></a><span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-4"><a href="#L-4"><span class="linenos"> 4</span></a><span class="n">__docformat__</span> <span class="o">=</span> <span class="s2">&quot;numpy&quot;</span>
+</span><span id="L-5"><a href="#L-5"><span class="linenos"> 5</span></a>
+</span><span id="L-6"><a href="#L-6"><span class="linenos"> 6</span></a><span class="kn">from</span> <span class="nn">algebra_with_sympy.algebraic_equation</span> <span class="kn">import</span> <span class="o">*</span>
+</span><span id="L-7"><a href="#L-7"><span class="linenos"> 7</span></a><span class="kn">from</span> <span class="nn">algebra_with_sympy.preparser</span> <span class="kn">import</span> <span class="o">*</span>
+</span><span id="L-8"><a href="#L-8"><span class="linenos"> 8</span></a><span class="c1"># Set the output formatting defaults</span>
+</span><span id="L-9"><a href="#L-9"><span class="linenos"> 9</span></a><span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span> <span class="o">=</span> <span class="kc">False</span>
+</span><span id="L-10"><a href="#L-10"><span class="linenos">10</span></a><span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span> <span class="o">=</span> <span class="kc">False</span>
+</span><span id="L-11"><a href="#L-11"><span class="linenos">11</span></a><span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="kc">True</span>
+</span></pre></div>
+
 
             </section>
     </main>
diff --git a/docs/algebra_with_sympy/algebraic_equation.html b/docs/algebra_with_sympy/algebraic_equation.html
index 23adfe3..afca1b9 100644
--- a/docs/algebra_with_sympy/algebraic_equation.html
+++ b/docs/algebra_with_sympy/algebraic_equation.html
@@ -3,14 +3,14 @@
 <head>
     <meta charset="utf-8">
     <meta name="viewport" content="width=device-width, initial-scale=1">
-    <meta name="generator" content="pdoc 10.0.3"/>
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     <title>algebra_with_sympy.algebraic_equation API documentation</title>
 
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         tex: {
@@ -224,6 +224,24 @@ <h2>API Documentation</h2>
             <li>
                     <a class="function" href="#collect">collect</a>
             </li>
+            <li>
+                    <a class="class" href="#Eq">Eq</a>
+                            <ul class="memberlist">
+                        <li>
+                                <a class="function" href="#Eq.__init__">Eq</a>
+                        </li>
+                        <li>
+                                <a class="function" href="#Eq.to_Equation">to_Equation</a>
+                        </li>
+                        <li>
+                                <a class="function" href="#Eq.to_Eqn">to_Eqn</a>
+                        </li>
+                        <li>
+                                <a class="variable" href="#Eq.default_assumptions">default_assumptions</a>
+                        </li>
+                </ul>
+
+            </li>
             <li>
                     <a class="class" href="#EqnFunction">EqnFunction</a>
                             <ul class="memberlist">
@@ -501,10 +519,10 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#re.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#re.is_commutative">is_commutative</a>
+                                <a class="variable" href="#re.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
-                                <a class="variable" href="#re.is_imaginary">is_imaginary</a>
+                                <a class="variable" href="#re.is_commutative">is_commutative</a>
                         </li>
                 </ul>
 
@@ -522,10 +540,10 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#im.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#im.is_commutative">is_commutative</a>
+                                <a class="variable" href="#im.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
-                                <a class="variable" href="#im.is_imaginary">is_imaginary</a>
+                                <a class="variable" href="#im.is_commutative">is_commutative</a>
                         </li>
                 </ul>
 
@@ -542,15 +560,15 @@ <h2>API Documentation</h2>
                         <li>
                                 <a class="variable" href="#sign.default_assumptions">default_assumptions</a>
                         </li>
-                        <li>
-                                <a class="variable" href="#sign.is_finite">is_finite</a>
-                        </li>
                         <li>
                                 <a class="variable" href="#sign.is_commutative">is_commutative</a>
                         </li>
                         <li>
                                 <a class="variable" href="#sign.is_infinite">is_infinite</a>
                         </li>
+                        <li>
+                                <a class="variable" href="#sign.is_finite">is_finite</a>
+                        </li>
                 </ul>
 
             </li>
@@ -573,13 +591,13 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#Abs.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#Abs.is_commutative">is_commutative</a>
+                                <a class="variable" href="#Abs.is_negative">is_negative</a>
                         </li>
                         <li>
                                 <a class="variable" href="#Abs.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
-                                <a class="variable" href="#Abs.is_negative">is_negative</a>
+                                <a class="variable" href="#Abs.is_commutative">is_commutative</a>
                         </li>
                 </ul>
 
@@ -615,19 +633,19 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#arg.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#arg.is_commutative">is_commutative</a>
+                                <a class="variable" href="#arg.is_infinite">is_infinite</a>
                         </li>
                         <li>
                                 <a class="variable" href="#arg.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
-                                <a class="variable" href="#arg.is_hermitian">is_hermitian</a>
+                                <a class="variable" href="#arg.is_commutative">is_commutative</a>
                         </li>
                         <li>
-                                <a class="variable" href="#arg.is_complex">is_complex</a>
+                                <a class="variable" href="#arg.is_hermitian">is_hermitian</a>
                         </li>
                         <li>
-                                <a class="variable" href="#arg.is_infinite">is_infinite</a>
+                                <a class="variable" href="#arg.is_complex">is_complex</a>
                         </li>
                 </ul>
 
@@ -1494,43 +1512,43 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#LeviCivita.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_hermitian">is_hermitian</a>
+                                <a class="variable" href="#LeviCivita.is_transcendental">is_transcendental</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_extended_real">is_extended_real</a>
+                                <a class="variable" href="#LeviCivita.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
                                 <a class="variable" href="#LeviCivita.is_commutative">is_commutative</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_complex">is_complex</a>
+                                <a class="variable" href="#LeviCivita.is_infinite">is_infinite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_real">is_real</a>
+                                <a class="variable" href="#LeviCivita.is_irrational">is_irrational</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_transcendental">is_transcendental</a>
+                                <a class="variable" href="#LeviCivita.is_real">is_real</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_infinite">is_infinite</a>
+                                <a class="variable" href="#LeviCivita.is_hermitian">is_hermitian</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_imaginary">is_imaginary</a>
+                                <a class="variable" href="#LeviCivita.is_finite">is_finite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_algebraic">is_algebraic</a>
+                                <a class="variable" href="#LeviCivita.is_extended_real">is_extended_real</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_finite">is_finite</a>
+                                <a class="variable" href="#LeviCivita.is_algebraic">is_algebraic</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_rational">is_rational</a>
+                                <a class="variable" href="#LeviCivita.is_complex">is_complex</a>
                         </li>
                         <li>
                                 <a class="variable" href="#LeviCivita.is_noninteger">is_noninteger</a>
                         </li>
                         <li>
-                                <a class="variable" href="#LeviCivita.is_irrational">is_irrational</a>
+                                <a class="variable" href="#LeviCivita.is_rational">is_rational</a>
                         </li>
                 </ul>
 
@@ -1548,43 +1566,43 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#KroneckerDelta.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_hermitian">is_hermitian</a>
+                                <a class="variable" href="#KroneckerDelta.is_transcendental">is_transcendental</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_extended_real">is_extended_real</a>
+                                <a class="variable" href="#KroneckerDelta.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
                                 <a class="variable" href="#KroneckerDelta.is_commutative">is_commutative</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_complex">is_complex</a>
+                                <a class="variable" href="#KroneckerDelta.is_infinite">is_infinite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_real">is_real</a>
+                                <a class="variable" href="#KroneckerDelta.is_irrational">is_irrational</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_transcendental">is_transcendental</a>
+                                <a class="variable" href="#KroneckerDelta.is_real">is_real</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_infinite">is_infinite</a>
+                                <a class="variable" href="#KroneckerDelta.is_hermitian">is_hermitian</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_imaginary">is_imaginary</a>
+                                <a class="variable" href="#KroneckerDelta.is_finite">is_finite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_algebraic">is_algebraic</a>
+                                <a class="variable" href="#KroneckerDelta.is_extended_real">is_extended_real</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_finite">is_finite</a>
+                                <a class="variable" href="#KroneckerDelta.is_algebraic">is_algebraic</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_rational">is_rational</a>
+                                <a class="variable" href="#KroneckerDelta.is_complex">is_complex</a>
                         </li>
                         <li>
                                 <a class="variable" href="#KroneckerDelta.is_noninteger">is_noninteger</a>
                         </li>
                         <li>
-                                <a class="variable" href="#KroneckerDelta.is_irrational">is_irrational</a>
+                                <a class="variable" href="#KroneckerDelta.is_rational">is_rational</a>
                         </li>
                 </ul>
 
@@ -1602,25 +1620,25 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#SingularityFunction.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#SingularityFunction.is_hermitian">is_hermitian</a>
+                                <a class="variable" href="#SingularityFunction.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
                                 <a class="variable" href="#SingularityFunction.is_commutative">is_commutative</a>
                         </li>
                         <li>
-                                <a class="variable" href="#SingularityFunction.is_extended_real">is_extended_real</a>
+                                <a class="variable" href="#SingularityFunction.is_infinite">is_infinite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#SingularityFunction.is_complex">is_complex</a>
+                                <a class="variable" href="#SingularityFunction.is_finite">is_finite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#SingularityFunction.is_infinite">is_infinite</a>
+                                <a class="variable" href="#SingularityFunction.is_hermitian">is_hermitian</a>
                         </li>
                         <li>
-                                <a class="variable" href="#SingularityFunction.is_imaginary">is_imaginary</a>
+                                <a class="variable" href="#SingularityFunction.is_extended_real">is_extended_real</a>
                         </li>
                         <li>
-                                <a class="variable" href="#SingularityFunction.is_finite">is_finite</a>
+                                <a class="variable" href="#SingularityFunction.is_complex">is_complex</a>
                         </li>
                 </ul>
 
@@ -1638,25 +1656,25 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#DiracDelta.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#DiracDelta.is_hermitian">is_hermitian</a>
+                                <a class="variable" href="#DiracDelta.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
                                 <a class="variable" href="#DiracDelta.is_commutative">is_commutative</a>
                         </li>
                         <li>
-                                <a class="variable" href="#DiracDelta.is_extended_real">is_extended_real</a>
+                                <a class="variable" href="#DiracDelta.is_infinite">is_infinite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#DiracDelta.is_complex">is_complex</a>
+                                <a class="variable" href="#DiracDelta.is_finite">is_finite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#DiracDelta.is_infinite">is_infinite</a>
+                                <a class="variable" href="#DiracDelta.is_hermitian">is_hermitian</a>
                         </li>
                         <li>
-                                <a class="variable" href="#DiracDelta.is_imaginary">is_imaginary</a>
+                                <a class="variable" href="#DiracDelta.is_extended_real">is_extended_real</a>
                         </li>
                         <li>
-                                <a class="variable" href="#DiracDelta.is_finite">is_finite</a>
+                                <a class="variable" href="#DiracDelta.is_complex">is_complex</a>
                         </li>
                 </ul>
 
@@ -1674,25 +1692,25 @@ <h2>API Documentation</h2>
                                 <a class="variable" href="#Heaviside.default_assumptions">default_assumptions</a>
                         </li>
                         <li>
-                                <a class="variable" href="#Heaviside.is_hermitian">is_hermitian</a>
+                                <a class="variable" href="#Heaviside.is_imaginary">is_imaginary</a>
                         </li>
                         <li>
                                 <a class="variable" href="#Heaviside.is_commutative">is_commutative</a>
                         </li>
                         <li>
-                                <a class="variable" href="#Heaviside.is_extended_real">is_extended_real</a>
+                                <a class="variable" href="#Heaviside.is_infinite">is_infinite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#Heaviside.is_complex">is_complex</a>
+                                <a class="variable" href="#Heaviside.is_finite">is_finite</a>
                         </li>
                         <li>
-                                <a class="variable" href="#Heaviside.is_infinite">is_infinite</a>
+                                <a class="variable" href="#Heaviside.is_hermitian">is_hermitian</a>
                         </li>
                         <li>
-                                <a class="variable" href="#Heaviside.is_imaginary">is_imaginary</a>
+                                <a class="variable" href="#Heaviside.is_extended_real">is_extended_real</a>
                         </li>
                         <li>
-                                <a class="variable" href="#Heaviside.is_finite">is_finite</a>
+                                <a class="variable" href="#Heaviside.is_complex">is_complex</a>
                         </li>
                 </ul>
 
@@ -2222,9 +2240,9 @@ <h2>API Documentation</h2>
     </ul>
 
 
-            <footer>Algebra with Sympy v0.9.1</footer>
+            <footer>Algebra with Sympy v0.9.2</footer>
 
-        <a class="attribution" title="pdoc: Python API documentation generator" href="https://pdoc.dev">
+        <a class="attribution" title="pdoc: Python API documentation generator" href="https://pdoc.dev" target="_blank">
             built with <span class="visually-hidden">pdoc</span><img
                 alt="pdoc logo"
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@@ -2232,7 +2250,7 @@ <h2>API Documentation</h2>
 </div>
     </nav>
     <main class="pdoc">
-            <section>
+            <section class="module-info">
                     <h1 class="modulename">
 <a href="./../algebra_with_sympy.html">algebra_with_sympy</a><wbr>.algebraic_equation    </h1>
 
@@ -2245,7 +2263,7 @@ <h1 class="modulename">
 not be set.</p>
 
 <p>This class should not be confused with the Boolean class <code>Equality</code>
-(abbreviated <code>Eq</code>) which specifies that the equality of two objects is
+(abbreviated <code><a href="#Eq">Eq</a></code>) which specifies that the equality of two objects is
 <code>True</code>.</p>
 
 <p>This tool applies operations to both sides of the equation simultaneously, just
@@ -2261,960 +2279,993 @@ <h1 class="modulename">
 <a href="http://maxima.sourceforge.net/">Maxima</a>.</p>
 </div>
 
-                        <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Algebraic Equations with SymPy</span>
-<span class="sd">==============================</span>
-
-<span class="sd">These tools define relations that all high school and college students would</span>
-<span class="sd">recognize as mathematical equations. They consist of a left hand side (lhs)</span>
-<span class="sd">and a right hand side (rhs) connected by a relation operator such as &quot;=&quot;. At</span>
-<span class="sd">present the &quot;=&quot; relation operator is the only option. The relation operator may</span>
-<span class="sd">not be set.</span>
-
-<span class="sd">This class should not be confused with the Boolean class ``Equality``</span>
-<span class="sd">(abbreviated ``Eq``) which specifies that the equality of two objects is</span>
-<span class="sd">``True``.</span>
-
-<span class="sd">This tool applies operations to both sides of the equation simultaneously, just</span>
-<span class="sd">as students are taught to do when attempting to isolate (solve for) a</span>
-<span class="sd">variable. Thus the statement ``Equation/b`` yields a new equation</span>
-<span class="sd">``Equation.lhs/b = Equation.rhs/b``</span>
-
-<span class="sd">The intent is to allow using the mathematical tools in SymPy to rearrange</span>
-<span class="sd">equations and perform algebra in a stepwise fashion. In this way more people</span>
-<span class="sd">can successfully perform algebraic rearrangements without stumbling over</span>
-<span class="sd">missed details such as a negative sign. This mimics the capabilities available</span>
-<span class="sd">in [SageMath](https://www.sagemath.org/) and</span>
-<span class="sd">[Maxima](http://maxima.sourceforge.net/).</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.evalf</span> <span class="kn">import</span> <span class="n">EvalfMixin</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">_sympify</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">functions</span>
-<span class="kn">import</span> <span class="nn">functools</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-
-
-<span class="k">class</span> <span class="nc">algwsym_config</span><span class="p">():</span>
-
-    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This is a class to hold parameters that control behavior of</span>
-<span class="sd">        the algebra_with_sympy package.</span>
-
-<span class="sd">        Settings</span>
-<span class="sd">        ========</span>
-<span class="sd">        Printing</span>
-<span class="sd">        --------</span>
-<span class="sd">        In interactive environments the default output of an equation is a</span>
-<span class="sd">        human readable string with the two sides connected by an equals</span>
-<span class="sd">        sign or a typeset equation with the two sides connected by an equals sign.</span>
-<span class="sd">        `print(Eqn)` or `str(Eqn)` will return this human readable text version of</span>
-<span class="sd">        the equation as well. This is consistent with python standards, but not</span>
-<span class="sd">        sympy, where `str()` is supposed to return something that can be</span>
-<span class="sd">        copy-pasted into code. If the equation has a declared name as in `eq1 =</span>
-<span class="sd">        Eqn(a,b/c)` the name will be displayed to the right of the equation in</span>
-<span class="sd">        parentheses (eg. `a = b/c    (eq1)`). Use `print(repr(Eqn))` instead of</span>
-<span class="sd">        `print(Eqn)` or `repr(Eqn)` instead of `str(Eqn)` to get a code</span>
-<span class="sd">        compatible version of the equation.</span>
-
-<span class="sd">        You can adjust this behvior using some flags that impact output:</span>
-<span class="sd">        * `algwsym_config.output.show_code` default is `False`.</span>
-<span class="sd">        * `algwsym_config.output.human_text` default is `False`.</span>
-<span class="sd">        * `algwsym_config.output.label` default is `True`.</span>
-
-<span class="sd">        In interactive environments you can get both types of output by setting</span>
-<span class="sd">        the `algwsym_config.output.show_code` flag. If this flag is true</span>
-<span class="sd">        calls to `latex` and `str` will also print an additional line &quot;code</span>
-<span class="sd">        version: `repr(Eqn)`&quot;. Thus in Jupyter you will get a line of typeset</span>
-<span class="sd">        mathematics output preceded by the code version that can be copy-pasted.</span>
-<span class="sd">        Default is `False`.</span>
-
-<span class="sd">        A second flag `algwsym_config.output.human_text` is useful in</span>
-<span class="sd">        text-based interactive environments such as command line python or</span>
-<span class="sd">        ipython. If this flag is true `repr` will return `str`. Thus the human</span>
-<span class="sd">        readable text will be printed as the output of a line that is an</span>
-<span class="sd">        expression containing an equation.</span>
-<span class="sd">        Default is `False`.</span>
-
-<span class="sd">        Setting both of these flags to true in a command line or ipython</span>
-<span class="sd">        environment will show both the code version and the human readable text.</span>
-<span class="sd">        These flags impact the behavior of the `print(Eqn)` statement.</span>
-
-<span class="sd">        The third flag `algwsym_config.output.label` has a default value of</span>
-<span class="sd">        `True`. Setting this to `False` suppresses the labeling of an equation</span>
-<span class="sd">        with its python name off to the right of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">pass</span>
-
-    <span class="k">class</span> <span class="nc">output</span><span class="p">():</span>
-
-        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;This holds settings that impact output.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">pass</span>
-
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">show_code</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` code versions of the equation expression will be</span>
-<span class="sd">            output in interactive environments. Default = `False`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">show_code</span>
-
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">human_text</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` the human readable equation expression will be</span>
-<span class="sd">            output in text interactive environments. Default = `False`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">human_text</span>
-
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` the human readable equation will be followed by the</span>
-<span class="sd">            python name it is assigned to. Default = `True`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span>
-
-
-<span class="k">class</span> <span class="nc">Equation</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class defines an equation with a left-hand-side (lhs) and a right-</span>
-<span class="sd">    hand-side (rhs) connected by the &quot;=&quot; operator (e.g. `p*V = n*R*T`).</span>
-
-<span class="sd">    Explanation</span>
-<span class="sd">    ===========</span>
-<span class="sd">    This class defines relations that all high school and college students</span>
-<span class="sd">    would recognize as mathematical equations. At present only the &quot;=&quot; relation</span>
-<span class="sd">    operator is recognized.</span>
-
-<span class="sd">    This class is intended to allow using the mathematical tools in SymPy to</span>
-<span class="sd">    rearrange equations and perform algebra in a stepwise fashion. In this</span>
-<span class="sd">    way more people can successfully perform algebraic rearrangements without</span>
-<span class="sd">    stumbling over missed details such as a negative sign.</span>
-
-<span class="sd">    Create an equation with the call ``Equation(lhs,rhs)``, where ``lhs`` and</span>
-<span class="sd">    ``rhs`` are any valid Sympy expression. ``Eqn(...)`` is a synonym for</span>
-<span class="sd">    ``Equation(...)``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-<span class="sd">    lhs: sympy expression, ``class Expr``.</span>
-<span class="sd">    rhs: sympy expression, ``class Expr``.</span>
-<span class="sd">    kwargs:</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from algebra_with_sympy import *</span>
-<span class="sd">    &gt;&gt;&gt; a, b, c, x = var(&#39;a b c x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Equation(a,b/c)</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t=Eqn(a,b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t*c</span>
-<span class="sd">    Equation(a*c, b)</span>
-<span class="sd">    &gt;&gt;&gt; c*t</span>
-<span class="sd">    Equation(a*c, b)</span>
-<span class="sd">    &gt;&gt;&gt; exp(t)</span>
-<span class="sd">    Equation(exp(a), exp(b/c))</span>
-<span class="sd">    &gt;&gt;&gt; exp(log(t))</span>
-<span class="sd">    Equation(a, b/c)</span>
-
-<span class="sd">    Simplification and Expansion</span>
-<span class="sd">    &gt;&gt;&gt; f = Eqn(x**2 - 1, c)</span>
-<span class="sd">    &gt;&gt;&gt; f</span>
-<span class="sd">    Equation(x**2 - 1, c)</span>
-<span class="sd">    &gt;&gt;&gt; f/(x+1)</span>
-<span class="sd">    Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; (f/(x+1)).simplify()</span>
-<span class="sd">    Equation(x - 1, c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; simplify(f/(x+1))</span>
-<span class="sd">    Equation(x - 1, c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; (f/(x+1)).expand()</span>
-<span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; expand(f/(x+1))</span>
-<span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; factor(f)</span>
-<span class="sd">    Equation((x - 1)*(x + 1), c)</span>
-<span class="sd">    &gt;&gt;&gt; f.factor()</span>
-<span class="sd">    Equation((x - 1)*(x + 1), c)</span>
-<span class="sd">    &gt;&gt;&gt; f2 = f+a*x**2+b*x +c</span>
-<span class="sd">    &gt;&gt;&gt; f2</span>
-<span class="sd">    Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>
-<span class="sd">    &gt;&gt;&gt; collect(f2,x)</span>
-<span class="sd">    Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>
-
-<span class="sd">    Apply operation to only one side</span>
-<span class="sd">    &gt;&gt;&gt; poly = Eqn(a*x**2 + b*x + c*x**2, a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.applyrhs(factor,x)</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>
-<span class="sd">    &gt;&gt;&gt; poly.applylhs(factor)</span>
-<span class="sd">    Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.applylhs(collect,x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
-
-<span class="sd">    ``.apply...`` also works with user defined python functions</span>
-<span class="sd">    &gt;&gt;&gt; def addsquare(eqn):</span>
-<span class="sd">    ...     return eqn+eqn**2</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; t.apply(addsquare)</span>
-<span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.applyrhs(addsquare)</span>
-<span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.apply(addsquare, side = &#39;rhs&#39;)</span>
-<span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.applylhs(addsquare)</span>
-<span class="sd">    Equation(a**2 + a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; addsquare(t)</span>
-<span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
-
-<span class="sd">    Inaddition to ``.apply...`` there is also the less general ``.do``,</span>
-<span class="sd">    ``.dolhs``, ``.dorhs``, which only works for operations defined on the</span>
-<span class="sd">    ``Expr`` class (e.g.``.collect(), .factor(), .expand()``, etc...).</span>
-<span class="sd">    &gt;&gt;&gt; poly.dolhs.collect(x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.dorhs.collect(x)</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>
-<span class="sd">    &gt;&gt;&gt; poly.do.collect(x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>
-<span class="sd">    &gt;&gt;&gt; poly.dorhs.factor()</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>
-
-<span class="sd">    ``poly.do.exp()`` or other sympy math functions will raise an error.</span>
-
-<span class="sd">    Rearranging an equation (simple example made complicated as illustration)</span>
-<span class="sd">    &gt;&gt;&gt; p, V, n, R, T = var(&#39;p V n R T&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; eq1=Eqn(p*V,n*R*T)</span>
-<span class="sd">    &gt;&gt;&gt; eq1</span>
-<span class="sd">    Equation(V*p, R*T*n)</span>
-<span class="sd">    &gt;&gt;&gt; eq2 =eq1/V</span>
-<span class="sd">    &gt;&gt;&gt; eq2</span>
-<span class="sd">    Equation(p, R*T*n/V)</span>
-<span class="sd">    &gt;&gt;&gt; eq3 = eq2/R/T</span>
-<span class="sd">    &gt;&gt;&gt; eq3</span>
-<span class="sd">    Equation(p/(R*T), n/V)</span>
-<span class="sd">    &gt;&gt;&gt; eq4 = eq3*R/p</span>
-<span class="sd">    &gt;&gt;&gt; eq4</span>
-<span class="sd">    Equation(1/T, R*n/(V*p))</span>
-<span class="sd">    &gt;&gt;&gt; 1/eq4</span>
-<span class="sd">    Equation(T, V*p/(R*n))</span>
-<span class="sd">    &gt;&gt;&gt; eq5 = 1/eq4 - T</span>
-<span class="sd">    &gt;&gt;&gt; eq5</span>
-<span class="sd">    Equation(0, -T + V*p/(R*n))</span>
-
-<span class="sd">    Substitution (#&#39;s and units)</span>
-<span class="sd">    &gt;&gt;&gt; L, atm, mol, K = var(&#39;L atm mol K&#39;, positive=True, real=True) # units</span>
-<span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L})</span>
-<span class="sd">    Equation(p, 0.9334325*atm)</span>
-<span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L}).evalf(4)</span>
-<span class="sd">    Equation(p, 0.9334*atm)</span>
-
-<span class="sd">    Combining equations (Math with equations: lhs with lhs and rhs with rhs)</span>
-<span class="sd">    &gt;&gt;&gt; q = Eqn(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q</span>
-<span class="sd">    Equation(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; t</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; q+t</span>
-<span class="sd">    Equation(a*c + a, b/c + b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q/t</span>
-<span class="sd">    Equation(c, 1/c)</span>
-<span class="sd">    &gt;&gt;&gt; t**q</span>
-<span class="sd">    Equation(a**(a*c), (b/c)**(b/c**2))</span>
-
-<span class="sd">    Utility operations</span>
-<span class="sd">    &gt;&gt;&gt; t.reversed</span>
-<span class="sd">    Equation(b/c, a)</span>
-<span class="sd">    &gt;&gt;&gt; t.swap</span>
-<span class="sd">    Equation(b/c, a)</span>
-<span class="sd">    &gt;&gt;&gt; t.lhs</span>
-<span class="sd">    a</span>
-<span class="sd">    &gt;&gt;&gt; t.rhs</span>
-<span class="sd">    b/c</span>
-<span class="sd">    &gt;&gt;&gt; t.as_Boolean()</span>
-<span class="sd">    Eq(a, b/c)</span>
-
-<span class="sd">    `.check()` convenience method for `.as_Boolean().simplify()`</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import I, pi</span>
-<span class="sd">    &gt;&gt;&gt; Equation(pi*(I+2), pi*I+2*pi).check()</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; Eqn(a,a+1).check()</span>
-<span class="sd">    False</span>
-
-<span class="sd">    Differentiation</span>
-<span class="sd">    Differentiation is applied to both sides if the wrt variable appears on</span>
-<span class="sd">    both sides.</span>
-<span class="sd">    &gt;&gt;&gt; q=Eqn(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q</span>
-<span class="sd">    Equation(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,b)</span>
-<span class="sd">    Equation(Derivative(a*c, b), c**(-2))</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c)</span>
-<span class="sd">    Equation(a, -2*b/c**3)</span>
-<span class="sd">    &gt;&gt;&gt; diff(log(q),b)</span>
-<span class="sd">    Equation(Derivative(log(a*c), b), 1/b)</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c,2)</span>
-<span class="sd">    Equation(Derivative(a, c), 6*b/c**4)</span>
-
-<span class="sd">    If you specify multiple differentiation all at once the assumption</span>
-<span class="sd">    is order of differentiation matters and the lhs will not be</span>
-<span class="sd">    evaluated.</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c,b)</span>
-<span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
-
-<span class="sd">    To overcome this specify the order of operations.</span>
-<span class="sd">    &gt;&gt;&gt; diff(diff(q,c),b)</span>
-<span class="sd">    Equation(Derivative(a, b), -2/c**3)</span>
-
-<span class="sd">    But the reverse order returns an unevaulated lhs (a may depend on b).</span>
-<span class="sd">    &gt;&gt;&gt; diff(diff(q,b),c)</span>
-<span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
-
-<span class="sd">    Integration can only be performed on one side at a time.</span>
-<span class="sd">    &gt;&gt;&gt; q=Eqn(a*c,b/c)</span>
-<span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;rhs&#39;)</span>
-<span class="sd">    b**2/(2*c)</span>
-<span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;lhs&#39;)</span>
-<span class="sd">    a*b*c</span>
-
-<span class="sd">    Make a pretty statement of integration from an equation</span>
-<span class="sd">    &gt;&gt;&gt; Eqn(Integral(q.lhs,b),integrate(q,b,side=&#39;rhs&#39;))</span>
-<span class="sd">    Equation(Integral(a*c, b), b**2/(2*c))</span>
-
-<span class="sd">    Integration of each side with respect to different variables</span>
-<span class="sd">    &gt;&gt;&gt; q.dorhs.integrate(b).dolhs.integrate(a)</span>
-<span class="sd">    Equation(a**2*c/2, b**2/(2*c))</span>
-
-<span class="sd">    SymPy&#39;s solvers do not understand these equations. They expect an</span>
-<span class="sd">    expression that the solver assumes = 0. Thus to use the solver the</span>
-<span class="sd">    equation must be rearranged so that all non-zero symbols are on one side.</span>
-<span class="sd">    Then just the non-zero symbolic side is passed to ``solve()``.</span>
-<span class="sd">    &gt;&gt;&gt; t2 = t-t.rhs</span>
-<span class="sd">    &gt;&gt;&gt; t2</span>
-<span class="sd">    Equation(a - b/c, 0)</span>
-<span class="sd">    &gt;&gt;&gt; solve(t2.lhs,c)</span>
-<span class="sd">    [b/a]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">lhs</span><span class="p">)</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;lhs and rhs must be valid sympy expressions.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_get_eqn_name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">IPython</span> <span class="kn">import</span> <span class="n">get_ipython</span>
-        <span class="n">global_dict</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">get_ipython</span><span class="p">(),</span> <span class="s1">&#39;user_ns&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-        <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
-        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
-        <span class="k">if</span> <span class="n">global_dict</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">var_name</span> <span class="ow">in</span> <span class="n">global_dict</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">])</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">==</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> \
-                            <span class="n">var_name</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s1">&#39;_&#39;</span><span class="p">):</span>
-                        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
-                        <span class="k">return</span> <span class="n">var_name</span>
-        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span> <span class="o">=</span> <span class="n">human_text</span>
-        <span class="k">return</span> <span class="s1">&#39;&#39;</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">lhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the lhs of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the rhs of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Converts the equation to an Equality.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
-<span class="sd">        Parameters</span>
-<span class="sd">        ----------</span>
-<span class="sd">        kwargs any appropriate for `Equality`.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        -------</span>
-<span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">reversed</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Swaps the lhs and the rhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">swap</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Synonym for `.reversed`</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">reversed</span>
-
-    <span class="k">def</span> <span class="nf">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># Applies a function to an expression checking whether there</span>
-        <span class="c1"># is a specialized version associated with the particular type of</span>
-        <span class="c1"># expression. Errors will be raised if the function cannot be</span>
-        <span class="c1"># applied to an expression.</span>
-        <span class="n">funcname</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s1">&#39;__name__&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">funcname</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-            <span class="n">localfunc</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">funcname</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">localfunc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">localfunc</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply an operation/function/method to the equation returning the</span>
-<span class="sd">        resulting equation.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        func: object</span>
-<span class="sd">            object to apply usually a function</span>
-
-<span class="sd">        args: as necessary for the function</span>
-
-<span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
-<span class="sd">            Specifies which side of the equation the operation will be applied</span>
-<span class="sd">            to. Default is &#39;both&#39;.</span>
-
-<span class="sd">        kwargs: as necessary for the function</span>
-<span class="sd">         &quot;&quot;&quot;</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the lhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the rhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
-
-    <span class="k">class</span> <span class="nc">_sides</span><span class="p">:</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Helper class for the `.do.`, `.dolhs.`, `.dorhs.` syntax for applying</span>
-<span class="sd">        submethods of expressions.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eqn</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">eqn</span> <span class="o">=</span> <span class="n">eqn</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="o">=</span> <span class="n">side</span>
-
-        <span class="k">def</span> <span class="fm">__getattr__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="kc">None</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;rhs&#39;</span><span class="p">,</span> <span class="s1">&#39;both&#39;</span><span class="p">):</span>
-                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;Expressions in the equation have no &#39;</span>
-                                     <span class="s1">&#39;attribute `&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span>
-                    <span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;`. Try `.apply(&#39;</span>
-                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;, *args)` or &#39;</span>
-                                                   <span class="s1">&#39;pass the equation as a parameter to `&#39;</span>
-                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;()`.&#39;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">apply</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">side</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">do</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dolhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dorhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Overrides of binary math operations</span>
-    <span class="c1">#####</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_binary_op</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">opfunc_ab</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rtruediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rpow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__pow__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Operation helper functions</span>
-    <span class="c1">#####</span>
-    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># TODO: cancel out factors common to both sides.</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># TODO Find why diff and Derivative do not appear to pass through</span>
-        <span class="c1">#  kwargs to this. Since we cannot set evaluation of lhs manually</span>
-        <span class="c1">#  try to be intelligent about when to do it.</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Derivative</span>
-        <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">False</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)):</span>
-            <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">sym</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span>
-                        <span class="n">_sympify</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">):</span>
-                    <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">True</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="n">eval_lhs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">side</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s1">&#39;side&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>  <span class="c1"># Could not seem to pass values for</span>
-        <span class="c1"># `evaluate` through to here.</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;You must specify `side=&quot;lhs&quot;` or `side=&quot;rhs&quot;` &#39;</span>
-                             <span class="s1">&#39;when integrating an Equation&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;`side` must equal &quot;lhs&quot; or &quot;rhs&quot;.&#39;</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Output helper functions</span>
-    <span class="c1">#####</span>
-    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">repstr</span> <span class="o">=</span> <span class="s1">&#39;Equation(</span><span class="si">%s</span><span class="s1">, </span><span class="si">%s</span><span class="s1">)&#39;</span> <span class="o">%</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">())</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__str__</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">repstr</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span> <span class="ow">and</span> <span class="ow">not</span> \
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span><span class="s1">&#39;</span><span class="se">\\</span><span class="s1">text{code version: &#39;</span><span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">+</span><span class="s1">&#39;} </span><span class="se">\\</span><span class="s1">newline &#39;</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;=&#39;</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span><span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,&#39;</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;(</span><span class="se">\\</span><span class="s1">text{&#39;</span><span class="o">+</span><span class="n">namestr</span><span class="o">+</span><span class="s1">&#39;})&#39;</span>
-        <span class="k">return</span> <span class="n">tempstr</span>
-
-    <span class="k">def</span> <span class="fm">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span><span class="p">:</span>
-            <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;code version: &#39;</span><span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="o">+</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39; = &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span> <span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;          (&#39;</span> <span class="o">+</span> <span class="n">namestr</span> <span class="o">+</span> <span class="s1">&#39;)&#39;</span>
-        <span class="k">return</span> <span class="n">tempstr</span>
-
-
-<span class="n">Eqn</span> <span class="o">=</span> <span class="n">Equation</span>
-
-<span class="k">def</span> <span class="nf">solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Override of sympy `solve()`. If it is passed an equation it will</span>
-<span class="sd">    output the solutions as typeset equations and return</span>
-<span class="sd">    the answer as a list of equations that can be accessed for additional</span>
-<span class="sd">    manipulations.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.solvers.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-    <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-        <span class="n">flags</span><span class="p">[</span><span class="s1">&#39;dict&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">f</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-        <span class="n">solns</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c1"># return result</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">k</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-                <span class="n">val</span> <span class="o">=</span> <span class="n">k</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-                <span class="n">tempeqn</span> <span class="o">=</span> <span class="n">Eqn</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
-                <span class="n">display</span><span class="p">(</span><span class="n">tempeqn</span><span class="p">)</span>
-                <span class="n">solns</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tempeqn</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">solns</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-
-<span class="k">def</span> <span class="nf">sqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">evaluate</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Override of sympy convenience function `sqrt`. Simply divides equations</span>
-<span class="sd">    into two sides if `arg` is an instance of `Equation`. This avoids an</span>
-<span class="sd">    issue with the way sympy is delaying specialized applications of _Pow_ on</span>
-<span class="sd">    objects that are not basic sympy expressions.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span> <span class="k">as</span> <span class="n">symsqrt</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">),</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span><span class="n">evaluate</span><span class="p">)</span>
-
-<span class="c1"># Pick up the docstring for sqrt from sympy</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span> <span class="k">as</span> <span class="n">symsqrt</span>
-<span class="n">sqrt</span><span class="o">.</span><span class="vm">__doc__</span><span class="o">+=</span><span class="n">symsqrt</span><span class="o">.</span><span class="vm">__doc__</span>
-<span class="k">del</span> <span class="n">symsqrt</span>
-
-<span class="k">def</span> <span class="nf">root</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">evaluate</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Override of sympy convenience function `root`. Simply divides equations</span>
-<span class="sd">    into two sides if `arg` is an instance of `Equation`. This avoids an</span>
-<span class="sd">    issue with the way sympy is delaying specialized applications of _Pow_ on</span>
-<span class="sd">    objects that are not basic sympy expressions.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">root</span> <span class="k">as</span> <span class="n">symroot</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">),</span> <span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span>
-                                                            <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">)</span>
-
-<span class="c1"># pick up the docstring for root from sympy</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">root</span> <span class="k">as</span> <span class="n">symroot</span>
-<span class="n">root</span><span class="o">.</span><span class="vm">__doc__</span><span class="o">+=</span><span class="n">symroot</span><span class="o">.</span><span class="vm">__doc__</span>
-<span class="k">del</span> <span class="n">symroot</span>
-
-<span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
-            <span class="n">distribute_order_term</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Override of sympy `collect()`.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
-    <span class="n">_eval_collect</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s1">&#39;_eval_collect&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">_eval_collect</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_eval_collect</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span>
-                             <span class="n">exact</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span> <span class="n">exact</span><span class="p">,</span>
-                       <span class="n">distribute_order_term</span><span class="p">)</span>
-
-
-<span class="c1">#####</span>
-<span class="c1"># Extension of the Function class. For incorporation into SymPy this should</span>
-<span class="c1"># become part of the class</span>
-<span class="c1">#####</span>
-<span class="k">class</span> <span class="nc">EqnFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Extension of the sympy Function class to understand equations. Each</span>
-<span class="sd">    sympy function impacted by this extension is listed in the documentation</span>
-<span class="sd">    that follows.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">eqnloc</span> <span class="o">=</span> <span class="kc">None</span>
-        <span class="n">neqns</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
-                    <span class="n">neqns</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">eqnloc</span> <span class="o">=</span> <span class="n">i</span>
-            <span class="k">if</span> <span class="n">neqns</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Function calls with more than one &#39;</span>
-                                          <span class="s1">&#39;Equation as a parameter are not &#39;</span>
-                                          <span class="s1">&#39;supported. You may be able to get &#39;</span>
-                                          <span class="s1">&#39;your desired outcome using .applyrhs&#39;</span>
-                                          <span class="s1">&#39; and .applylhs.&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">neqns</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span><span class="o">.</span><span class="n">lhs</span>
-                <span class="n">lhs</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-                <span class="n">newargs</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span><span class="o">.</span><span class="n">rhs</span>
-                <span class="n">rhs</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-
-<span class="k">for</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">functions</span><span class="o">.</span><span class="n">__all__</span><span class="p">:</span>
-    <span class="c1"># TODO: This will not be needed when incorporated into SymPy listed in</span>
-    <span class="c1">#  `skip` cannot be extended because of `mro` error or `metaclass</span>
-    <span class="c1">#  conflict`. Seems to reflect expectation that a helper function will be</span>
-    <span class="c1">#  defined within the object (e.g. `_eval_power()` for all the flavors of</span>
-    <span class="c1">#  `root`).</span>
-    <span class="n">skip</span> <span class="o">=</span> <span class="p">(</span><span class="s1">&#39;sqrt&#39;</span><span class="p">,</span> <span class="s1">&#39;root&#39;</span><span class="p">,</span> <span class="s1">&#39;Min&#39;</span><span class="p">,</span> <span class="s1">&#39;Max&#39;</span><span class="p">,</span> <span class="s1">&#39;Id&#39;</span><span class="p">,</span> <span class="s1">&#39;real_root&#39;</span><span class="p">,</span> <span class="s1">&#39;cbrt&#39;</span><span class="p">,</span>
-            <span class="s1">&#39;unbranched_argument&#39;</span><span class="p">,</span> <span class="s1">&#39;polarify&#39;</span><span class="p">,</span> <span class="s1">&#39;unpolarify&#39;</span><span class="p">,</span>
-            <span class="s1">&#39;piecewise_fold&#39;</span><span class="p">,</span> <span class="s1">&#39;E1&#39;</span><span class="p">,</span> <span class="s1">&#39;Eijk&#39;</span><span class="p">,</span> <span class="s1">&#39;bspline_basis&#39;</span><span class="p">,</span>
-            <span class="s1">&#39;bspline_basis_set&#39;</span><span class="p">,</span> <span class="s1">&#39;interpolating_spline&#39;</span><span class="p">,</span> <span class="s1">&#39;jn_zeros&#39;</span><span class="p">,</span>
-            <span class="s1">&#39;jacobi_normalized&#39;</span><span class="p">,</span> <span class="s1">&#39;Ynm_c&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">func</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">skip</span><span class="p">:</span>
-        <span class="n">execstr</span> <span class="o">=</span> <span class="s1">&#39;class &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">func</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;(&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span>
-            <span class="n">func</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;,EqnFunction):</span><span class="se">\n</span><span class="s1">    pass</span><span class="se">\n</span><span class="s1">&#39;</span>
-        <span class="n">exec</span><span class="p">(</span><span class="n">execstr</span><span class="p">,</span> <span class="nb">globals</span><span class="p">(),</span> <span class="nb">locals</span><span class="p">())</span>
-</pre></div>
-
-        </details>
+                        <input id="algebraic_equation-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+
+                        <label class="view-source-button" for="algebraic_equation-view-source"><span>View Source</span></label>
+
+                        <div class="pdoc-code codehilite"><pre><span></span><span id="L-1"><a href="#L-1"><span class="linenos">  1</span></a><span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-2"><a href="#L-2"><span class="linenos">  2</span></a><span class="sd">Algebraic Equations with SymPy</span>
+</span><span id="L-3"><a href="#L-3"><span class="linenos">  3</span></a><span class="sd">==============================</span>
+</span><span id="L-4"><a href="#L-4"><span class="linenos">  4</span></a>
+</span><span id="L-5"><a href="#L-5"><span class="linenos">  5</span></a><span class="sd">These tools define relations that all high school and college students would</span>
+</span><span id="L-6"><a href="#L-6"><span class="linenos">  6</span></a><span class="sd">recognize as mathematical equations. They consist of a left hand side (lhs)</span>
+</span><span id="L-7"><a href="#L-7"><span class="linenos">  7</span></a><span class="sd">and a right hand side (rhs) connected by a relation operator such as &quot;=&quot;. At</span>
+</span><span id="L-8"><a href="#L-8"><span class="linenos">  8</span></a><span class="sd">present the &quot;=&quot; relation operator is the only option. The relation operator may</span>
+</span><span id="L-9"><a href="#L-9"><span class="linenos">  9</span></a><span class="sd">not be set.</span>
+</span><span id="L-10"><a href="#L-10"><span class="linenos"> 10</span></a>
+</span><span id="L-11"><a href="#L-11"><span class="linenos"> 11</span></a><span class="sd">This class should not be confused with the Boolean class ``Equality``</span>
+</span><span id="L-12"><a href="#L-12"><span class="linenos"> 12</span></a><span class="sd">(abbreviated ``Eq``) which specifies that the equality of two objects is</span>
+</span><span id="L-13"><a href="#L-13"><span class="linenos"> 13</span></a><span class="sd">``True``.</span>
+</span><span id="L-14"><a href="#L-14"><span class="linenos"> 14</span></a>
+</span><span id="L-15"><a href="#L-15"><span class="linenos"> 15</span></a><span class="sd">This tool applies operations to both sides of the equation simultaneously, just</span>
+</span><span id="L-16"><a href="#L-16"><span class="linenos"> 16</span></a><span class="sd">as students are taught to do when attempting to isolate (solve for) a</span>
+</span><span id="L-17"><a href="#L-17"><span class="linenos"> 17</span></a><span class="sd">variable. Thus the statement ``Equation/b`` yields a new equation</span>
+</span><span id="L-18"><a href="#L-18"><span class="linenos"> 18</span></a><span class="sd">``Equation.lhs/b = Equation.rhs/b``</span>
+</span><span id="L-19"><a href="#L-19"><span class="linenos"> 19</span></a>
+</span><span id="L-20"><a href="#L-20"><span class="linenos"> 20</span></a><span class="sd">The intent is to allow using the mathematical tools in SymPy to rearrange</span>
+</span><span id="L-21"><a href="#L-21"><span class="linenos"> 21</span></a><span class="sd">equations and perform algebra in a stepwise fashion. In this way more people</span>
+</span><span id="L-22"><a href="#L-22"><span class="linenos"> 22</span></a><span class="sd">can successfully perform algebraic rearrangements without stumbling over</span>
+</span><span id="L-23"><a href="#L-23"><span class="linenos"> 23</span></a><span class="sd">missed details such as a negative sign. This mimics the capabilities available</span>
+</span><span id="L-24"><a href="#L-24"><span class="linenos"> 24</span></a><span class="sd">in [SageMath](https://www.sagemath.org/) and</span>
+</span><span id="L-25"><a href="#L-25"><span class="linenos"> 25</span></a><span class="sd">[Maxima](http://maxima.sourceforge.net/).</span>
+</span><span id="L-26"><a href="#L-26"><span class="linenos"> 26</span></a><span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-27"><a href="#L-27"><span class="linenos"> 27</span></a>
+</span><span id="L-28"><a href="#L-28"><span class="linenos"> 28</span></a><span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
+</span><span id="L-29"><a href="#L-29"><span class="linenos"> 29</span></a><span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
+</span><span id="L-30"><a href="#L-30"><span class="linenos"> 30</span></a><span class="kn">from</span> <span class="nn">sympy.core.evalf</span> <span class="kn">import</span> <span class="n">EvalfMixin</span>
+</span><span id="L-31"><a href="#L-31"><span class="linenos"> 31</span></a><span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">_sympify</span>
+</span><span id="L-32"><a href="#L-32"><span class="linenos"> 32</span></a><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">functions</span>
+</span><span id="L-33"><a href="#L-33"><span class="linenos"> 33</span></a><span class="kn">import</span> <span class="nn">functools</span>
+</span><span id="L-34"><a href="#L-34"><span class="linenos"> 34</span></a><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
+</span><span id="L-35"><a href="#L-35"><span class="linenos"> 35</span></a>
+</span><span id="L-36"><a href="#L-36"><span class="linenos"> 36</span></a>
+</span><span id="L-37"><a href="#L-37"><span class="linenos"> 37</span></a><span class="k">class</span> <span class="nc">algwsym_config</span><span class="p">():</span>
+</span><span id="L-38"><a href="#L-38"><span class="linenos"> 38</span></a>
+</span><span id="L-39"><a href="#L-39"><span class="linenos"> 39</span></a>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-40"><a href="#L-40"><span class="linenos"> 40</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-41"><a href="#L-41"><span class="linenos"> 41</span></a><span class="sd">        This is a class to hold parameters that control behavior of</span>
+</span><span id="L-42"><a href="#L-42"><span class="linenos"> 42</span></a><span class="sd">        the algebra_with_sympy package.</span>
+</span><span id="L-43"><a href="#L-43"><span class="linenos"> 43</span></a>
+</span><span id="L-44"><a href="#L-44"><span class="linenos"> 44</span></a><span class="sd">        Settings</span>
+</span><span id="L-45"><a href="#L-45"><span class="linenos"> 45</span></a><span class="sd">        ========</span>
+</span><span id="L-46"><a href="#L-46"><span class="linenos"> 46</span></a><span class="sd">        Printing</span>
+</span><span id="L-47"><a href="#L-47"><span class="linenos"> 47</span></a><span class="sd">        --------</span>
+</span><span id="L-48"><a href="#L-48"><span class="linenos"> 48</span></a><span class="sd">        In interactive environments the default output of an equation is a</span>
+</span><span id="L-49"><a href="#L-49"><span class="linenos"> 49</span></a><span class="sd">        human readable string with the two sides connected by an equals</span>
+</span><span id="L-50"><a href="#L-50"><span class="linenos"> 50</span></a><span class="sd">        sign or a typeset equation with the two sides connected by an equals sign.</span>
+</span><span id="L-51"><a href="#L-51"><span class="linenos"> 51</span></a><span class="sd">        `print(Eqn)` or `str(Eqn)` will return this human readable text version of</span>
+</span><span id="L-52"><a href="#L-52"><span class="linenos"> 52</span></a><span class="sd">        the equation as well. This is consistent with python standards, but not</span>
+</span><span id="L-53"><a href="#L-53"><span class="linenos"> 53</span></a><span class="sd">        sympy, where `str()` is supposed to return something that can be</span>
+</span><span id="L-54"><a href="#L-54"><span class="linenos"> 54</span></a><span class="sd">        copy-pasted into code. If the equation has a declared name as in `eq1 =</span>
+</span><span id="L-55"><a href="#L-55"><span class="linenos"> 55</span></a><span class="sd">        Eqn(a,b/c)` the name will be displayed to the right of the equation in</span>
+</span><span id="L-56"><a href="#L-56"><span class="linenos"> 56</span></a><span class="sd">        parentheses (eg. `a = b/c    (eq1)`). Use `print(repr(Eqn))` instead of</span>
+</span><span id="L-57"><a href="#L-57"><span class="linenos"> 57</span></a><span class="sd">        `print(Eqn)` or `repr(Eqn)` instead of `str(Eqn)` to get a code</span>
+</span><span id="L-58"><a href="#L-58"><span class="linenos"> 58</span></a><span class="sd">        compatible version of the equation.</span>
+</span><span id="L-59"><a href="#L-59"><span class="linenos"> 59</span></a>
+</span><span id="L-60"><a href="#L-60"><span class="linenos"> 60</span></a><span class="sd">        You can adjust this behvior using some flags that impact output:</span>
+</span><span id="L-61"><a href="#L-61"><span class="linenos"> 61</span></a><span class="sd">        * `algwsym_config.output.show_code` default is `False`.</span>
+</span><span id="L-62"><a href="#L-62"><span class="linenos"> 62</span></a><span class="sd">        * `algwsym_config.output.human_text` default is `False`.</span>
+</span><span id="L-63"><a href="#L-63"><span class="linenos"> 63</span></a><span class="sd">        * `algwsym_config.output.label` default is `True`.</span>
+</span><span id="L-64"><a href="#L-64"><span class="linenos"> 64</span></a>
+</span><span id="L-65"><a href="#L-65"><span class="linenos"> 65</span></a><span class="sd">        In interactive environments you can get both types of output by setting</span>
+</span><span id="L-66"><a href="#L-66"><span class="linenos"> 66</span></a><span class="sd">        the `algwsym_config.output.show_code` flag. If this flag is true</span>
+</span><span id="L-67"><a href="#L-67"><span class="linenos"> 67</span></a><span class="sd">        calls to `latex` and `str` will also print an additional line &quot;code</span>
+</span><span id="L-68"><a href="#L-68"><span class="linenos"> 68</span></a><span class="sd">        version: `repr(Eqn)`&quot;. Thus in Jupyter you will get a line of typeset</span>
+</span><span id="L-69"><a href="#L-69"><span class="linenos"> 69</span></a><span class="sd">        mathematics output preceded by the code version that can be copy-pasted.</span>
+</span><span id="L-70"><a href="#L-70"><span class="linenos"> 70</span></a><span class="sd">        Default is `False`.</span>
+</span><span id="L-71"><a href="#L-71"><span class="linenos"> 71</span></a>
+</span><span id="L-72"><a href="#L-72"><span class="linenos"> 72</span></a><span class="sd">        A second flag `algwsym_config.output.human_text` is useful in</span>
+</span><span id="L-73"><a href="#L-73"><span class="linenos"> 73</span></a><span class="sd">        text-based interactive environments such as command line python or</span>
+</span><span id="L-74"><a href="#L-74"><span class="linenos"> 74</span></a><span class="sd">        ipython. If this flag is true `repr` will return `str`. Thus the human</span>
+</span><span id="L-75"><a href="#L-75"><span class="linenos"> 75</span></a><span class="sd">        readable text will be printed as the output of a line that is an</span>
+</span><span id="L-76"><a href="#L-76"><span class="linenos"> 76</span></a><span class="sd">        expression containing an equation.</span>
+</span><span id="L-77"><a href="#L-77"><span class="linenos"> 77</span></a><span class="sd">        Default is `False`.</span>
+</span><span id="L-78"><a href="#L-78"><span class="linenos"> 78</span></a>
+</span><span id="L-79"><a href="#L-79"><span class="linenos"> 79</span></a><span class="sd">        Setting both of these flags to true in a command line or ipython</span>
+</span><span id="L-80"><a href="#L-80"><span class="linenos"> 80</span></a><span class="sd">        environment will show both the code version and the human readable text.</span>
+</span><span id="L-81"><a href="#L-81"><span class="linenos"> 81</span></a><span class="sd">        These flags impact the behavior of the `print(Eqn)` statement.</span>
+</span><span id="L-82"><a href="#L-82"><span class="linenos"> 82</span></a>
+</span><span id="L-83"><a href="#L-83"><span class="linenos"> 83</span></a><span class="sd">        The third flag `algwsym_config.output.label` has a default value of</span>
+</span><span id="L-84"><a href="#L-84"><span class="linenos"> 84</span></a><span class="sd">        `True`. Setting this to `False` suppresses the labeling of an equation</span>
+</span><span id="L-85"><a href="#L-85"><span class="linenos"> 85</span></a><span class="sd">        with its python name off to the right of the equation.</span>
+</span><span id="L-86"><a href="#L-86"><span class="linenos"> 86</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-87"><a href="#L-87"><span class="linenos"> 87</span></a>        <span class="k">pass</span>
+</span><span id="L-88"><a href="#L-88"><span class="linenos"> 88</span></a>
+</span><span id="L-89"><a href="#L-89"><span class="linenos"> 89</span></a>    <span class="k">class</span> <span class="nc">output</span><span class="p">():</span>
+</span><span id="L-90"><a href="#L-90"><span class="linenos"> 90</span></a>
+</span><span id="L-91"><a href="#L-91"><span class="linenos"> 91</span></a>        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-92"><a href="#L-92"><span class="linenos"> 92</span></a>            <span class="sd">&quot;&quot;&quot;This holds settings that impact output.</span>
+</span><span id="L-93"><a href="#L-93"><span class="linenos"> 93</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="L-94"><a href="#L-94"><span class="linenos"> 94</span></a>            <span class="k">pass</span>
+</span><span id="L-95"><a href="#L-95"><span class="linenos"> 95</span></a>
+</span><span id="L-96"><a href="#L-96"><span class="linenos"> 96</span></a>        <span class="nd">@property</span>
+</span><span id="L-97"><a href="#L-97"><span class="linenos"> 97</span></a>        <span class="k">def</span> <span class="nf">show_code</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-98"><a href="#L-98"><span class="linenos"> 98</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-99"><a href="#L-99"><span class="linenos"> 99</span></a><span class="sd">            If `True` code versions of the equation expression will be</span>
+</span><span id="L-100"><a href="#L-100"><span class="linenos">100</span></a><span class="sd">            output in interactive environments. Default = `False`.</span>
+</span><span id="L-101"><a href="#L-101"><span class="linenos">101</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="L-102"><a href="#L-102"><span class="linenos">102</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">show_code</span>
+</span><span id="L-103"><a href="#L-103"><span class="linenos">103</span></a>
+</span><span id="L-104"><a href="#L-104"><span class="linenos">104</span></a>        <span class="nd">@property</span>
+</span><span id="L-105"><a href="#L-105"><span class="linenos">105</span></a>        <span class="k">def</span> <span class="nf">human_text</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-106"><a href="#L-106"><span class="linenos">106</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-107"><a href="#L-107"><span class="linenos">107</span></a><span class="sd">            If `True` the human readable equation expression will be</span>
+</span><span id="L-108"><a href="#L-108"><span class="linenos">108</span></a><span class="sd">            output in text interactive environments. Default = `False`.</span>
+</span><span id="L-109"><a href="#L-109"><span class="linenos">109</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="L-110"><a href="#L-110"><span class="linenos">110</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="L-111"><a href="#L-111"><span class="linenos">111</span></a>
+</span><span id="L-112"><a href="#L-112"><span class="linenos">112</span></a>        <span class="nd">@property</span>
+</span><span id="L-113"><a href="#L-113"><span class="linenos">113</span></a>        <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-114"><a href="#L-114"><span class="linenos">114</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-115"><a href="#L-115"><span class="linenos">115</span></a><span class="sd">            If `True` the human readable equation will be followed by the</span>
+</span><span id="L-116"><a href="#L-116"><span class="linenos">116</span></a><span class="sd">            python name it is assigned to. Default = `True`.</span>
+</span><span id="L-117"><a href="#L-117"><span class="linenos">117</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="L-118"><a href="#L-118"><span class="linenos">118</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span>
+</span><span id="L-119"><a href="#L-119"><span class="linenos">119</span></a>
+</span><span id="L-120"><a href="#L-120"><span class="linenos">120</span></a>
+</span><span id="L-121"><a href="#L-121"><span class="linenos">121</span></a><span class="k">class</span> <span class="nc">Equation</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
+</span><span id="L-122"><a href="#L-122"><span class="linenos">122</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-123"><a href="#L-123"><span class="linenos">123</span></a><span class="sd">    This class defines an equation with a left-hand-side (lhs) and a right-</span>
+</span><span id="L-124"><a href="#L-124"><span class="linenos">124</span></a><span class="sd">    hand-side (rhs) connected by the &quot;=&quot; operator (e.g. `p*V = n*R*T`).</span>
+</span><span id="L-125"><a href="#L-125"><span class="linenos">125</span></a>
+</span><span id="L-126"><a href="#L-126"><span class="linenos">126</span></a><span class="sd">    Explanation</span>
+</span><span id="L-127"><a href="#L-127"><span class="linenos">127</span></a><span class="sd">    ===========</span>
+</span><span id="L-128"><a href="#L-128"><span class="linenos">128</span></a><span class="sd">    This class defines relations that all high school and college students</span>
+</span><span id="L-129"><a href="#L-129"><span class="linenos">129</span></a><span class="sd">    would recognize as mathematical equations. At present only the &quot;=&quot; relation</span>
+</span><span id="L-130"><a href="#L-130"><span class="linenos">130</span></a><span class="sd">    operator is recognized.</span>
+</span><span id="L-131"><a href="#L-131"><span class="linenos">131</span></a>
+</span><span id="L-132"><a href="#L-132"><span class="linenos">132</span></a><span class="sd">    This class is intended to allow using the mathematical tools in SymPy to</span>
+</span><span id="L-133"><a href="#L-133"><span class="linenos">133</span></a><span class="sd">    rearrange equations and perform algebra in a stepwise fashion. In this</span>
+</span><span id="L-134"><a href="#L-134"><span class="linenos">134</span></a><span class="sd">    way more people can successfully perform algebraic rearrangements without</span>
+</span><span id="L-135"><a href="#L-135"><span class="linenos">135</span></a><span class="sd">    stumbling over missed details such as a negative sign.</span>
+</span><span id="L-136"><a href="#L-136"><span class="linenos">136</span></a>
+</span><span id="L-137"><a href="#L-137"><span class="linenos">137</span></a><span class="sd">    Create an equation with the call ``Equation(lhs,rhs)``, where ``lhs`` and</span>
+</span><span id="L-138"><a href="#L-138"><span class="linenos">138</span></a><span class="sd">    ``rhs`` are any valid Sympy expression. ``Eqn(...)`` is a synonym for</span>
+</span><span id="L-139"><a href="#L-139"><span class="linenos">139</span></a><span class="sd">    ``Equation(...)``.</span>
+</span><span id="L-140"><a href="#L-140"><span class="linenos">140</span></a>
+</span><span id="L-141"><a href="#L-141"><span class="linenos">141</span></a><span class="sd">    Parameters</span>
+</span><span id="L-142"><a href="#L-142"><span class="linenos">142</span></a><span class="sd">    ==========</span>
+</span><span id="L-143"><a href="#L-143"><span class="linenos">143</span></a><span class="sd">    lhs: sympy expression, ``class Expr``.</span>
+</span><span id="L-144"><a href="#L-144"><span class="linenos">144</span></a><span class="sd">    rhs: sympy expression, ``class Expr``.</span>
+</span><span id="L-145"><a href="#L-145"><span class="linenos">145</span></a><span class="sd">    kwargs:</span>
+</span><span id="L-146"><a href="#L-146"><span class="linenos">146</span></a>
+</span><span id="L-147"><a href="#L-147"><span class="linenos">147</span></a><span class="sd">    Examples</span>
+</span><span id="L-148"><a href="#L-148"><span class="linenos">148</span></a><span class="sd">    ========</span>
+</span><span id="L-149"><a href="#L-149"><span class="linenos">149</span></a><span class="sd">    &gt;&gt;&gt; from algebra_with_sympy import *</span>
+</span><span id="L-150"><a href="#L-150"><span class="linenos">150</span></a><span class="sd">    &gt;&gt;&gt; a, b, c, x = var(&#39;a b c x&#39;)</span>
+</span><span id="L-151"><a href="#L-151"><span class="linenos">151</span></a><span class="sd">    &gt;&gt;&gt; Equation(a,b/c)</span>
+</span><span id="L-152"><a href="#L-152"><span class="linenos">152</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="L-153"><a href="#L-153"><span class="linenos">153</span></a><span class="sd">    &gt;&gt;&gt; t=Eqn(a,b/c)</span>
+</span><span id="L-154"><a href="#L-154"><span class="linenos">154</span></a><span class="sd">    &gt;&gt;&gt; t</span>
+</span><span id="L-155"><a href="#L-155"><span class="linenos">155</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="L-156"><a href="#L-156"><span class="linenos">156</span></a><span class="sd">    &gt;&gt;&gt; t*c</span>
+</span><span id="L-157"><a href="#L-157"><span class="linenos">157</span></a><span class="sd">    Equation(a*c, b)</span>
+</span><span id="L-158"><a href="#L-158"><span class="linenos">158</span></a><span class="sd">    &gt;&gt;&gt; c*t</span>
+</span><span id="L-159"><a href="#L-159"><span class="linenos">159</span></a><span class="sd">    Equation(a*c, b)</span>
+</span><span id="L-160"><a href="#L-160"><span class="linenos">160</span></a><span class="sd">    &gt;&gt;&gt; exp(t)</span>
+</span><span id="L-161"><a href="#L-161"><span class="linenos">161</span></a><span class="sd">    Equation(exp(a), exp(b/c))</span>
+</span><span id="L-162"><a href="#L-162"><span class="linenos">162</span></a><span class="sd">    &gt;&gt;&gt; exp(log(t))</span>
+</span><span id="L-163"><a href="#L-163"><span class="linenos">163</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="L-164"><a href="#L-164"><span class="linenos">164</span></a>
+</span><span id="L-165"><a href="#L-165"><span class="linenos">165</span></a><span class="sd">    Simplification and Expansion</span>
+</span><span id="L-166"><a href="#L-166"><span class="linenos">166</span></a><span class="sd">    &gt;&gt;&gt; f = Eqn(x**2 - 1, c)</span>
+</span><span id="L-167"><a href="#L-167"><span class="linenos">167</span></a><span class="sd">    &gt;&gt;&gt; f</span>
+</span><span id="L-168"><a href="#L-168"><span class="linenos">168</span></a><span class="sd">    Equation(x**2 - 1, c)</span>
+</span><span id="L-169"><a href="#L-169"><span class="linenos">169</span></a><span class="sd">    &gt;&gt;&gt; f/(x+1)</span>
+</span><span id="L-170"><a href="#L-170"><span class="linenos">170</span></a><span class="sd">    Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>
+</span><span id="L-171"><a href="#L-171"><span class="linenos">171</span></a><span class="sd">    &gt;&gt;&gt; (f/(x+1)).simplify()</span>
+</span><span id="L-172"><a href="#L-172"><span class="linenos">172</span></a><span class="sd">    Equation(x - 1, c/(x + 1))</span>
+</span><span id="L-173"><a href="#L-173"><span class="linenos">173</span></a><span class="sd">    &gt;&gt;&gt; simplify(f/(x+1))</span>
+</span><span id="L-174"><a href="#L-174"><span class="linenos">174</span></a><span class="sd">    Equation(x - 1, c/(x + 1))</span>
+</span><span id="L-175"><a href="#L-175"><span class="linenos">175</span></a><span class="sd">    &gt;&gt;&gt; (f/(x+1)).expand()</span>
+</span><span id="L-176"><a href="#L-176"><span class="linenos">176</span></a><span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
+</span><span id="L-177"><a href="#L-177"><span class="linenos">177</span></a><span class="sd">    &gt;&gt;&gt; expand(f/(x+1))</span>
+</span><span id="L-178"><a href="#L-178"><span class="linenos">178</span></a><span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
+</span><span id="L-179"><a href="#L-179"><span class="linenos">179</span></a><span class="sd">    &gt;&gt;&gt; factor(f)</span>
+</span><span id="L-180"><a href="#L-180"><span class="linenos">180</span></a><span class="sd">    Equation((x - 1)*(x + 1), c)</span>
+</span><span id="L-181"><a href="#L-181"><span class="linenos">181</span></a><span class="sd">    &gt;&gt;&gt; f.factor()</span>
+</span><span id="L-182"><a href="#L-182"><span class="linenos">182</span></a><span class="sd">    Equation((x - 1)*(x + 1), c)</span>
+</span><span id="L-183"><a href="#L-183"><span class="linenos">183</span></a><span class="sd">    &gt;&gt;&gt; f2 = f+a*x**2+b*x +c</span>
+</span><span id="L-184"><a href="#L-184"><span class="linenos">184</span></a><span class="sd">    &gt;&gt;&gt; f2</span>
+</span><span id="L-185"><a href="#L-185"><span class="linenos">185</span></a><span class="sd">    Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>
+</span><span id="L-186"><a href="#L-186"><span class="linenos">186</span></a><span class="sd">    &gt;&gt;&gt; collect(f2,x)</span>
+</span><span id="L-187"><a href="#L-187"><span class="linenos">187</span></a><span class="sd">    Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>
+</span><span id="L-188"><a href="#L-188"><span class="linenos">188</span></a>
+</span><span id="L-189"><a href="#L-189"><span class="linenos">189</span></a><span class="sd">    Apply operation to only one side</span>
+</span><span id="L-190"><a href="#L-190"><span class="linenos">190</span></a><span class="sd">    &gt;&gt;&gt; poly = Eqn(a*x**2 + b*x + c*x**2, a*x**3 + b*x**3 + c*x)</span>
+</span><span id="L-191"><a href="#L-191"><span class="linenos">191</span></a><span class="sd">    &gt;&gt;&gt; poly.applyrhs(factor,x)</span>
+</span><span id="L-192"><a href="#L-192"><span class="linenos">192</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>
+</span><span id="L-193"><a href="#L-193"><span class="linenos">193</span></a><span class="sd">    &gt;&gt;&gt; poly.applylhs(factor)</span>
+</span><span id="L-194"><a href="#L-194"><span class="linenos">194</span></a><span class="sd">    Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="L-195"><a href="#L-195"><span class="linenos">195</span></a><span class="sd">    &gt;&gt;&gt; poly.applylhs(collect,x)</span>
+</span><span id="L-196"><a href="#L-196"><span class="linenos">196</span></a><span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="L-197"><a href="#L-197"><span class="linenos">197</span></a>
+</span><span id="L-198"><a href="#L-198"><span class="linenos">198</span></a><span class="sd">    ``.apply...`` also works with user defined python functions</span>
+</span><span id="L-199"><a href="#L-199"><span class="linenos">199</span></a><span class="sd">    &gt;&gt;&gt; def addsquare(eqn):</span>
+</span><span id="L-200"><a href="#L-200"><span class="linenos">200</span></a><span class="sd">    ...     return eqn+eqn**2</span>
+</span><span id="L-201"><a href="#L-201"><span class="linenos">201</span></a><span class="sd">    ...</span>
+</span><span id="L-202"><a href="#L-202"><span class="linenos">202</span></a><span class="sd">    &gt;&gt;&gt; t.apply(addsquare)</span>
+</span><span id="L-203"><a href="#L-203"><span class="linenos">203</span></a><span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
+</span><span id="L-204"><a href="#L-204"><span class="linenos">204</span></a><span class="sd">    &gt;&gt;&gt; t.applyrhs(addsquare)</span>
+</span><span id="L-205"><a href="#L-205"><span class="linenos">205</span></a><span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
+</span><span id="L-206"><a href="#L-206"><span class="linenos">206</span></a><span class="sd">    &gt;&gt;&gt; t.apply(addsquare, side = &#39;rhs&#39;)</span>
+</span><span id="L-207"><a href="#L-207"><span class="linenos">207</span></a><span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
+</span><span id="L-208"><a href="#L-208"><span class="linenos">208</span></a><span class="sd">    &gt;&gt;&gt; t.applylhs(addsquare)</span>
+</span><span id="L-209"><a href="#L-209"><span class="linenos">209</span></a><span class="sd">    Equation(a**2 + a, b/c)</span>
+</span><span id="L-210"><a href="#L-210"><span class="linenos">210</span></a><span class="sd">    &gt;&gt;&gt; addsquare(t)</span>
+</span><span id="L-211"><a href="#L-211"><span class="linenos">211</span></a><span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
+</span><span id="L-212"><a href="#L-212"><span class="linenos">212</span></a>
+</span><span id="L-213"><a href="#L-213"><span class="linenos">213</span></a><span class="sd">    Inaddition to ``.apply...`` there is also the less general ``.do``,</span>
+</span><span id="L-214"><a href="#L-214"><span class="linenos">214</span></a><span class="sd">    ``.dolhs``, ``.dorhs``, which only works for operations defined on the</span>
+</span><span id="L-215"><a href="#L-215"><span class="linenos">215</span></a><span class="sd">    ``Expr`` class (e.g.``.collect(), .factor(), .expand()``, etc...).</span>
+</span><span id="L-216"><a href="#L-216"><span class="linenos">216</span></a><span class="sd">    &gt;&gt;&gt; poly.dolhs.collect(x)</span>
+</span><span id="L-217"><a href="#L-217"><span class="linenos">217</span></a><span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="L-218"><a href="#L-218"><span class="linenos">218</span></a><span class="sd">    &gt;&gt;&gt; poly.dorhs.collect(x)</span>
+</span><span id="L-219"><a href="#L-219"><span class="linenos">219</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>
+</span><span id="L-220"><a href="#L-220"><span class="linenos">220</span></a><span class="sd">    &gt;&gt;&gt; poly.do.collect(x)</span>
+</span><span id="L-221"><a href="#L-221"><span class="linenos">221</span></a><span class="sd">    Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>
+</span><span id="L-222"><a href="#L-222"><span class="linenos">222</span></a><span class="sd">    &gt;&gt;&gt; poly.dorhs.factor()</span>
+</span><span id="L-223"><a href="#L-223"><span class="linenos">223</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>
+</span><span id="L-224"><a href="#L-224"><span class="linenos">224</span></a>
+</span><span id="L-225"><a href="#L-225"><span class="linenos">225</span></a><span class="sd">    ``poly.do.exp()`` or other sympy math functions will raise an error.</span>
+</span><span id="L-226"><a href="#L-226"><span class="linenos">226</span></a>
+</span><span id="L-227"><a href="#L-227"><span class="linenos">227</span></a><span class="sd">    Rearranging an equation (simple example made complicated as illustration)</span>
+</span><span id="L-228"><a href="#L-228"><span class="linenos">228</span></a><span class="sd">    &gt;&gt;&gt; p, V, n, R, T = var(&#39;p V n R T&#39;)</span>
+</span><span id="L-229"><a href="#L-229"><span class="linenos">229</span></a><span class="sd">    &gt;&gt;&gt; eq1=Eqn(p*V,n*R*T)</span>
+</span><span id="L-230"><a href="#L-230"><span class="linenos">230</span></a><span class="sd">    &gt;&gt;&gt; eq1</span>
+</span><span id="L-231"><a href="#L-231"><span class="linenos">231</span></a><span class="sd">    Equation(V*p, R*T*n)</span>
+</span><span id="L-232"><a href="#L-232"><span class="linenos">232</span></a><span class="sd">    &gt;&gt;&gt; eq2 =eq1/V</span>
+</span><span id="L-233"><a href="#L-233"><span class="linenos">233</span></a><span class="sd">    &gt;&gt;&gt; eq2</span>
+</span><span id="L-234"><a href="#L-234"><span class="linenos">234</span></a><span class="sd">    Equation(p, R*T*n/V)</span>
+</span><span id="L-235"><a href="#L-235"><span class="linenos">235</span></a><span class="sd">    &gt;&gt;&gt; eq3 = eq2/R/T</span>
+</span><span id="L-236"><a href="#L-236"><span class="linenos">236</span></a><span class="sd">    &gt;&gt;&gt; eq3</span>
+</span><span id="L-237"><a href="#L-237"><span class="linenos">237</span></a><span class="sd">    Equation(p/(R*T), n/V)</span>
+</span><span id="L-238"><a href="#L-238"><span class="linenos">238</span></a><span class="sd">    &gt;&gt;&gt; eq4 = eq3*R/p</span>
+</span><span id="L-239"><a href="#L-239"><span class="linenos">239</span></a><span class="sd">    &gt;&gt;&gt; eq4</span>
+</span><span id="L-240"><a href="#L-240"><span class="linenos">240</span></a><span class="sd">    Equation(1/T, R*n/(V*p))</span>
+</span><span id="L-241"><a href="#L-241"><span class="linenos">241</span></a><span class="sd">    &gt;&gt;&gt; 1/eq4</span>
+</span><span id="L-242"><a href="#L-242"><span class="linenos">242</span></a><span class="sd">    Equation(T, V*p/(R*n))</span>
+</span><span id="L-243"><a href="#L-243"><span class="linenos">243</span></a><span class="sd">    &gt;&gt;&gt; eq5 = 1/eq4 - T</span>
+</span><span id="L-244"><a href="#L-244"><span class="linenos">244</span></a><span class="sd">    &gt;&gt;&gt; eq5</span>
+</span><span id="L-245"><a href="#L-245"><span class="linenos">245</span></a><span class="sd">    Equation(0, -T + V*p/(R*n))</span>
+</span><span id="L-246"><a href="#L-246"><span class="linenos">246</span></a>
+</span><span id="L-247"><a href="#L-247"><span class="linenos">247</span></a><span class="sd">    Substitution (#&#39;s and units)</span>
+</span><span id="L-248"><a href="#L-248"><span class="linenos">248</span></a><span class="sd">    &gt;&gt;&gt; L, atm, mol, K = var(&#39;L atm mol K&#39;, positive=True, real=True) # units</span>
+</span><span id="L-249"><a href="#L-249"><span class="linenos">249</span></a><span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L})</span>
+</span><span id="L-250"><a href="#L-250"><span class="linenos">250</span></a><span class="sd">    Equation(p, 0.9334325*atm)</span>
+</span><span id="L-251"><a href="#L-251"><span class="linenos">251</span></a><span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L}).evalf(4)</span>
+</span><span id="L-252"><a href="#L-252"><span class="linenos">252</span></a><span class="sd">    Equation(p, 0.9334*atm)</span>
+</span><span id="L-253"><a href="#L-253"><span class="linenos">253</span></a>
+</span><span id="L-254"><a href="#L-254"><span class="linenos">254</span></a><span class="sd">    Combining equations (Math with equations: lhs with lhs and rhs with rhs)</span>
+</span><span id="L-255"><a href="#L-255"><span class="linenos">255</span></a><span class="sd">    &gt;&gt;&gt; q = Eqn(a*c, b/c**2)</span>
+</span><span id="L-256"><a href="#L-256"><span class="linenos">256</span></a><span class="sd">    &gt;&gt;&gt; q</span>
+</span><span id="L-257"><a href="#L-257"><span class="linenos">257</span></a><span class="sd">    Equation(a*c, b/c**2)</span>
+</span><span id="L-258"><a href="#L-258"><span class="linenos">258</span></a><span class="sd">    &gt;&gt;&gt; t</span>
+</span><span id="L-259"><a href="#L-259"><span class="linenos">259</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="L-260"><a href="#L-260"><span class="linenos">260</span></a><span class="sd">    &gt;&gt;&gt; q+t</span>
+</span><span id="L-261"><a href="#L-261"><span class="linenos">261</span></a><span class="sd">    Equation(a*c + a, b/c + b/c**2)</span>
+</span><span id="L-262"><a href="#L-262"><span class="linenos">262</span></a><span class="sd">    &gt;&gt;&gt; q/t</span>
+</span><span id="L-263"><a href="#L-263"><span class="linenos">263</span></a><span class="sd">    Equation(c, 1/c)</span>
+</span><span id="L-264"><a href="#L-264"><span class="linenos">264</span></a><span class="sd">    &gt;&gt;&gt; t**q</span>
+</span><span id="L-265"><a href="#L-265"><span class="linenos">265</span></a><span class="sd">    Equation(a**(a*c), (b/c)**(b/c**2))</span>
+</span><span id="L-266"><a href="#L-266"><span class="linenos">266</span></a>
+</span><span id="L-267"><a href="#L-267"><span class="linenos">267</span></a><span class="sd">    Utility operations</span>
+</span><span id="L-268"><a href="#L-268"><span class="linenos">268</span></a><span class="sd">    &gt;&gt;&gt; t.reversed</span>
+</span><span id="L-269"><a href="#L-269"><span class="linenos">269</span></a><span class="sd">    Equation(b/c, a)</span>
+</span><span id="L-270"><a href="#L-270"><span class="linenos">270</span></a><span class="sd">    &gt;&gt;&gt; t.swap</span>
+</span><span id="L-271"><a href="#L-271"><span class="linenos">271</span></a><span class="sd">    Equation(b/c, a)</span>
+</span><span id="L-272"><a href="#L-272"><span class="linenos">272</span></a><span class="sd">    &gt;&gt;&gt; t.lhs</span>
+</span><span id="L-273"><a href="#L-273"><span class="linenos">273</span></a><span class="sd">    a</span>
+</span><span id="L-274"><a href="#L-274"><span class="linenos">274</span></a><span class="sd">    &gt;&gt;&gt; t.rhs</span>
+</span><span id="L-275"><a href="#L-275"><span class="linenos">275</span></a><span class="sd">    b/c</span>
+</span><span id="L-276"><a href="#L-276"><span class="linenos">276</span></a><span class="sd">    &gt;&gt;&gt; t.as_Boolean()</span>
+</span><span id="L-277"><a href="#L-277"><span class="linenos">277</span></a><span class="sd">    Eq(a, b/c)</span>
+</span><span id="L-278"><a href="#L-278"><span class="linenos">278</span></a>
+</span><span id="L-279"><a href="#L-279"><span class="linenos">279</span></a><span class="sd">    `.check()` convenience method for `.as_Boolean().simplify()`</span>
+</span><span id="L-280"><a href="#L-280"><span class="linenos">280</span></a><span class="sd">    &gt;&gt;&gt; from sympy import I, pi</span>
+</span><span id="L-281"><a href="#L-281"><span class="linenos">281</span></a><span class="sd">    &gt;&gt;&gt; Equation(pi*(I+2), pi*I+2*pi).check()</span>
+</span><span id="L-282"><a href="#L-282"><span class="linenos">282</span></a><span class="sd">    True</span>
+</span><span id="L-283"><a href="#L-283"><span class="linenos">283</span></a><span class="sd">    &gt;&gt;&gt; Eqn(a,a+1).check()</span>
+</span><span id="L-284"><a href="#L-284"><span class="linenos">284</span></a><span class="sd">    False</span>
+</span><span id="L-285"><a href="#L-285"><span class="linenos">285</span></a>
+</span><span id="L-286"><a href="#L-286"><span class="linenos">286</span></a><span class="sd">    Differentiation</span>
+</span><span id="L-287"><a href="#L-287"><span class="linenos">287</span></a><span class="sd">    Differentiation is applied to both sides if the wrt variable appears on</span>
+</span><span id="L-288"><a href="#L-288"><span class="linenos">288</span></a><span class="sd">    both sides.</span>
+</span><span id="L-289"><a href="#L-289"><span class="linenos">289</span></a><span class="sd">    &gt;&gt;&gt; q=Eqn(a*c, b/c**2)</span>
+</span><span id="L-290"><a href="#L-290"><span class="linenos">290</span></a><span class="sd">    &gt;&gt;&gt; q</span>
+</span><span id="L-291"><a href="#L-291"><span class="linenos">291</span></a><span class="sd">    Equation(a*c, b/c**2)</span>
+</span><span id="L-292"><a href="#L-292"><span class="linenos">292</span></a><span class="sd">    &gt;&gt;&gt; diff(q,b)</span>
+</span><span id="L-293"><a href="#L-293"><span class="linenos">293</span></a><span class="sd">    Equation(Derivative(a*c, b), c**(-2))</span>
+</span><span id="L-294"><a href="#L-294"><span class="linenos">294</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c)</span>
+</span><span id="L-295"><a href="#L-295"><span class="linenos">295</span></a><span class="sd">    Equation(a, -2*b/c**3)</span>
+</span><span id="L-296"><a href="#L-296"><span class="linenos">296</span></a><span class="sd">    &gt;&gt;&gt; diff(log(q),b)</span>
+</span><span id="L-297"><a href="#L-297"><span class="linenos">297</span></a><span class="sd">    Equation(Derivative(log(a*c), b), 1/b)</span>
+</span><span id="L-298"><a href="#L-298"><span class="linenos">298</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c,2)</span>
+</span><span id="L-299"><a href="#L-299"><span class="linenos">299</span></a><span class="sd">    Equation(Derivative(a, c), 6*b/c**4)</span>
+</span><span id="L-300"><a href="#L-300"><span class="linenos">300</span></a>
+</span><span id="L-301"><a href="#L-301"><span class="linenos">301</span></a><span class="sd">    If you specify multiple differentiation all at once the assumption</span>
+</span><span id="L-302"><a href="#L-302"><span class="linenos">302</span></a><span class="sd">    is order of differentiation matters and the lhs will not be</span>
+</span><span id="L-303"><a href="#L-303"><span class="linenos">303</span></a><span class="sd">    evaluated.</span>
+</span><span id="L-304"><a href="#L-304"><span class="linenos">304</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c,b)</span>
+</span><span id="L-305"><a href="#L-305"><span class="linenos">305</span></a><span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
+</span><span id="L-306"><a href="#L-306"><span class="linenos">306</span></a>
+</span><span id="L-307"><a href="#L-307"><span class="linenos">307</span></a><span class="sd">    To overcome this specify the order of operations.</span>
+</span><span id="L-308"><a href="#L-308"><span class="linenos">308</span></a><span class="sd">    &gt;&gt;&gt; diff(diff(q,c),b)</span>
+</span><span id="L-309"><a href="#L-309"><span class="linenos">309</span></a><span class="sd">    Equation(Derivative(a, b), -2/c**3)</span>
+</span><span id="L-310"><a href="#L-310"><span class="linenos">310</span></a>
+</span><span id="L-311"><a href="#L-311"><span class="linenos">311</span></a><span class="sd">    But the reverse order returns an unevaulated lhs (a may depend on b).</span>
+</span><span id="L-312"><a href="#L-312"><span class="linenos">312</span></a><span class="sd">    &gt;&gt;&gt; diff(diff(q,b),c)</span>
+</span><span id="L-313"><a href="#L-313"><span class="linenos">313</span></a><span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
+</span><span id="L-314"><a href="#L-314"><span class="linenos">314</span></a>
+</span><span id="L-315"><a href="#L-315"><span class="linenos">315</span></a><span class="sd">    Integration can only be performed on one side at a time.</span>
+</span><span id="L-316"><a href="#L-316"><span class="linenos">316</span></a><span class="sd">    &gt;&gt;&gt; q=Eqn(a*c,b/c)</span>
+</span><span id="L-317"><a href="#L-317"><span class="linenos">317</span></a><span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;rhs&#39;)</span>
+</span><span id="L-318"><a href="#L-318"><span class="linenos">318</span></a><span class="sd">    b**2/(2*c)</span>
+</span><span id="L-319"><a href="#L-319"><span class="linenos">319</span></a><span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;lhs&#39;)</span>
+</span><span id="L-320"><a href="#L-320"><span class="linenos">320</span></a><span class="sd">    a*b*c</span>
+</span><span id="L-321"><a href="#L-321"><span class="linenos">321</span></a>
+</span><span id="L-322"><a href="#L-322"><span class="linenos">322</span></a><span class="sd">    Make a pretty statement of integration from an equation</span>
+</span><span id="L-323"><a href="#L-323"><span class="linenos">323</span></a><span class="sd">    &gt;&gt;&gt; Eqn(Integral(q.lhs,b),integrate(q,b,side=&#39;rhs&#39;))</span>
+</span><span id="L-324"><a href="#L-324"><span class="linenos">324</span></a><span class="sd">    Equation(Integral(a*c, b), b**2/(2*c))</span>
+</span><span id="L-325"><a href="#L-325"><span class="linenos">325</span></a>
+</span><span id="L-326"><a href="#L-326"><span class="linenos">326</span></a><span class="sd">    Integration of each side with respect to different variables</span>
+</span><span id="L-327"><a href="#L-327"><span class="linenos">327</span></a><span class="sd">    &gt;&gt;&gt; q.dorhs.integrate(b).dolhs.integrate(a)</span>
+</span><span id="L-328"><a href="#L-328"><span class="linenos">328</span></a><span class="sd">    Equation(a**2*c/2, b**2/(2*c))</span>
+</span><span id="L-329"><a href="#L-329"><span class="linenos">329</span></a>
+</span><span id="L-330"><a href="#L-330"><span class="linenos">330</span></a><span class="sd">    SymPy&#39;s solvers do not understand these equations. They expect an</span>
+</span><span id="L-331"><a href="#L-331"><span class="linenos">331</span></a><span class="sd">    expression that the solver assumes = 0. Thus to use the solver the</span>
+</span><span id="L-332"><a href="#L-332"><span class="linenos">332</span></a><span class="sd">    equation must be rearranged so that all non-zero symbols are on one side.</span>
+</span><span id="L-333"><a href="#L-333"><span class="linenos">333</span></a><span class="sd">    Then just the non-zero symbolic side is passed to ``solve()``.</span>
+</span><span id="L-334"><a href="#L-334"><span class="linenos">334</span></a><span class="sd">    &gt;&gt;&gt; t2 = t-t.rhs</span>
+</span><span id="L-335"><a href="#L-335"><span class="linenos">335</span></a><span class="sd">    &gt;&gt;&gt; t2</span>
+</span><span id="L-336"><a href="#L-336"><span class="linenos">336</span></a><span class="sd">    Equation(a - b/c, 0)</span>
+</span><span id="L-337"><a href="#L-337"><span class="linenos">337</span></a><span class="sd">    &gt;&gt;&gt; solve(t2.lhs,c)</span>
+</span><span id="L-338"><a href="#L-338"><span class="linenos">338</span></a><span class="sd">    [b/a]</span>
+</span><span id="L-339"><a href="#L-339"><span class="linenos">339</span></a>
+</span><span id="L-340"><a href="#L-340"><span class="linenos">340</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="L-341"><a href="#L-341"><span class="linenos">341</span></a>
+</span><span id="L-342"><a href="#L-342"><span class="linenos">342</span></a>    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-343"><a href="#L-343"><span class="linenos">343</span></a>        <span class="n">lhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">lhs</span><span class="p">)</span>
+</span><span id="L-344"><a href="#L-344"><span class="linenos">344</span></a>        <span class="n">rhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-345"><a href="#L-345"><span class="linenos">345</span></a>        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
+</span><span id="L-346"><a href="#L-346"><span class="linenos">346</span></a>            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;lhs and rhs must be valid sympy expressions.&#39;</span><span class="p">)</span>
+</span><span id="L-347"><a href="#L-347"><span class="linenos">347</span></a>        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-348"><a href="#L-348"><span class="linenos">348</span></a>
+</span><span id="L-349"><a href="#L-349"><span class="linenos">349</span></a>    <span class="k">def</span> <span class="nf">_get_eqn_name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-350"><a href="#L-350"><span class="linenos">350</span></a>        <span class="kn">from</span> <span class="nn">IPython</span> <span class="kn">import</span> <span class="n">get_ipython</span>
+</span><span id="L-351"><a href="#L-351"><span class="linenos">351</span></a>        <span class="n">global_dict</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">get_ipython</span><span class="p">(),</span> <span class="s1">&#39;user_ns&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="L-352"><a href="#L-352"><span class="linenos">352</span></a>        <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="L-353"><a href="#L-353"><span class="linenos">353</span></a>        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
+</span><span id="L-354"><a href="#L-354"><span class="linenos">354</span></a>        <span class="k">if</span> <span class="n">global_dict</span><span class="p">:</span>
+</span><span id="L-355"><a href="#L-355"><span class="linenos">355</span></a>            <span class="k">for</span> <span class="n">var_name</span> <span class="ow">in</span> <span class="n">global_dict</span><span class="p">:</span>
+</span><span id="L-356"><a href="#L-356"><span class="linenos">356</span></a>                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-357"><a href="#L-357"><span class="linenos">357</span></a>                    <span class="k">if</span> <span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">])</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">==</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> \
+</span><span id="L-358"><a href="#L-358"><span class="linenos">358</span></a>                            <span class="n">var_name</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s1">&#39;_&#39;</span><span class="p">):</span>
+</span><span id="L-359"><a href="#L-359"><span class="linenos">359</span></a>                        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
+</span><span id="L-360"><a href="#L-360"><span class="linenos">360</span></a>                        <span class="k">return</span> <span class="n">var_name</span>
+</span><span id="L-361"><a href="#L-361"><span class="linenos">361</span></a>        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span> <span class="o">=</span> <span class="n">human_text</span>
+</span><span id="L-362"><a href="#L-362"><span class="linenos">362</span></a>        <span class="k">return</span> <span class="s1">&#39;&#39;</span>
+</span><span id="L-363"><a href="#L-363"><span class="linenos">363</span></a>
+</span><span id="L-364"><a href="#L-364"><span class="linenos">364</span></a>    <span class="nd">@property</span>
+</span><span id="L-365"><a href="#L-365"><span class="linenos">365</span></a>    <span class="k">def</span> <span class="nf">lhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-366"><a href="#L-366"><span class="linenos">366</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-367"><a href="#L-367"><span class="linenos">367</span></a><span class="sd">        Returns the lhs of the equation.</span>
+</span><span id="L-368"><a href="#L-368"><span class="linenos">368</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-369"><a href="#L-369"><span class="linenos">369</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+</span><span id="L-370"><a href="#L-370"><span class="linenos">370</span></a>
+</span><span id="L-371"><a href="#L-371"><span class="linenos">371</span></a>    <span class="nd">@property</span>
+</span><span id="L-372"><a href="#L-372"><span class="linenos">372</span></a>    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-373"><a href="#L-373"><span class="linenos">373</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-374"><a href="#L-374"><span class="linenos">374</span></a><span class="sd">        Returns the rhs of the equation.</span>
+</span><span id="L-375"><a href="#L-375"><span class="linenos">375</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-376"><a href="#L-376"><span class="linenos">376</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span><span id="L-377"><a href="#L-377"><span class="linenos">377</span></a>
+</span><span id="L-378"><a href="#L-378"><span class="linenos">378</span></a>    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-379"><a href="#L-379"><span class="linenos">379</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-380"><a href="#L-380"><span class="linenos">380</span></a><span class="sd">        Converts the equation to an Equality.</span>
+</span><span id="L-381"><a href="#L-381"><span class="linenos">381</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-382"><a href="#L-382"><span class="linenos">382</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="L-383"><a href="#L-383"><span class="linenos">383</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-384"><a href="#L-384"><span class="linenos">384</span></a>
+</span><span id="L-385"><a href="#L-385"><span class="linenos">385</span></a>    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-386"><a href="#L-386"><span class="linenos">386</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-387"><a href="#L-387"><span class="linenos">387</span></a><span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
+</span><span id="L-388"><a href="#L-388"><span class="linenos">388</span></a><span class="sd">        Parameters</span>
+</span><span id="L-389"><a href="#L-389"><span class="linenos">389</span></a><span class="sd">        ----------</span>
+</span><span id="L-390"><a href="#L-390"><span class="linenos">390</span></a><span class="sd">        kwargs any appropriate for `Equality`.</span>
+</span><span id="L-391"><a href="#L-391"><span class="linenos">391</span></a>
+</span><span id="L-392"><a href="#L-392"><span class="linenos">392</span></a><span class="sd">        Returns</span>
+</span><span id="L-393"><a href="#L-393"><span class="linenos">393</span></a><span class="sd">        -------</span>
+</span><span id="L-394"><a href="#L-394"><span class="linenos">394</span></a><span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
+</span><span id="L-395"><a href="#L-395"><span class="linenos">395</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-396"><a href="#L-396"><span class="linenos">396</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="L-397"><a href="#L-397"><span class="linenos">397</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
+</span><span id="L-398"><a href="#L-398"><span class="linenos">398</span></a>
+</span><span id="L-399"><a href="#L-399"><span class="linenos">399</span></a>    <span class="nd">@property</span>
+</span><span id="L-400"><a href="#L-400"><span class="linenos">400</span></a>    <span class="k">def</span> <span class="nf">reversed</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-401"><a href="#L-401"><span class="linenos">401</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-402"><a href="#L-402"><span class="linenos">402</span></a><span class="sd">        Swaps the lhs and the rhs.</span>
+</span><span id="L-403"><a href="#L-403"><span class="linenos">403</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-404"><a href="#L-404"><span class="linenos">404</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
+</span><span id="L-405"><a href="#L-405"><span class="linenos">405</span></a>
+</span><span id="L-406"><a href="#L-406"><span class="linenos">406</span></a>    <span class="nd">@property</span>
+</span><span id="L-407"><a href="#L-407"><span class="linenos">407</span></a>    <span class="k">def</span> <span class="nf">swap</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-408"><a href="#L-408"><span class="linenos">408</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-409"><a href="#L-409"><span class="linenos">409</span></a><span class="sd">        Synonym for `.reversed`</span>
+</span><span id="L-410"><a href="#L-410"><span class="linenos">410</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-411"><a href="#L-411"><span class="linenos">411</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">reversed</span>
+</span><span id="L-412"><a href="#L-412"><span class="linenos">412</span></a>
+</span><span id="L-413"><a href="#L-413"><span class="linenos">413</span></a>    <span class="k">def</span> <span class="nf">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-414"><a href="#L-414"><span class="linenos">414</span></a>        <span class="c1"># Applies a function to an expression checking whether there</span>
+</span><span id="L-415"><a href="#L-415"><span class="linenos">415</span></a>        <span class="c1"># is a specialized version associated with the particular type of</span>
+</span><span id="L-416"><a href="#L-416"><span class="linenos">416</span></a>        <span class="c1"># expression. Errors will be raised if the function cannot be</span>
+</span><span id="L-417"><a href="#L-417"><span class="linenos">417</span></a>        <span class="c1"># applied to an expression.</span>
+</span><span id="L-418"><a href="#L-418"><span class="linenos">418</span></a>        <span class="n">funcname</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s1">&#39;__name__&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="L-419"><a href="#L-419"><span class="linenos">419</span></a>        <span class="k">if</span> <span class="n">funcname</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="L-420"><a href="#L-420"><span class="linenos">420</span></a>            <span class="n">localfunc</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">funcname</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="L-421"><a href="#L-421"><span class="linenos">421</span></a>            <span class="k">if</span> <span class="n">localfunc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="L-422"><a href="#L-422"><span class="linenos">422</span></a>                <span class="k">return</span> <span class="n">localfunc</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-423"><a href="#L-423"><span class="linenos">423</span></a>        <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-424"><a href="#L-424"><span class="linenos">424</span></a>
+</span><span id="L-425"><a href="#L-425"><span class="linenos">425</span></a>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-426"><a href="#L-426"><span class="linenos">426</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-427"><a href="#L-427"><span class="linenos">427</span></a><span class="sd">        Apply an operation/function/method to the equation returning the</span>
+</span><span id="L-428"><a href="#L-428"><span class="linenos">428</span></a><span class="sd">        resulting equation.</span>
+</span><span id="L-429"><a href="#L-429"><span class="linenos">429</span></a>
+</span><span id="L-430"><a href="#L-430"><span class="linenos">430</span></a><span class="sd">        Parameters</span>
+</span><span id="L-431"><a href="#L-431"><span class="linenos">431</span></a><span class="sd">        ==========</span>
+</span><span id="L-432"><a href="#L-432"><span class="linenos">432</span></a>
+</span><span id="L-433"><a href="#L-433"><span class="linenos">433</span></a><span class="sd">        func: object</span>
+</span><span id="L-434"><a href="#L-434"><span class="linenos">434</span></a><span class="sd">            object to apply usually a function</span>
+</span><span id="L-435"><a href="#L-435"><span class="linenos">435</span></a>
+</span><span id="L-436"><a href="#L-436"><span class="linenos">436</span></a><span class="sd">        args: as necessary for the function</span>
+</span><span id="L-437"><a href="#L-437"><span class="linenos">437</span></a>
+</span><span id="L-438"><a href="#L-438"><span class="linenos">438</span></a><span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
+</span><span id="L-439"><a href="#L-439"><span class="linenos">439</span></a><span class="sd">            Specifies which side of the equation the operation will be applied</span>
+</span><span id="L-440"><a href="#L-440"><span class="linenos">440</span></a><span class="sd">            to. Default is &#39;both&#39;.</span>
+</span><span id="L-441"><a href="#L-441"><span class="linenos">441</span></a>
+</span><span id="L-442"><a href="#L-442"><span class="linenos">442</span></a><span class="sd">        kwargs: as necessary for the function</span>
+</span><span id="L-443"><a href="#L-443"><span class="linenos">443</span></a><span class="sd">         &quot;&quot;&quot;</span>
+</span><span id="L-444"><a href="#L-444"><span class="linenos">444</span></a>        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
+</span><span id="L-445"><a href="#L-445"><span class="linenos">445</span></a>        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
+</span><span id="L-446"><a href="#L-446"><span class="linenos">446</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
+</span><span id="L-447"><a href="#L-447"><span class="linenos">447</span></a>            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-448"><a href="#L-448"><span class="linenos">448</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
+</span><span id="L-449"><a href="#L-449"><span class="linenos">449</span></a>            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-450"><a href="#L-450"><span class="linenos">450</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-451"><a href="#L-451"><span class="linenos">451</span></a>
+</span><span id="L-452"><a href="#L-452"><span class="linenos">452</span></a>    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-453"><a href="#L-453"><span class="linenos">453</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-454"><a href="#L-454"><span class="linenos">454</span></a><span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="L-455"><a href="#L-455"><span class="linenos">455</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="L-456"><a href="#L-456"><span class="linenos">456</span></a><span class="sd">        The operation is applied to only the lhs.</span>
+</span><span id="L-457"><a href="#L-457"><span class="linenos">457</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-458"><a href="#L-458"><span class="linenos">458</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
+</span><span id="L-459"><a href="#L-459"><span class="linenos">459</span></a>
+</span><span id="L-460"><a href="#L-460"><span class="linenos">460</span></a>    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-461"><a href="#L-461"><span class="linenos">461</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-462"><a href="#L-462"><span class="linenos">462</span></a><span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="L-463"><a href="#L-463"><span class="linenos">463</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="L-464"><a href="#L-464"><span class="linenos">464</span></a><span class="sd">        The operation is applied to only the rhs.</span>
+</span><span id="L-465"><a href="#L-465"><span class="linenos">465</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-466"><a href="#L-466"><span class="linenos">466</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
+</span><span id="L-467"><a href="#L-467"><span class="linenos">467</span></a>
+</span><span id="L-468"><a href="#L-468"><span class="linenos">468</span></a>    <span class="k">class</span> <span class="nc">_sides</span><span class="p">:</span>
+</span><span id="L-469"><a href="#L-469"><span class="linenos">469</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-470"><a href="#L-470"><span class="linenos">470</span></a><span class="sd">        Helper class for the `.do.`, `.dolhs.`, `.dorhs.` syntax for applying</span>
+</span><span id="L-471"><a href="#L-471"><span class="linenos">471</span></a><span class="sd">        submethods of expressions.</span>
+</span><span id="L-472"><a href="#L-472"><span class="linenos">472</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-473"><a href="#L-473"><span class="linenos">473</span></a>
+</span><span id="L-474"><a href="#L-474"><span class="linenos">474</span></a>        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eqn</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">):</span>
+</span><span id="L-475"><a href="#L-475"><span class="linenos">475</span></a>            <span class="bp">self</span><span class="o">.</span><span class="n">eqn</span> <span class="o">=</span> <span class="n">eqn</span>
+</span><span id="L-476"><a href="#L-476"><span class="linenos">476</span></a>            <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="o">=</span> <span class="n">side</span>
+</span><span id="L-477"><a href="#L-477"><span class="linenos">477</span></a>
+</span><span id="L-478"><a href="#L-478"><span class="linenos">478</span></a>        <span class="k">def</span> <span class="fm">__getattr__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
+</span><span id="L-479"><a href="#L-479"><span class="linenos">479</span></a>            <span class="n">func</span> <span class="o">=</span> <span class="kc">None</span>
+</span><span id="L-480"><a href="#L-480"><span class="linenos">480</span></a>            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;rhs&#39;</span><span class="p">,</span> <span class="s1">&#39;both&#39;</span><span class="p">):</span>
+</span><span id="L-481"><a href="#L-481"><span class="linenos">481</span></a>                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="L-482"><a href="#L-482"><span class="linenos">482</span></a>            <span class="k">else</span><span class="p">:</span>
+</span><span id="L-483"><a href="#L-483"><span class="linenos">483</span></a>                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="L-484"><a href="#L-484"><span class="linenos">484</span></a>            <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="L-485"><a href="#L-485"><span class="linenos">485</span></a>                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;Expressions in the equation have no &#39;</span>
+</span><span id="L-486"><a href="#L-486"><span class="linenos">486</span></a>                                     <span class="s1">&#39;attribute `&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span>
+</span><span id="L-487"><a href="#L-487"><span class="linenos">487</span></a>                    <span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;`. Try `.apply(&#39;</span>
+</span><span id="L-488"><a href="#L-488"><span class="linenos">488</span></a>                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;, *args)` or &#39;</span>
+</span><span id="L-489"><a href="#L-489"><span class="linenos">489</span></a>                                                   <span class="s1">&#39;pass the equation as a parameter to `&#39;</span>
+</span><span id="L-490"><a href="#L-490"><span class="linenos">490</span></a>                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;()`.&#39;</span><span class="p">)</span>
+</span><span id="L-491"><a href="#L-491"><span class="linenos">491</span></a>            <span class="k">return</span> <span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">apply</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">side</span><span class="p">)</span>
+</span><span id="L-492"><a href="#L-492"><span class="linenos">492</span></a>
+</span><span id="L-493"><a href="#L-493"><span class="linenos">493</span></a>    <span class="nd">@property</span>
+</span><span id="L-494"><a href="#L-494"><span class="linenos">494</span></a>    <span class="k">def</span> <span class="nf">do</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-495"><a href="#L-495"><span class="linenos">495</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">)</span>
+</span><span id="L-496"><a href="#L-496"><span class="linenos">496</span></a>
+</span><span id="L-497"><a href="#L-497"><span class="linenos">497</span></a>    <span class="nd">@property</span>
+</span><span id="L-498"><a href="#L-498"><span class="linenos">498</span></a>    <span class="k">def</span> <span class="nf">dolhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-499"><a href="#L-499"><span class="linenos">499</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
+</span><span id="L-500"><a href="#L-500"><span class="linenos">500</span></a>
+</span><span id="L-501"><a href="#L-501"><span class="linenos">501</span></a>    <span class="nd">@property</span>
+</span><span id="L-502"><a href="#L-502"><span class="linenos">502</span></a>    <span class="k">def</span> <span class="nf">dorhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-503"><a href="#L-503"><span class="linenos">503</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
+</span><span id="L-504"><a href="#L-504"><span class="linenos">504</span></a>
+</span><span id="L-505"><a href="#L-505"><span class="linenos">505</span></a>    <span class="c1">#####</span>
+</span><span id="L-506"><a href="#L-506"><span class="linenos">506</span></a>    <span class="c1"># Overrides of binary math operations</span>
+</span><span id="L-507"><a href="#L-507"><span class="linenos">507</span></a>    <span class="c1">#####</span>
+</span><span id="L-508"><a href="#L-508"><span class="linenos">508</span></a>
+</span><span id="L-509"><a href="#L-509"><span class="linenos">509</span></a>    <span class="nd">@classmethod</span>
+</span><span id="L-510"><a href="#L-510"><span class="linenos">510</span></a>    <span class="k">def</span> <span class="nf">_binary_op</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">opfunc_ab</span><span class="p">):</span>
+</span><span id="L-511"><a href="#L-511"><span class="linenos">511</span></a>        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-512"><a href="#L-512"><span class="linenos">512</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
+</span><span id="L-513"><a href="#L-513"><span class="linenos">513</span></a>        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-514"><a href="#L-514"><span class="linenos">514</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
+</span><span id="L-515"><a href="#L-515"><span class="linenos">515</span></a>        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-516"><a href="#L-516"><span class="linenos">516</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
+</span><span id="L-517"><a href="#L-517"><span class="linenos">517</span></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="L-518"><a href="#L-518"><span class="linenos">518</span></a>            <span class="k">return</span> <span class="bp">NotImplemented</span>
+</span><span id="L-519"><a href="#L-519"><span class="linenos">519</span></a>
+</span><span id="L-520"><a href="#L-520"><span class="linenos">520</span></a>    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-521"><a href="#L-521"><span class="linenos">521</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-522"><a href="#L-522"><span class="linenos">522</span></a>
+</span><span id="L-523"><a href="#L-523"><span class="linenos">523</span></a>    <span class="k">def</span> <span class="fm">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-524"><a href="#L-524"><span class="linenos">524</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-525"><a href="#L-525"><span class="linenos">525</span></a>
+</span><span id="L-526"><a href="#L-526"><span class="linenos">526</span></a>    <span class="k">def</span> <span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-527"><a href="#L-527"><span class="linenos">527</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-528"><a href="#L-528"><span class="linenos">528</span></a>
+</span><span id="L-529"><a href="#L-529"><span class="linenos">529</span></a>    <span class="k">def</span> <span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-530"><a href="#L-530"><span class="linenos">530</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-531"><a href="#L-531"><span class="linenos">531</span></a>
+</span><span id="L-532"><a href="#L-532"><span class="linenos">532</span></a>    <span class="k">def</span> <span class="fm">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-533"><a href="#L-533"><span class="linenos">533</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-534"><a href="#L-534"><span class="linenos">534</span></a>
+</span><span id="L-535"><a href="#L-535"><span class="linenos">535</span></a>    <span class="k">def</span> <span class="fm">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-536"><a href="#L-536"><span class="linenos">536</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-537"><a href="#L-537"><span class="linenos">537</span></a>
+</span><span id="L-538"><a href="#L-538"><span class="linenos">538</span></a>    <span class="k">def</span> <span class="fm">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-539"><a href="#L-539"><span class="linenos">539</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-540"><a href="#L-540"><span class="linenos">540</span></a>
+</span><span id="L-541"><a href="#L-541"><span class="linenos">541</span></a>    <span class="k">def</span> <span class="fm">__rtruediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-542"><a href="#L-542"><span class="linenos">542</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-543"><a href="#L-543"><span class="linenos">543</span></a>
+</span><span id="L-544"><a href="#L-544"><span class="linenos">544</span></a>    <span class="k">def</span> <span class="fm">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-545"><a href="#L-545"><span class="linenos">545</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-546"><a href="#L-546"><span class="linenos">546</span></a>
+</span><span id="L-547"><a href="#L-547"><span class="linenos">547</span></a>    <span class="k">def</span> <span class="fm">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-548"><a href="#L-548"><span class="linenos">548</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-549"><a href="#L-549"><span class="linenos">549</span></a>
+</span><span id="L-550"><a href="#L-550"><span class="linenos">550</span></a>    <span class="k">def</span> <span class="fm">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-551"><a href="#L-551"><span class="linenos">551</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-552"><a href="#L-552"><span class="linenos">552</span></a>
+</span><span id="L-553"><a href="#L-553"><span class="linenos">553</span></a>    <span class="k">def</span> <span class="fm">__rpow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-554"><a href="#L-554"><span class="linenos">554</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="L-555"><a href="#L-555"><span class="linenos">555</span></a>
+</span><span id="L-556"><a href="#L-556"><span class="linenos">556</span></a>    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="L-557"><a href="#L-557"><span class="linenos">557</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__pow__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
+</span><span id="L-558"><a href="#L-558"><span class="linenos">558</span></a>
+</span><span id="L-559"><a href="#L-559"><span class="linenos">559</span></a>    <span class="c1">#####</span>
+</span><span id="L-560"><a href="#L-560"><span class="linenos">560</span></a>    <span class="c1"># Operation helper functions</span>
+</span><span id="L-561"><a href="#L-561"><span class="linenos">561</span></a>    <span class="c1">#####</span>
+</span><span id="L-562"><a href="#L-562"><span class="linenos">562</span></a>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-563"><a href="#L-563"><span class="linenos">563</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
+</span><span id="L-564"><a href="#L-564"><span class="linenos">564</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="L-565"><a href="#L-565"><span class="linenos">565</span></a>
+</span><span id="L-566"><a href="#L-566"><span class="linenos">566</span></a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-567"><a href="#L-567"><span class="linenos">567</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-568"><a href="#L-568"><span class="linenos">568</span></a>
+</span><span id="L-569"><a href="#L-569"><span class="linenos">569</span></a>    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-570"><a href="#L-570"><span class="linenos">570</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span>
+</span><span id="L-571"><a href="#L-571"><span class="linenos">571</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="L-572"><a href="#L-572"><span class="linenos">572</span></a>
+</span><span id="L-573"><a href="#L-573"><span class="linenos">573</span></a>    <span class="k">def</span> <span class="nf">_eval_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-574"><a href="#L-574"><span class="linenos">574</span></a>        <span class="c1"># TODO: cancel out factors common to both sides.</span>
+</span><span id="L-575"><a href="#L-575"><span class="linenos">575</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span>
+</span><span id="L-576"><a href="#L-576"><span class="linenos">576</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="L-577"><a href="#L-577"><span class="linenos">577</span></a>
+</span><span id="L-578"><a href="#L-578"><span class="linenos">578</span></a>    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-579"><a href="#L-579"><span class="linenos">579</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-580"><a href="#L-580"><span class="linenos">580</span></a>
+</span><span id="L-581"><a href="#L-581"><span class="linenos">581</span></a>    <span class="k">def</span> <span class="nf">_eval_collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-582"><a href="#L-582"><span class="linenos">582</span></a>        <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
+</span><span id="L-583"><a href="#L-583"><span class="linenos">583</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="L-584"><a href="#L-584"><span class="linenos">584</span></a>                        <span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="L-585"><a href="#L-585"><span class="linenos">585</span></a>
+</span><span id="L-586"><a href="#L-586"><span class="linenos">586</span></a>    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-587"><a href="#L-587"><span class="linenos">587</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-588"><a href="#L-588"><span class="linenos">588</span></a>
+</span><span id="L-589"><a href="#L-589"><span class="linenos">589</span></a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-590"><a href="#L-590"><span class="linenos">590</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="L-591"><a href="#L-591"><span class="linenos">591</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="L-592"><a href="#L-592"><span class="linenos">592</span></a>
+</span><span id="L-593"><a href="#L-593"><span class="linenos">593</span></a>    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
+</span><span id="L-594"><a href="#L-594"><span class="linenos">594</span></a>
+</span><span id="L-595"><a href="#L-595"><span class="linenos">595</span></a>    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-596"><a href="#L-596"><span class="linenos">596</span></a>        <span class="c1"># TODO Find why diff and Derivative do not appear to pass through</span>
+</span><span id="L-597"><a href="#L-597"><span class="linenos">597</span></a>        <span class="c1">#  kwargs to this. Since we cannot set evaluation of lhs manually</span>
+</span><span id="L-598"><a href="#L-598"><span class="linenos">598</span></a>        <span class="c1">#  try to be intelligent about when to do it.</span>
+</span><span id="L-599"><a href="#L-599"><span class="linenos">599</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Derivative</span>
+</span><span id="L-600"><a href="#L-600"><span class="linenos">600</span></a>        <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">False</span>
+</span><span id="L-601"><a href="#L-601"><span class="linenos">601</span></a>        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)):</span>
+</span><span id="L-602"><a href="#L-602"><span class="linenos">602</span></a>            <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
+</span><span id="L-603"><a href="#L-603"><span class="linenos">603</span></a>                <span class="k">if</span> <span class="n">sym</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span>
+</span><span id="L-604"><a href="#L-604"><span class="linenos">604</span></a>                        <span class="n">_sympify</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">):</span>
+</span><span id="L-605"><a href="#L-605"><span class="linenos">605</span></a>                    <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="L-606"><a href="#L-606"><span class="linenos">606</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="n">eval_lhs</span><span class="p">),</span>
+</span><span id="L-607"><a href="#L-607"><span class="linenos">607</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="L-608"><a href="#L-608"><span class="linenos">608</span></a>
+</span><span id="L-609"><a href="#L-609"><span class="linenos">609</span></a>    <span class="k">def</span> <span class="nf">_eval_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-610"><a href="#L-610"><span class="linenos">610</span></a>        <span class="n">side</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s1">&#39;side&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>  <span class="c1"># Could not seem to pass values for</span>
+</span><span id="L-611"><a href="#L-611"><span class="linenos">611</span></a>        <span class="c1"># `evaluate` through to here.</span>
+</span><span id="L-612"><a href="#L-612"><span class="linenos">612</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="L-613"><a href="#L-613"><span class="linenos">613</span></a>            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;You must specify `side=&quot;lhs&quot;` or `side=&quot;rhs&quot;` &#39;</span>
+</span><span id="L-614"><a href="#L-614"><span class="linenos">614</span></a>                             <span class="s1">&#39;when integrating an Equation&#39;</span><span class="p">)</span>
+</span><span id="L-615"><a href="#L-615"><span class="linenos">615</span></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="L-616"><a href="#L-616"><span class="linenos">616</span></a>            <span class="k">try</span><span class="p">:</span>
+</span><span id="L-617"><a href="#L-617"><span class="linenos">617</span></a>                <span class="k">return</span> <span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="L-618"><a href="#L-618"><span class="linenos">618</span></a>            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
+</span><span id="L-619"><a href="#L-619"><span class="linenos">619</span></a>                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;`side` must equal &quot;lhs&quot; or &quot;rhs&quot;.&#39;</span><span class="p">)</span>
+</span><span id="L-620"><a href="#L-620"><span class="linenos">620</span></a>
+</span><span id="L-621"><a href="#L-621"><span class="linenos">621</span></a>    <span class="c1">#####</span>
+</span><span id="L-622"><a href="#L-622"><span class="linenos">622</span></a>    <span class="c1"># Output helper functions</span>
+</span><span id="L-623"><a href="#L-623"><span class="linenos">623</span></a>    <span class="c1">#####</span>
+</span><span id="L-624"><a href="#L-624"><span class="linenos">624</span></a>    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-625"><a href="#L-625"><span class="linenos">625</span></a>        <span class="n">repstr</span> <span class="o">=</span> <span class="s1">&#39;Equation(</span><span class="si">%s</span><span class="s1">, </span><span class="si">%s</span><span class="s1">)&#39;</span> <span class="o">%</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">())</span>
+</span><span id="L-626"><a href="#L-626"><span class="linenos">626</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
+</span><span id="L-627"><a href="#L-627"><span class="linenos">627</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__str__</span><span class="p">()</span>
+</span><span id="L-628"><a href="#L-628"><span class="linenos">628</span></a>        <span class="k">return</span> <span class="n">repstr</span>
+</span><span id="L-629"><a href="#L-629"><span class="linenos">629</span></a>
+</span><span id="L-630"><a href="#L-630"><span class="linenos">630</span></a>    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-631"><a href="#L-631"><span class="linenos">631</span></a>        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
+</span><span id="L-632"><a href="#L-632"><span class="linenos">632</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span> <span class="ow">and</span> <span class="ow">not</span> \
+</span><span id="L-633"><a href="#L-633"><span class="linenos">633</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
+</span><span id="L-634"><a href="#L-634"><span class="linenos">634</span></a>            <span class="n">tempstr</span> <span class="o">+=</span><span class="s1">&#39;</span><span class="se">\\</span><span class="s1">text{code version: &#39;</span><span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">+</span><span class="s1">&#39;} </span><span class="se">\\</span><span class="s1">newline &#39;</span>
+</span><span id="L-635"><a href="#L-635"><span class="linenos">635</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-636"><a href="#L-636"><span class="linenos">636</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;=&#39;</span>
+</span><span id="L-637"><a href="#L-637"><span class="linenos">637</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-638"><a href="#L-638"><span class="linenos">638</span></a>        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
+</span><span id="L-639"><a href="#L-639"><span class="linenos">639</span></a>        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span><span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
+</span><span id="L-640"><a href="#L-640"><span class="linenos">640</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,&#39;</span>
+</span><span id="L-641"><a href="#L-641"><span class="linenos">641</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;(</span><span class="se">\\</span><span class="s1">text{&#39;</span><span class="o">+</span><span class="n">namestr</span><span class="o">+</span><span class="s1">&#39;})&#39;</span>
+</span><span id="L-642"><a href="#L-642"><span class="linenos">642</span></a>        <span class="k">return</span> <span class="n">tempstr</span>
+</span><span id="L-643"><a href="#L-643"><span class="linenos">643</span></a>
+</span><span id="L-644"><a href="#L-644"><span class="linenos">644</span></a>    <span class="k">def</span> <span class="fm">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-645"><a href="#L-645"><span class="linenos">645</span></a>        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
+</span><span id="L-646"><a href="#L-646"><span class="linenos">646</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span><span class="p">:</span>
+</span><span id="L-647"><a href="#L-647"><span class="linenos">647</span></a>            <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="L-648"><a href="#L-648"><span class="linenos">648</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
+</span><span id="L-649"><a href="#L-649"><span class="linenos">649</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;code version: &#39;</span><span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="o">+</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
+</span><span id="L-650"><a href="#L-650"><span class="linenos">650</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
+</span><span id="L-651"><a href="#L-651"><span class="linenos">651</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39; = &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-652"><a href="#L-652"><span class="linenos">652</span></a>        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
+</span><span id="L-653"><a href="#L-653"><span class="linenos">653</span></a>        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span> <span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
+</span><span id="L-654"><a href="#L-654"><span class="linenos">654</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;          (&#39;</span> <span class="o">+</span> <span class="n">namestr</span> <span class="o">+</span> <span class="s1">&#39;)&#39;</span>
+</span><span id="L-655"><a href="#L-655"><span class="linenos">655</span></a>        <span class="k">return</span> <span class="n">tempstr</span>
+</span><span id="L-656"><a href="#L-656"><span class="linenos">656</span></a>
+</span><span id="L-657"><a href="#L-657"><span class="linenos">657</span></a>
+</span><span id="L-658"><a href="#L-658"><span class="linenos">658</span></a><span class="n">Eqn</span> <span class="o">=</span> <span class="n">Equation</span>
+</span><span id="L-659"><a href="#L-659"><span class="linenos">659</span></a>
+</span><span id="L-660"><a href="#L-660"><span class="linenos">660</span></a><span class="k">def</span> <span class="nf">solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
+</span><span id="L-661"><a href="#L-661"><span class="linenos">661</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-662"><a href="#L-662"><span class="linenos">662</span></a><span class="sd">    Override of sympy `solve()`. If it is passed an equation it will</span>
+</span><span id="L-663"><a href="#L-663"><span class="linenos">663</span></a><span class="sd">    output the solutions as typeset equations and return</span>
+</span><span id="L-664"><a href="#L-664"><span class="linenos">664</span></a><span class="sd">    the answer as a list of equations that can be accessed for additional</span>
+</span><span id="L-665"><a href="#L-665"><span class="linenos">665</span></a><span class="sd">    manipulations.</span>
+</span><span id="L-666"><a href="#L-666"><span class="linenos">666</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="L-667"><a href="#L-667"><span class="linenos">667</span></a>    <span class="kn">from</span> <span class="nn">sympy.solvers.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
+</span><span id="L-668"><a href="#L-668"><span class="linenos">668</span></a>    <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
+</span><span id="L-669"><a href="#L-669"><span class="linenos">669</span></a>    <span class="n">newf</span> <span class="o">=</span><span class="p">[]</span>
+</span><span id="L-670"><a href="#L-670"><span class="linenos">670</span></a>    <span class="n">solns</span> <span class="o">=</span> <span class="p">[]</span>
+</span><span id="L-671"><a href="#L-671"><span class="linenos">671</span></a>    <span class="n">contains_eqn</span> <span class="o">=</span> <span class="kc">False</span>
+</span><span id="L-672"><a href="#L-672"><span class="linenos">672</span></a>    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="s1">&#39;__iter__&#39;</span><span class="p">):</span>
+</span><span id="L-673"><a href="#L-673"><span class="linenos">673</span></a>        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
+</span><span id="L-674"><a href="#L-674"><span class="linenos">674</span></a>            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-675"><a href="#L-675"><span class="linenos">675</span></a>                <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="o">.</span><span class="n">lhs</span><span class="o">-</span><span class="n">k</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-676"><a href="#L-676"><span class="linenos">676</span></a>                <span class="n">contains_eqn</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="L-677"><a href="#L-677"><span class="linenos">677</span></a>            <span class="k">else</span><span class="p">:</span>
+</span><span id="L-678"><a href="#L-678"><span class="linenos">678</span></a>                <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
+</span><span id="L-679"><a href="#L-679"><span class="linenos">679</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="L-680"><a href="#L-680"><span class="linenos">680</span></a>        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-681"><a href="#L-681"><span class="linenos">681</span></a>            <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">f</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-682"><a href="#L-682"><span class="linenos">682</span></a>            <span class="n">contains_eqn</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="L-683"><a href="#L-683"><span class="linenos">683</span></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="L-684"><a href="#L-684"><span class="linenos">684</span></a>            <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
+</span><span id="L-685"><a href="#L-685"><span class="linenos">685</span></a>    <span class="n">flags</span><span class="p">[</span><span class="s1">&#39;dict&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="L-686"><a href="#L-686"><span class="linenos">686</span></a>    <span class="n">result</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">newf</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
+</span><span id="L-687"><a href="#L-687"><span class="linenos">687</span></a>    <span class="k">if</span> <span class="n">contains_eqn</span><span class="p">:</span>
+</span><span id="L-688"><a href="#L-688"><span class="linenos">688</span></a>        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
+</span><span id="L-689"><a href="#L-689"><span class="linenos">689</span></a>            <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">k</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
+</span><span id="L-690"><a href="#L-690"><span class="linenos">690</span></a>                <span class="n">val</span> <span class="o">=</span> <span class="n">k</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
+</span><span id="L-691"><a href="#L-691"><span class="linenos">691</span></a>                <span class="n">tempeqn</span> <span class="o">=</span> <span class="n">Eqn</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
+</span><span id="L-692"><a href="#L-692"><span class="linenos">692</span></a>                <span class="n">display</span><span class="p">(</span><span class="n">tempeqn</span><span class="p">)</span>
+</span><span id="L-693"><a href="#L-693"><span class="linenos">693</span></a>                <span class="n">solns</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tempeqn</span><span class="p">)</span>
+</span><span id="L-694"><a href="#L-694"><span class="linenos">694</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="L-695"><a href="#L-695"><span class="linenos">695</span></a>        <span class="n">solns</span> <span class="o">=</span> <span class="n">result</span>
+</span><span id="L-696"><a href="#L-696"><span class="linenos">696</span></a>    <span class="k">return</span> <span class="n">solns</span>
+</span><span id="L-697"><a href="#L-697"><span class="linenos">697</span></a>    <span class="c1"># else:</span>
+</span><span id="L-698"><a href="#L-698"><span class="linenos">698</span></a>    <span class="c1">#     return solve(f, *symbols, **flags)</span>
+</span><span id="L-699"><a href="#L-699"><span class="linenos">699</span></a>
+</span><span id="L-700"><a href="#L-700"><span class="linenos">700</span></a><span class="k">def</span> <span class="nf">sqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">evaluate</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span>
+</span><span id="L-701"><a href="#L-701"><span class="linenos">701</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-702"><a href="#L-702"><span class="linenos">702</span></a><span class="sd">    Override of sympy convenience function `sqrt`. Simply divides equations</span>
+</span><span id="L-703"><a href="#L-703"><span class="linenos">703</span></a><span class="sd">    into two sides if `arg` is an instance of `Equation`. This avoids an</span>
+</span><span id="L-704"><a href="#L-704"><span class="linenos">704</span></a><span class="sd">    issue with the way sympy is delaying specialized applications of _Pow_ on</span>
+</span><span id="L-705"><a href="#L-705"><span class="linenos">705</span></a><span class="sd">    objects that are not basic sympy expressions.</span>
+</span><span id="L-706"><a href="#L-706"><span class="linenos">706</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="L-707"><a href="#L-707"><span class="linenos">707</span></a>    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span> <span class="k">as</span> <span class="n">symsqrt</span>
+</span><span id="L-708"><a href="#L-708"><span class="linenos">708</span></a>    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-709"><a href="#L-709"><span class="linenos">709</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">),</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">))</span>
+</span><span id="L-710"><a href="#L-710"><span class="linenos">710</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="L-711"><a href="#L-711"><span class="linenos">711</span></a>        <span class="k">return</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span><span class="n">evaluate</span><span class="p">)</span>
+</span><span id="L-712"><a href="#L-712"><span class="linenos">712</span></a>
+</span><span id="L-713"><a href="#L-713"><span class="linenos">713</span></a><span class="c1"># Pick up the docstring for sqrt from sympy</span>
+</span><span id="L-714"><a href="#L-714"><span class="linenos">714</span></a><span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span> <span class="k">as</span> <span class="n">symsqrt</span>
+</span><span id="L-715"><a href="#L-715"><span class="linenos">715</span></a><span class="n">sqrt</span><span class="o">.</span><span class="vm">__doc__</span><span class="o">+=</span><span class="n">symsqrt</span><span class="o">.</span><span class="vm">__doc__</span>
+</span><span id="L-716"><a href="#L-716"><span class="linenos">716</span></a><span class="k">del</span> <span class="n">symsqrt</span>
+</span><span id="L-717"><a href="#L-717"><span class="linenos">717</span></a>
+</span><span id="L-718"><a href="#L-718"><span class="linenos">718</span></a><span class="k">def</span> <span class="nf">root</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">evaluate</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span>
+</span><span id="L-719"><a href="#L-719"><span class="linenos">719</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-720"><a href="#L-720"><span class="linenos">720</span></a><span class="sd">    Override of sympy convenience function `root`. Simply divides equations</span>
+</span><span id="L-721"><a href="#L-721"><span class="linenos">721</span></a><span class="sd">    into two sides if `arg` is an instance of `Equation`. This avoids an</span>
+</span><span id="L-722"><a href="#L-722"><span class="linenos">722</span></a><span class="sd">    issue with the way sympy is delaying specialized applications of _Pow_ on</span>
+</span><span id="L-723"><a href="#L-723"><span class="linenos">723</span></a><span class="sd">    objects that are not basic sympy expressions.</span>
+</span><span id="L-724"><a href="#L-724"><span class="linenos">724</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="L-725"><a href="#L-725"><span class="linenos">725</span></a>    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">root</span> <span class="k">as</span> <span class="n">symroot</span>
+</span><span id="L-726"><a href="#L-726"><span class="linenos">726</span></a>    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-727"><a href="#L-727"><span class="linenos">727</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">),</span> <span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span>
+</span><span id="L-728"><a href="#L-728"><span class="linenos">728</span></a>                                                            <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">))</span>
+</span><span id="L-729"><a href="#L-729"><span class="linenos">729</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="L-730"><a href="#L-730"><span class="linenos">730</span></a>        <span class="k">return</span> <span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">)</span>
+</span><span id="L-731"><a href="#L-731"><span class="linenos">731</span></a>
+</span><span id="L-732"><a href="#L-732"><span class="linenos">732</span></a><span class="c1"># pick up the docstring for root from sympy</span>
+</span><span id="L-733"><a href="#L-733"><span class="linenos">733</span></a><span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">root</span> <span class="k">as</span> <span class="n">symroot</span>
+</span><span id="L-734"><a href="#L-734"><span class="linenos">734</span></a><span class="n">root</span><span class="o">.</span><span class="vm">__doc__</span><span class="o">+=</span><span class="n">symroot</span><span class="o">.</span><span class="vm">__doc__</span>
+</span><span id="L-735"><a href="#L-735"><span class="linenos">735</span></a><span class="k">del</span> <span class="n">symroot</span>
+</span><span id="L-736"><a href="#L-736"><span class="linenos">736</span></a>
+</span><span id="L-737"><a href="#L-737"><span class="linenos">737</span></a><span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+</span><span id="L-738"><a href="#L-738"><span class="linenos">738</span></a>            <span class="n">distribute_order_term</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+</span><span id="L-739"><a href="#L-739"><span class="linenos">739</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-740"><a href="#L-740"><span class="linenos">740</span></a><span class="sd">    Override of sympy `collect()`.</span>
+</span><span id="L-741"><a href="#L-741"><span class="linenos">741</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="L-742"><a href="#L-742"><span class="linenos">742</span></a>    <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
+</span><span id="L-743"><a href="#L-743"><span class="linenos">743</span></a>    <span class="n">_eval_collect</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s1">&#39;_eval_collect&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="L-744"><a href="#L-744"><span class="linenos">744</span></a>    <span class="k">if</span> <span class="n">_eval_collect</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="L-745"><a href="#L-745"><span class="linenos">745</span></a>        <span class="k">return</span> <span class="n">_eval_collect</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span>
+</span><span id="L-746"><a href="#L-746"><span class="linenos">746</span></a>                             <span class="n">exact</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="p">)</span>
+</span><span id="L-747"><a href="#L-747"><span class="linenos">747</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="L-748"><a href="#L-748"><span class="linenos">748</span></a>        <span class="k">return</span> <span class="n">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span> <span class="n">exact</span><span class="p">,</span>
+</span><span id="L-749"><a href="#L-749"><span class="linenos">749</span></a>                       <span class="n">distribute_order_term</span><span class="p">)</span>
+</span><span id="L-750"><a href="#L-750"><span class="linenos">750</span></a>
+</span><span id="L-751"><a href="#L-751"><span class="linenos">751</span></a><span class="k">class</span> <span class="nc">Eq</span><span class="p">(</span><span class="n">Eq</span><span class="p">):</span>
+</span><span id="L-752"><a href="#L-752"><span class="linenos">752</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-753"><a href="#L-753"><span class="linenos">753</span></a><span class="sd">    Extension of Equality class to include the ability to convert it to an</span>
+</span><span id="L-754"><a href="#L-754"><span class="linenos">754</span></a><span class="sd">    Equation.</span>
+</span><span id="L-755"><a href="#L-755"><span class="linenos">755</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="L-756"><a href="#L-756"><span class="linenos">756</span></a>    <span class="k">def</span> <span class="nf">to_Equation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-757"><a href="#L-757"><span class="linenos">757</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-758"><a href="#L-758"><span class="linenos">758</span></a><span class="sd">        Return: recasts the Equality as an Equation.</span>
+</span><span id="L-759"><a href="#L-759"><span class="linenos">759</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-760"><a href="#L-760"><span class="linenos">760</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-761"><a href="#L-761"><span class="linenos">761</span></a>
+</span><span id="L-762"><a href="#L-762"><span class="linenos">762</span></a>    <span class="k">def</span> <span class="nf">to_Eqn</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="L-763"><a href="#L-763"><span class="linenos">763</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-764"><a href="#L-764"><span class="linenos">764</span></a><span class="sd">        Synonym for to_Equation.</span>
+</span><span id="L-765"><a href="#L-765"><span class="linenos">765</span></a><span class="sd">        Return: recasts the Equality as an Equation.</span>
+</span><span id="L-766"><a href="#L-766"><span class="linenos">766</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="L-767"><a href="#L-767"><span class="linenos">767</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_Equation</span><span class="p">()</span>
+</span><span id="L-768"><a href="#L-768"><span class="linenos">768</span></a>
+</span><span id="L-769"><a href="#L-769"><span class="linenos">769</span></a><span class="c1">#####</span>
+</span><span id="L-770"><a href="#L-770"><span class="linenos">770</span></a><span class="c1"># Extension of the Function class. For incorporation into SymPy this should</span>
+</span><span id="L-771"><a href="#L-771"><span class="linenos">771</span></a><span class="c1"># become part of the class</span>
+</span><span id="L-772"><a href="#L-772"><span class="linenos">772</span></a><span class="c1">#####</span>
+</span><span id="L-773"><a href="#L-773"><span class="linenos">773</span></a><span class="k">class</span> <span class="nc">EqnFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
+</span><span id="L-774"><a href="#L-774"><span class="linenos">774</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="L-775"><a href="#L-775"><span class="linenos">775</span></a><span class="sd">    Extension of the sympy Function class to understand equations. Each</span>
+</span><span id="L-776"><a href="#L-776"><span class="linenos">776</span></a><span class="sd">    sympy function impacted by this extension is listed in the documentation</span>
+</span><span id="L-777"><a href="#L-777"><span class="linenos">777</span></a><span class="sd">    that follows.</span>
+</span><span id="L-778"><a href="#L-778"><span class="linenos">778</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="L-779"><a href="#L-779"><span class="linenos">779</span></a>    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="L-780"><a href="#L-780"><span class="linenos">780</span></a>        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
+</span><span id="L-781"><a href="#L-781"><span class="linenos">781</span></a>        <span class="n">eqnloc</span> <span class="o">=</span> <span class="kc">None</span>
+</span><span id="L-782"><a href="#L-782"><span class="linenos">782</span></a>        <span class="n">neqns</span> <span class="o">=</span> <span class="mi">0</span>
+</span><span id="L-783"><a href="#L-783"><span class="linenos">783</span></a>        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
+</span><span id="L-784"><a href="#L-784"><span class="linenos">784</span></a>        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
+</span><span id="L-785"><a href="#L-785"><span class="linenos">785</span></a>            <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
+</span><span id="L-786"><a href="#L-786"><span class="linenos">786</span></a>        <span class="k">if</span> <span class="p">(</span><span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
+</span><span id="L-787"><a href="#L-787"><span class="linenos">787</span></a>            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
+</span><span id="L-788"><a href="#L-788"><span class="linenos">788</span></a>                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="L-789"><a href="#L-789"><span class="linenos">789</span></a>                    <span class="n">neqns</span> <span class="o">+=</span> <span class="mi">1</span>
+</span><span id="L-790"><a href="#L-790"><span class="linenos">790</span></a>                    <span class="n">eqnloc</span> <span class="o">=</span> <span class="n">i</span>
+</span><span id="L-791"><a href="#L-791"><span class="linenos">791</span></a>            <span class="k">if</span> <span class="n">neqns</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
+</span><span id="L-792"><a href="#L-792"><span class="linenos">792</span></a>                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Function calls with more than one &#39;</span>
+</span><span id="L-793"><a href="#L-793"><span class="linenos">793</span></a>                                          <span class="s1">&#39;Equation as a parameter are not &#39;</span>
+</span><span id="L-794"><a href="#L-794"><span class="linenos">794</span></a>                                          <span class="s1">&#39;supported. You may be able to get &#39;</span>
+</span><span id="L-795"><a href="#L-795"><span class="linenos">795</span></a>                                          <span class="s1">&#39;your desired outcome using .applyrhs&#39;</span>
+</span><span id="L-796"><a href="#L-796"><span class="linenos">796</span></a>                                          <span class="s1">&#39; and .applylhs.&#39;</span><span class="p">)</span>
+</span><span id="L-797"><a href="#L-797"><span class="linenos">797</span></a>            <span class="k">if</span> <span class="n">neqns</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+</span><span id="L-798"><a href="#L-798"><span class="linenos">798</span></a>                <span class="n">newargs</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span><span class="o">.</span><span class="n">lhs</span>
+</span><span id="L-799"><a href="#L-799"><span class="linenos">799</span></a>                <span class="n">lhs</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-800"><a href="#L-800"><span class="linenos">800</span></a>                <span class="n">newargs</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span><span class="o">.</span><span class="n">rhs</span>
+</span><span id="L-801"><a href="#L-801"><span class="linenos">801</span></a>                <span class="n">rhs</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-802"><a href="#L-802"><span class="linenos">802</span></a>                <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="L-803"><a href="#L-803"><span class="linenos">803</span></a>        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="L-804"><a href="#L-804"><span class="linenos">804</span></a>
+</span><span id="L-805"><a href="#L-805"><span class="linenos">805</span></a>
+</span><span id="L-806"><a href="#L-806"><span class="linenos">806</span></a><span class="k">for</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">functions</span><span class="o">.</span><span class="n">__all__</span><span class="p">:</span>
+</span><span id="L-807"><a href="#L-807"><span class="linenos">807</span></a>    <span class="c1"># TODO: This will not be needed when incorporated into SymPy listed in</span>
+</span><span id="L-808"><a href="#L-808"><span class="linenos">808</span></a>    <span class="c1">#  `skip` cannot be extended because of `mro` error or `metaclass</span>
+</span><span id="L-809"><a href="#L-809"><span class="linenos">809</span></a>    <span class="c1">#  conflict`. Seems to reflect expectation that a helper function will be</span>
+</span><span id="L-810"><a href="#L-810"><span class="linenos">810</span></a>    <span class="c1">#  defined within the object (e.g. `_eval_power()` for all the flavors of</span>
+</span><span id="L-811"><a href="#L-811"><span class="linenos">811</span></a>    <span class="c1">#  `root`).</span>
+</span><span id="L-812"><a href="#L-812"><span class="linenos">812</span></a>    <span class="n">skip</span> <span class="o">=</span> <span class="p">(</span><span class="s1">&#39;sqrt&#39;</span><span class="p">,</span> <span class="s1">&#39;root&#39;</span><span class="p">,</span> <span class="s1">&#39;Min&#39;</span><span class="p">,</span> <span class="s1">&#39;Max&#39;</span><span class="p">,</span> <span class="s1">&#39;Id&#39;</span><span class="p">,</span> <span class="s1">&#39;real_root&#39;</span><span class="p">,</span> <span class="s1">&#39;cbrt&#39;</span><span class="p">,</span>
+</span><span id="L-813"><a href="#L-813"><span class="linenos">813</span></a>            <span class="s1">&#39;unbranched_argument&#39;</span><span class="p">,</span> <span class="s1">&#39;polarify&#39;</span><span class="p">,</span> <span class="s1">&#39;unpolarify&#39;</span><span class="p">,</span>
+</span><span id="L-814"><a href="#L-814"><span class="linenos">814</span></a>            <span class="s1">&#39;piecewise_fold&#39;</span><span class="p">,</span> <span class="s1">&#39;E1&#39;</span><span class="p">,</span> <span class="s1">&#39;Eijk&#39;</span><span class="p">,</span> <span class="s1">&#39;bspline_basis&#39;</span><span class="p">,</span>
+</span><span id="L-815"><a href="#L-815"><span class="linenos">815</span></a>            <span class="s1">&#39;bspline_basis_set&#39;</span><span class="p">,</span> <span class="s1">&#39;interpolating_spline&#39;</span><span class="p">,</span> <span class="s1">&#39;jn_zeros&#39;</span><span class="p">,</span>
+</span><span id="L-816"><a href="#L-816"><span class="linenos">816</span></a>            <span class="s1">&#39;jacobi_normalized&#39;</span><span class="p">,</span> <span class="s1">&#39;Ynm_c&#39;</span><span class="p">)</span>
+</span><span id="L-817"><a href="#L-817"><span class="linenos">817</span></a>    <span class="k">if</span> <span class="n">func</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">skip</span><span class="p">:</span>
+</span><span id="L-818"><a href="#L-818"><span class="linenos">818</span></a>        <span class="n">execstr</span> <span class="o">=</span> <span class="s1">&#39;class &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">func</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;(&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span>
+</span><span id="L-819"><a href="#L-819"><span class="linenos">819</span></a>            <span class="n">func</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;,EqnFunction):</span><span class="se">\n</span><span class="s1">    pass</span><span class="se">\n</span><span class="s1">&#39;</span>
+</span><span id="L-820"><a href="#L-820"><span class="linenos">820</span></a>        <span class="n">exec</span><span class="p">(</span><span class="n">execstr</span><span class="p">,</span> <span class="nb">globals</span><span class="p">(),</span> <span class="nb">locals</span><span class="p">())</span>
+</span></pre></div>
+
 
             </section>
                 <section id="algwsym_config">
-                                <div class="attr class">
-        <a class="headerlink" href="#algwsym_config">#&nbsp;&nbsp</a>
-
-        
-        <span class="def">class</span>
-        <span class="name">algwsym_config</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="k">class</span> <span class="nc">algwsym_config</span><span class="p">():</span>
-
-    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This is a class to hold parameters that control behavior of</span>
-<span class="sd">        the algebra_with_sympy package.</span>
-
-<span class="sd">        Settings</span>
-<span class="sd">        ========</span>
-<span class="sd">        Printing</span>
-<span class="sd">        --------</span>
-<span class="sd">        In interactive environments the default output of an equation is a</span>
-<span class="sd">        human readable string with the two sides connected by an equals</span>
-<span class="sd">        sign or a typeset equation with the two sides connected by an equals sign.</span>
-<span class="sd">        `print(Eqn)` or `str(Eqn)` will return this human readable text version of</span>
-<span class="sd">        the equation as well. This is consistent with python standards, but not</span>
-<span class="sd">        sympy, where `str()` is supposed to return something that can be</span>
-<span class="sd">        copy-pasted into code. If the equation has a declared name as in `eq1 =</span>
-<span class="sd">        Eqn(a,b/c)` the name will be displayed to the right of the equation in</span>
-<span class="sd">        parentheses (eg. `a = b/c    (eq1)`). Use `print(repr(Eqn))` instead of</span>
-<span class="sd">        `print(Eqn)` or `repr(Eqn)` instead of `str(Eqn)` to get a code</span>
-<span class="sd">        compatible version of the equation.</span>
-
-<span class="sd">        You can adjust this behvior using some flags that impact output:</span>
-<span class="sd">        * `algwsym_config.output.show_code` default is `False`.</span>
-<span class="sd">        * `algwsym_config.output.human_text` default is `False`.</span>
-<span class="sd">        * `algwsym_config.output.label` default is `True`.</span>
-
-<span class="sd">        In interactive environments you can get both types of output by setting</span>
-<span class="sd">        the `algwsym_config.output.show_code` flag. If this flag is true</span>
-<span class="sd">        calls to `latex` and `str` will also print an additional line &quot;code</span>
-<span class="sd">        version: `repr(Eqn)`&quot;. Thus in Jupyter you will get a line of typeset</span>
-<span class="sd">        mathematics output preceded by the code version that can be copy-pasted.</span>
-<span class="sd">        Default is `False`.</span>
-
-<span class="sd">        A second flag `algwsym_config.output.human_text` is useful in</span>
-<span class="sd">        text-based interactive environments such as command line python or</span>
-<span class="sd">        ipython. If this flag is true `repr` will return `str`. Thus the human</span>
-<span class="sd">        readable text will be printed as the output of a line that is an</span>
-<span class="sd">        expression containing an equation.</span>
-<span class="sd">        Default is `False`.</span>
-
-<span class="sd">        Setting both of these flags to true in a command line or ipython</span>
-<span class="sd">        environment will show both the code version and the human readable text.</span>
-<span class="sd">        These flags impact the behavior of the `print(Eqn)` statement.</span>
-
-<span class="sd">        The third flag `algwsym_config.output.label` has a default value of</span>
-<span class="sd">        `True`. Setting this to `False` suppresses the labeling of an equation</span>
-<span class="sd">        with its python name off to the right of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">pass</span>
-
-    <span class="k">class</span> <span class="nc">output</span><span class="p">():</span>
-
-        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;This holds settings that impact output.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">pass</span>
-
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">show_code</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` code versions of the equation expression will be</span>
-<span class="sd">            output in interactive environments. Default = `False`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">show_code</span>
-
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">human_text</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` the human readable equation expression will be</span>
-<span class="sd">            output in text interactive environments. Default = `False`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">human_text</span>
-
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` the human readable equation will be followed by the</span>
-<span class="sd">            python name it is assigned to. Default = `True`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span>
-</pre></div>
-
-        </details>
+                            <input id="algwsym_config-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">algwsym_config</span>:
+
+                <label class="view-source-button" for="algwsym_config-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#algwsym_config"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="algwsym_config-38"><a href="#algwsym_config-38"><span class="linenos"> 38</span></a><span class="k">class</span> <span class="nc">algwsym_config</span><span class="p">():</span>
+</span><span id="algwsym_config-39"><a href="#algwsym_config-39"><span class="linenos"> 39</span></a>
+</span><span id="algwsym_config-40"><a href="#algwsym_config-40"><span class="linenos"> 40</span></a>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config-41"><a href="#algwsym_config-41"><span class="linenos"> 41</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="algwsym_config-42"><a href="#algwsym_config-42"><span class="linenos"> 42</span></a><span class="sd">        This is a class to hold parameters that control behavior of</span>
+</span><span id="algwsym_config-43"><a href="#algwsym_config-43"><span class="linenos"> 43</span></a><span class="sd">        the algebra_with_sympy package.</span>
+</span><span id="algwsym_config-44"><a href="#algwsym_config-44"><span class="linenos"> 44</span></a>
+</span><span id="algwsym_config-45"><a href="#algwsym_config-45"><span class="linenos"> 45</span></a><span class="sd">        Settings</span>
+</span><span id="algwsym_config-46"><a href="#algwsym_config-46"><span class="linenos"> 46</span></a><span class="sd">        ========</span>
+</span><span id="algwsym_config-47"><a href="#algwsym_config-47"><span class="linenos"> 47</span></a><span class="sd">        Printing</span>
+</span><span id="algwsym_config-48"><a href="#algwsym_config-48"><span class="linenos"> 48</span></a><span class="sd">        --------</span>
+</span><span id="algwsym_config-49"><a href="#algwsym_config-49"><span class="linenos"> 49</span></a><span class="sd">        In interactive environments the default output of an equation is a</span>
+</span><span id="algwsym_config-50"><a href="#algwsym_config-50"><span class="linenos"> 50</span></a><span class="sd">        human readable string with the two sides connected by an equals</span>
+</span><span id="algwsym_config-51"><a href="#algwsym_config-51"><span class="linenos"> 51</span></a><span class="sd">        sign or a typeset equation with the two sides connected by an equals sign.</span>
+</span><span id="algwsym_config-52"><a href="#algwsym_config-52"><span class="linenos"> 52</span></a><span class="sd">        `print(Eqn)` or `str(Eqn)` will return this human readable text version of</span>
+</span><span id="algwsym_config-53"><a href="#algwsym_config-53"><span class="linenos"> 53</span></a><span class="sd">        the equation as well. This is consistent with python standards, but not</span>
+</span><span id="algwsym_config-54"><a href="#algwsym_config-54"><span class="linenos"> 54</span></a><span class="sd">        sympy, where `str()` is supposed to return something that can be</span>
+</span><span id="algwsym_config-55"><a href="#algwsym_config-55"><span class="linenos"> 55</span></a><span class="sd">        copy-pasted into code. If the equation has a declared name as in `eq1 =</span>
+</span><span id="algwsym_config-56"><a href="#algwsym_config-56"><span class="linenos"> 56</span></a><span class="sd">        Eqn(a,b/c)` the name will be displayed to the right of the equation in</span>
+</span><span id="algwsym_config-57"><a href="#algwsym_config-57"><span class="linenos"> 57</span></a><span class="sd">        parentheses (eg. `a = b/c    (eq1)`). Use `print(repr(Eqn))` instead of</span>
+</span><span id="algwsym_config-58"><a href="#algwsym_config-58"><span class="linenos"> 58</span></a><span class="sd">        `print(Eqn)` or `repr(Eqn)` instead of `str(Eqn)` to get a code</span>
+</span><span id="algwsym_config-59"><a href="#algwsym_config-59"><span class="linenos"> 59</span></a><span class="sd">        compatible version of the equation.</span>
+</span><span id="algwsym_config-60"><a href="#algwsym_config-60"><span class="linenos"> 60</span></a>
+</span><span id="algwsym_config-61"><a href="#algwsym_config-61"><span class="linenos"> 61</span></a><span class="sd">        You can adjust this behvior using some flags that impact output:</span>
+</span><span id="algwsym_config-62"><a href="#algwsym_config-62"><span class="linenos"> 62</span></a><span class="sd">        * `algwsym_config.output.show_code` default is `False`.</span>
+</span><span id="algwsym_config-63"><a href="#algwsym_config-63"><span class="linenos"> 63</span></a><span class="sd">        * `algwsym_config.output.human_text` default is `False`.</span>
+</span><span id="algwsym_config-64"><a href="#algwsym_config-64"><span class="linenos"> 64</span></a><span class="sd">        * `algwsym_config.output.label` default is `True`.</span>
+</span><span id="algwsym_config-65"><a href="#algwsym_config-65"><span class="linenos"> 65</span></a>
+</span><span id="algwsym_config-66"><a href="#algwsym_config-66"><span class="linenos"> 66</span></a><span class="sd">        In interactive environments you can get both types of output by setting</span>
+</span><span id="algwsym_config-67"><a href="#algwsym_config-67"><span class="linenos"> 67</span></a><span class="sd">        the `algwsym_config.output.show_code` flag. If this flag is true</span>
+</span><span id="algwsym_config-68"><a href="#algwsym_config-68"><span class="linenos"> 68</span></a><span class="sd">        calls to `latex` and `str` will also print an additional line &quot;code</span>
+</span><span id="algwsym_config-69"><a href="#algwsym_config-69"><span class="linenos"> 69</span></a><span class="sd">        version: `repr(Eqn)`&quot;. Thus in Jupyter you will get a line of typeset</span>
+</span><span id="algwsym_config-70"><a href="#algwsym_config-70"><span class="linenos"> 70</span></a><span class="sd">        mathematics output preceded by the code version that can be copy-pasted.</span>
+</span><span id="algwsym_config-71"><a href="#algwsym_config-71"><span class="linenos"> 71</span></a><span class="sd">        Default is `False`.</span>
+</span><span id="algwsym_config-72"><a href="#algwsym_config-72"><span class="linenos"> 72</span></a>
+</span><span id="algwsym_config-73"><a href="#algwsym_config-73"><span class="linenos"> 73</span></a><span class="sd">        A second flag `algwsym_config.output.human_text` is useful in</span>
+</span><span id="algwsym_config-74"><a href="#algwsym_config-74"><span class="linenos"> 74</span></a><span class="sd">        text-based interactive environments such as command line python or</span>
+</span><span id="algwsym_config-75"><a href="#algwsym_config-75"><span class="linenos"> 75</span></a><span class="sd">        ipython. If this flag is true `repr` will return `str`. Thus the human</span>
+</span><span id="algwsym_config-76"><a href="#algwsym_config-76"><span class="linenos"> 76</span></a><span class="sd">        readable text will be printed as the output of a line that is an</span>
+</span><span id="algwsym_config-77"><a href="#algwsym_config-77"><span class="linenos"> 77</span></a><span class="sd">        expression containing an equation.</span>
+</span><span id="algwsym_config-78"><a href="#algwsym_config-78"><span class="linenos"> 78</span></a><span class="sd">        Default is `False`.</span>
+</span><span id="algwsym_config-79"><a href="#algwsym_config-79"><span class="linenos"> 79</span></a>
+</span><span id="algwsym_config-80"><a href="#algwsym_config-80"><span class="linenos"> 80</span></a><span class="sd">        Setting both of these flags to true in a command line or ipython</span>
+</span><span id="algwsym_config-81"><a href="#algwsym_config-81"><span class="linenos"> 81</span></a><span class="sd">        environment will show both the code version and the human readable text.</span>
+</span><span id="algwsym_config-82"><a href="#algwsym_config-82"><span class="linenos"> 82</span></a><span class="sd">        These flags impact the behavior of the `print(Eqn)` statement.</span>
+</span><span id="algwsym_config-83"><a href="#algwsym_config-83"><span class="linenos"> 83</span></a>
+</span><span id="algwsym_config-84"><a href="#algwsym_config-84"><span class="linenos"> 84</span></a><span class="sd">        The third flag `algwsym_config.output.label` has a default value of</span>
+</span><span id="algwsym_config-85"><a href="#algwsym_config-85"><span class="linenos"> 85</span></a><span class="sd">        `True`. Setting this to `False` suppresses the labeling of an equation</span>
+</span><span id="algwsym_config-86"><a href="#algwsym_config-86"><span class="linenos"> 86</span></a><span class="sd">        with its python name off to the right of the equation.</span>
+</span><span id="algwsym_config-87"><a href="#algwsym_config-87"><span class="linenos"> 87</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="algwsym_config-88"><a href="#algwsym_config-88"><span class="linenos"> 88</span></a>        <span class="k">pass</span>
+</span><span id="algwsym_config-89"><a href="#algwsym_config-89"><span class="linenos"> 89</span></a>
+</span><span id="algwsym_config-90"><a href="#algwsym_config-90"><span class="linenos"> 90</span></a>    <span class="k">class</span> <span class="nc">output</span><span class="p">():</span>
+</span><span id="algwsym_config-91"><a href="#algwsym_config-91"><span class="linenos"> 91</span></a>
+</span><span id="algwsym_config-92"><a href="#algwsym_config-92"><span class="linenos"> 92</span></a>        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config-93"><a href="#algwsym_config-93"><span class="linenos"> 93</span></a>            <span class="sd">&quot;&quot;&quot;This holds settings that impact output.</span>
+</span><span id="algwsym_config-94"><a href="#algwsym_config-94"><span class="linenos"> 94</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config-95"><a href="#algwsym_config-95"><span class="linenos"> 95</span></a>            <span class="k">pass</span>
+</span><span id="algwsym_config-96"><a href="#algwsym_config-96"><span class="linenos"> 96</span></a>
+</span><span id="algwsym_config-97"><a href="#algwsym_config-97"><span class="linenos"> 97</span></a>        <span class="nd">@property</span>
+</span><span id="algwsym_config-98"><a href="#algwsym_config-98"><span class="linenos"> 98</span></a>        <span class="k">def</span> <span class="nf">show_code</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config-99"><a href="#algwsym_config-99"><span class="linenos"> 99</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="algwsym_config-100"><a href="#algwsym_config-100"><span class="linenos">100</span></a><span class="sd">            If `True` code versions of the equation expression will be</span>
+</span><span id="algwsym_config-101"><a href="#algwsym_config-101"><span class="linenos">101</span></a><span class="sd">            output in interactive environments. Default = `False`.</span>
+</span><span id="algwsym_config-102"><a href="#algwsym_config-102"><span class="linenos">102</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config-103"><a href="#algwsym_config-103"><span class="linenos">103</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">show_code</span>
+</span><span id="algwsym_config-104"><a href="#algwsym_config-104"><span class="linenos">104</span></a>
+</span><span id="algwsym_config-105"><a href="#algwsym_config-105"><span class="linenos">105</span></a>        <span class="nd">@property</span>
+</span><span id="algwsym_config-106"><a href="#algwsym_config-106"><span class="linenos">106</span></a>        <span class="k">def</span> <span class="nf">human_text</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config-107"><a href="#algwsym_config-107"><span class="linenos">107</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="algwsym_config-108"><a href="#algwsym_config-108"><span class="linenos">108</span></a><span class="sd">            If `True` the human readable equation expression will be</span>
+</span><span id="algwsym_config-109"><a href="#algwsym_config-109"><span class="linenos">109</span></a><span class="sd">            output in text interactive environments. Default = `False`.</span>
+</span><span id="algwsym_config-110"><a href="#algwsym_config-110"><span class="linenos">110</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config-111"><a href="#algwsym_config-111"><span class="linenos">111</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="algwsym_config-112"><a href="#algwsym_config-112"><span class="linenos">112</span></a>
+</span><span id="algwsym_config-113"><a href="#algwsym_config-113"><span class="linenos">113</span></a>        <span class="nd">@property</span>
+</span><span id="algwsym_config-114"><a href="#algwsym_config-114"><span class="linenos">114</span></a>        <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config-115"><a href="#algwsym_config-115"><span class="linenos">115</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="algwsym_config-116"><a href="#algwsym_config-116"><span class="linenos">116</span></a><span class="sd">            If `True` the human readable equation will be followed by the</span>
+</span><span id="algwsym_config-117"><a href="#algwsym_config-117"><span class="linenos">117</span></a><span class="sd">            python name it is assigned to. Default = `True`.</span>
+</span><span id="algwsym_config-118"><a href="#algwsym_config-118"><span class="linenos">118</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config-119"><a href="#algwsym_config-119"><span class="linenos">119</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span>
+</span></pre></div>
+
 
     
 
                             <div id="algwsym_config.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#algwsym_config.__init__">#&nbsp;&nbsp</a>
-
-        
-            <span class="name">algwsym_config</span><span class="signature">()</span>
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This is a class to hold parameters that control behavior of</span>
-<span class="sd">        the algebra_with_sympy package.</span>
-
-<span class="sd">        Settings</span>
-<span class="sd">        ========</span>
-<span class="sd">        Printing</span>
-<span class="sd">        --------</span>
-<span class="sd">        In interactive environments the default output of an equation is a</span>
-<span class="sd">        human readable string with the two sides connected by an equals</span>
-<span class="sd">        sign or a typeset equation with the two sides connected by an equals sign.</span>
-<span class="sd">        `print(Eqn)` or `str(Eqn)` will return this human readable text version of</span>
-<span class="sd">        the equation as well. This is consistent with python standards, but not</span>
-<span class="sd">        sympy, where `str()` is supposed to return something that can be</span>
-<span class="sd">        copy-pasted into code. If the equation has a declared name as in `eq1 =</span>
-<span class="sd">        Eqn(a,b/c)` the name will be displayed to the right of the equation in</span>
-<span class="sd">        parentheses (eg. `a = b/c    (eq1)`). Use `print(repr(Eqn))` instead of</span>
-<span class="sd">        `print(Eqn)` or `repr(Eqn)` instead of `str(Eqn)` to get a code</span>
-<span class="sd">        compatible version of the equation.</span>
-
-<span class="sd">        You can adjust this behvior using some flags that impact output:</span>
-<span class="sd">        * `algwsym_config.output.show_code` default is `False`.</span>
-<span class="sd">        * `algwsym_config.output.human_text` default is `False`.</span>
-<span class="sd">        * `algwsym_config.output.label` default is `True`.</span>
-
-<span class="sd">        In interactive environments you can get both types of output by setting</span>
-<span class="sd">        the `algwsym_config.output.show_code` flag. If this flag is true</span>
-<span class="sd">        calls to `latex` and `str` will also print an additional line &quot;code</span>
-<span class="sd">        version: `repr(Eqn)`&quot;. Thus in Jupyter you will get a line of typeset</span>
-<span class="sd">        mathematics output preceded by the code version that can be copy-pasted.</span>
-<span class="sd">        Default is `False`.</span>
-
-<span class="sd">        A second flag `algwsym_config.output.human_text` is useful in</span>
-<span class="sd">        text-based interactive environments such as command line python or</span>
-<span class="sd">        ipython. If this flag is true `repr` will return `str`. Thus the human</span>
-<span class="sd">        readable text will be printed as the output of a line that is an</span>
-<span class="sd">        expression containing an equation.</span>
-<span class="sd">        Default is `False`.</span>
-
-<span class="sd">        Setting both of these flags to true in a command line or ipython</span>
-<span class="sd">        environment will show both the code version and the human readable text.</span>
-<span class="sd">        These flags impact the behavior of the `print(Eqn)` statement.</span>
-
-<span class="sd">        The third flag `algwsym_config.output.label` has a default value of</span>
-<span class="sd">        `True`. Setting this to `False` suppresses the labeling of an equation</span>
-<span class="sd">        with its python name off to the right of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">pass</span>
-</pre></div>
-
-        </details>
+                                        <input id="algwsym_config.__init__-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="name">algwsym_config</span><span class="signature pdoc-code condensed">()</span>
+
+                <label class="view-source-button" for="algwsym_config.__init__-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#algwsym_config.__init__"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="algwsym_config.__init__-40"><a href="#algwsym_config.__init__-40"><span class="linenos">40</span></a>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config.__init__-41"><a href="#algwsym_config.__init__-41"><span class="linenos">41</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="algwsym_config.__init__-42"><a href="#algwsym_config.__init__-42"><span class="linenos">42</span></a><span class="sd">        This is a class to hold parameters that control behavior of</span>
+</span><span id="algwsym_config.__init__-43"><a href="#algwsym_config.__init__-43"><span class="linenos">43</span></a><span class="sd">        the algebra_with_sympy package.</span>
+</span><span id="algwsym_config.__init__-44"><a href="#algwsym_config.__init__-44"><span class="linenos">44</span></a>
+</span><span id="algwsym_config.__init__-45"><a href="#algwsym_config.__init__-45"><span class="linenos">45</span></a><span class="sd">        Settings</span>
+</span><span id="algwsym_config.__init__-46"><a href="#algwsym_config.__init__-46"><span class="linenos">46</span></a><span class="sd">        ========</span>
+</span><span id="algwsym_config.__init__-47"><a href="#algwsym_config.__init__-47"><span class="linenos">47</span></a><span class="sd">        Printing</span>
+</span><span id="algwsym_config.__init__-48"><a href="#algwsym_config.__init__-48"><span class="linenos">48</span></a><span class="sd">        --------</span>
+</span><span id="algwsym_config.__init__-49"><a href="#algwsym_config.__init__-49"><span class="linenos">49</span></a><span class="sd">        In interactive environments the default output of an equation is a</span>
+</span><span id="algwsym_config.__init__-50"><a href="#algwsym_config.__init__-50"><span class="linenos">50</span></a><span class="sd">        human readable string with the two sides connected by an equals</span>
+</span><span id="algwsym_config.__init__-51"><a href="#algwsym_config.__init__-51"><span class="linenos">51</span></a><span class="sd">        sign or a typeset equation with the two sides connected by an equals sign.</span>
+</span><span id="algwsym_config.__init__-52"><a href="#algwsym_config.__init__-52"><span class="linenos">52</span></a><span class="sd">        `print(Eqn)` or `str(Eqn)` will return this human readable text version of</span>
+</span><span id="algwsym_config.__init__-53"><a href="#algwsym_config.__init__-53"><span class="linenos">53</span></a><span class="sd">        the equation as well. This is consistent with python standards, but not</span>
+</span><span id="algwsym_config.__init__-54"><a href="#algwsym_config.__init__-54"><span class="linenos">54</span></a><span class="sd">        sympy, where `str()` is supposed to return something that can be</span>
+</span><span id="algwsym_config.__init__-55"><a href="#algwsym_config.__init__-55"><span class="linenos">55</span></a><span class="sd">        copy-pasted into code. If the equation has a declared name as in `eq1 =</span>
+</span><span id="algwsym_config.__init__-56"><a href="#algwsym_config.__init__-56"><span class="linenos">56</span></a><span class="sd">        Eqn(a,b/c)` the name will be displayed to the right of the equation in</span>
+</span><span id="algwsym_config.__init__-57"><a href="#algwsym_config.__init__-57"><span class="linenos">57</span></a><span class="sd">        parentheses (eg. `a = b/c    (eq1)`). Use `print(repr(Eqn))` instead of</span>
+</span><span id="algwsym_config.__init__-58"><a href="#algwsym_config.__init__-58"><span class="linenos">58</span></a><span class="sd">        `print(Eqn)` or `repr(Eqn)` instead of `str(Eqn)` to get a code</span>
+</span><span id="algwsym_config.__init__-59"><a href="#algwsym_config.__init__-59"><span class="linenos">59</span></a><span class="sd">        compatible version of the equation.</span>
+</span><span id="algwsym_config.__init__-60"><a href="#algwsym_config.__init__-60"><span class="linenos">60</span></a>
+</span><span id="algwsym_config.__init__-61"><a href="#algwsym_config.__init__-61"><span class="linenos">61</span></a><span class="sd">        You can adjust this behvior using some flags that impact output:</span>
+</span><span id="algwsym_config.__init__-62"><a href="#algwsym_config.__init__-62"><span class="linenos">62</span></a><span class="sd">        * `algwsym_config.output.show_code` default is `False`.</span>
+</span><span id="algwsym_config.__init__-63"><a href="#algwsym_config.__init__-63"><span class="linenos">63</span></a><span class="sd">        * `algwsym_config.output.human_text` default is `False`.</span>
+</span><span id="algwsym_config.__init__-64"><a href="#algwsym_config.__init__-64"><span class="linenos">64</span></a><span class="sd">        * `algwsym_config.output.label` default is `True`.</span>
+</span><span id="algwsym_config.__init__-65"><a href="#algwsym_config.__init__-65"><span class="linenos">65</span></a>
+</span><span id="algwsym_config.__init__-66"><a href="#algwsym_config.__init__-66"><span class="linenos">66</span></a><span class="sd">        In interactive environments you can get both types of output by setting</span>
+</span><span id="algwsym_config.__init__-67"><a href="#algwsym_config.__init__-67"><span class="linenos">67</span></a><span class="sd">        the `algwsym_config.output.show_code` flag. If this flag is true</span>
+</span><span id="algwsym_config.__init__-68"><a href="#algwsym_config.__init__-68"><span class="linenos">68</span></a><span class="sd">        calls to `latex` and `str` will also print an additional line &quot;code</span>
+</span><span id="algwsym_config.__init__-69"><a href="#algwsym_config.__init__-69"><span class="linenos">69</span></a><span class="sd">        version: `repr(Eqn)`&quot;. Thus in Jupyter you will get a line of typeset</span>
+</span><span id="algwsym_config.__init__-70"><a href="#algwsym_config.__init__-70"><span class="linenos">70</span></a><span class="sd">        mathematics output preceded by the code version that can be copy-pasted.</span>
+</span><span id="algwsym_config.__init__-71"><a href="#algwsym_config.__init__-71"><span class="linenos">71</span></a><span class="sd">        Default is `False`.</span>
+</span><span id="algwsym_config.__init__-72"><a href="#algwsym_config.__init__-72"><span class="linenos">72</span></a>
+</span><span id="algwsym_config.__init__-73"><a href="#algwsym_config.__init__-73"><span class="linenos">73</span></a><span class="sd">        A second flag `algwsym_config.output.human_text` is useful in</span>
+</span><span id="algwsym_config.__init__-74"><a href="#algwsym_config.__init__-74"><span class="linenos">74</span></a><span class="sd">        text-based interactive environments such as command line python or</span>
+</span><span id="algwsym_config.__init__-75"><a href="#algwsym_config.__init__-75"><span class="linenos">75</span></a><span class="sd">        ipython. If this flag is true `repr` will return `str`. Thus the human</span>
+</span><span id="algwsym_config.__init__-76"><a href="#algwsym_config.__init__-76"><span class="linenos">76</span></a><span class="sd">        readable text will be printed as the output of a line that is an</span>
+</span><span id="algwsym_config.__init__-77"><a href="#algwsym_config.__init__-77"><span class="linenos">77</span></a><span class="sd">        expression containing an equation.</span>
+</span><span id="algwsym_config.__init__-78"><a href="#algwsym_config.__init__-78"><span class="linenos">78</span></a><span class="sd">        Default is `False`.</span>
+</span><span id="algwsym_config.__init__-79"><a href="#algwsym_config.__init__-79"><span class="linenos">79</span></a>
+</span><span id="algwsym_config.__init__-80"><a href="#algwsym_config.__init__-80"><span class="linenos">80</span></a><span class="sd">        Setting both of these flags to true in a command line or ipython</span>
+</span><span id="algwsym_config.__init__-81"><a href="#algwsym_config.__init__-81"><span class="linenos">81</span></a><span class="sd">        environment will show both the code version and the human readable text.</span>
+</span><span id="algwsym_config.__init__-82"><a href="#algwsym_config.__init__-82"><span class="linenos">82</span></a><span class="sd">        These flags impact the behavior of the `print(Eqn)` statement.</span>
+</span><span id="algwsym_config.__init__-83"><a href="#algwsym_config.__init__-83"><span class="linenos">83</span></a>
+</span><span id="algwsym_config.__init__-84"><a href="#algwsym_config.__init__-84"><span class="linenos">84</span></a><span class="sd">        The third flag `algwsym_config.output.label` has a default value of</span>
+</span><span id="algwsym_config.__init__-85"><a href="#algwsym_config.__init__-85"><span class="linenos">85</span></a><span class="sd">        `True`. Setting this to `False` suppresses the labeling of an equation</span>
+</span><span id="algwsym_config.__init__-86"><a href="#algwsym_config.__init__-86"><span class="linenos">86</span></a><span class="sd">        with its python name off to the right of the equation.</span>
+</span><span id="algwsym_config.__init__-87"><a href="#algwsym_config.__init__-87"><span class="linenos">87</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="algwsym_config.__init__-88"><a href="#algwsym_config.__init__-88"><span class="linenos">88</span></a>        <span class="k">pass</span>
+</span></pre></div>
+
 
             <div class="docstring"><p>This is a class to hold parameters that control behavior of
 the algebra_with_sympy package.</p>
@@ -3270,68 +3321,67 @@ <h2 id="printing">Printing</h2>
                             </div>
                 </section>
                 <section id="algwsym_config.output">
-                                <div class="attr class">
-        <a class="headerlink" href="#algwsym_config.output">#&nbsp;&nbsp</a>
-
-        
-        <span class="def">class</span>
-        <span class="name">algwsym_config.output</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">class</span> <span class="nc">output</span><span class="p">():</span>
-
-        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;This holds settings that impact output.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">pass</span>
-
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">show_code</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` code versions of the equation expression will be</span>
-<span class="sd">            output in interactive environments. Default = `False`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">show_code</span>
+                            <input id="algwsym_config.output-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">algwsym_config.output</span>:
+
+                <label class="view-source-button" for="algwsym_config.output-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#algwsym_config.output"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="algwsym_config.output-90"><a href="#algwsym_config.output-90"><span class="linenos"> 90</span></a>    <span class="k">class</span> <span class="nc">output</span><span class="p">():</span>
+</span><span id="algwsym_config.output-91"><a href="#algwsym_config.output-91"><span class="linenos"> 91</span></a>
+</span><span id="algwsym_config.output-92"><a href="#algwsym_config.output-92"><span class="linenos"> 92</span></a>        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config.output-93"><a href="#algwsym_config.output-93"><span class="linenos"> 93</span></a>            <span class="sd">&quot;&quot;&quot;This holds settings that impact output.</span>
+</span><span id="algwsym_config.output-94"><a href="#algwsym_config.output-94"><span class="linenos"> 94</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config.output-95"><a href="#algwsym_config.output-95"><span class="linenos"> 95</span></a>            <span class="k">pass</span>
+</span><span id="algwsym_config.output-96"><a href="#algwsym_config.output-96"><span class="linenos"> 96</span></a>
+</span><span id="algwsym_config.output-97"><a href="#algwsym_config.output-97"><span class="linenos"> 97</span></a>        <span class="nd">@property</span>
+</span><span id="algwsym_config.output-98"><a href="#algwsym_config.output-98"><span class="linenos"> 98</span></a>        <span class="k">def</span> <span class="nf">show_code</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config.output-99"><a href="#algwsym_config.output-99"><span class="linenos"> 99</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="algwsym_config.output-100"><a href="#algwsym_config.output-100"><span class="linenos">100</span></a><span class="sd">            If `True` code versions of the equation expression will be</span>
+</span><span id="algwsym_config.output-101"><a href="#algwsym_config.output-101"><span class="linenos">101</span></a><span class="sd">            output in interactive environments. Default = `False`.</span>
+</span><span id="algwsym_config.output-102"><a href="#algwsym_config.output-102"><span class="linenos">102</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config.output-103"><a href="#algwsym_config.output-103"><span class="linenos">103</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">show_code</span>
+</span><span id="algwsym_config.output-104"><a href="#algwsym_config.output-104"><span class="linenos">104</span></a>
+</span><span id="algwsym_config.output-105"><a href="#algwsym_config.output-105"><span class="linenos">105</span></a>        <span class="nd">@property</span>
+</span><span id="algwsym_config.output-106"><a href="#algwsym_config.output-106"><span class="linenos">106</span></a>        <span class="k">def</span> <span class="nf">human_text</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config.output-107"><a href="#algwsym_config.output-107"><span class="linenos">107</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="algwsym_config.output-108"><a href="#algwsym_config.output-108"><span class="linenos">108</span></a><span class="sd">            If `True` the human readable equation expression will be</span>
+</span><span id="algwsym_config.output-109"><a href="#algwsym_config.output-109"><span class="linenos">109</span></a><span class="sd">            output in text interactive environments. Default = `False`.</span>
+</span><span id="algwsym_config.output-110"><a href="#algwsym_config.output-110"><span class="linenos">110</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config.output-111"><a href="#algwsym_config.output-111"><span class="linenos">111</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="algwsym_config.output-112"><a href="#algwsym_config.output-112"><span class="linenos">112</span></a>
+</span><span id="algwsym_config.output-113"><a href="#algwsym_config.output-113"><span class="linenos">113</span></a>        <span class="nd">@property</span>
+</span><span id="algwsym_config.output-114"><a href="#algwsym_config.output-114"><span class="linenos">114</span></a>        <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config.output-115"><a href="#algwsym_config.output-115"><span class="linenos">115</span></a>            <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="algwsym_config.output-116"><a href="#algwsym_config.output-116"><span class="linenos">116</span></a><span class="sd">            If `True` the human readable equation will be followed by the</span>
+</span><span id="algwsym_config.output-117"><a href="#algwsym_config.output-117"><span class="linenos">117</span></a><span class="sd">            python name it is assigned to. Default = `True`.</span>
+</span><span id="algwsym_config.output-118"><a href="#algwsym_config.output-118"><span class="linenos">118</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config.output-119"><a href="#algwsym_config.output-119"><span class="linenos">119</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span>
+</span></pre></div>
 
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">human_text</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` the human readable equation expression will be</span>
-<span class="sd">            output in text interactive environments. Default = `False`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">human_text</span>
-
-        <span class="nd">@property</span>
-        <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If `True` the human readable equation will be followed by the</span>
-<span class="sd">            python name it is assigned to. Default = `True`.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span>
-</pre></div>
-
-        </details>
 
     
 
                             <div id="algwsym_config.output.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#algwsym_config.output.__init__">#&nbsp;&nbsp</a>
+                                        <input id="algwsym_config.output.__init__-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="name">algwsym_config.output</span><span class="signature pdoc-code condensed">()</span>
 
-        
-            <span class="name">algwsym_config.output</span><span class="signature">()</span>
-    </div>
+                <label class="view-source-button" for="algwsym_config.output.__init__-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;This holds settings that impact output.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">pass</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#algwsym_config.output.__init__"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="algwsym_config.output.__init__-92"><a href="#algwsym_config.output.__init__-92"><span class="linenos">92</span></a>        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="algwsym_config.output.__init__-93"><a href="#algwsym_config.output.__init__-93"><span class="linenos">93</span></a>            <span class="sd">&quot;&quot;&quot;This holds settings that impact output.</span>
+</span><span id="algwsym_config.output.__init__-94"><a href="#algwsym_config.output.__init__-94"><span class="linenos">94</span></a><span class="sd">            &quot;&quot;&quot;</span>
+</span><span id="algwsym_config.output.__init__-95"><a href="#algwsym_config.output.__init__-95"><span class="linenos">95</span></a>            <span class="k">pass</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>This holds settings that impact output.</p>
 </div>
@@ -3339,585 +3389,587 @@ <h2 id="printing">Printing</h2>
 
                             </div>
                             <div id="algwsym_config.output.show_code" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#algwsym_config.output.show_code">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">show_code</span><span class="default_value"> = False</span>
 
-        <span class="name">show_code</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#algwsym_config.output.show_code"></a>
     
     
 
                             </div>
                             <div id="algwsym_config.output.human_text" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#algwsym_config.output.human_text">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">human_text</span><span class="default_value"> = False</span>
 
-        <span class="name">human_text</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#algwsym_config.output.human_text"></a>
     
     
 
                             </div>
                             <div id="algwsym_config.output.label" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#algwsym_config.output.label">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">label</span><span class="default_value"> = True</span>
 
-        <span class="name">label</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#algwsym_config.output.label"></a>
     
     
 
                             </div>
                 </section>
                 <section id="Equation">
-                                <div class="attr class">
-        <a class="headerlink" href="#Equation">#&nbsp;&nbsp</a>
-
-        
-        <span class="def">class</span>
-        <span class="name">Equation</span><wbr>(<span class="base">sympy.core.basic.Basic</span>, <span class="base">sympy.core.evalf.EvalfMixin</span>):
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="k">class</span> <span class="nc">Equation</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class defines an equation with a left-hand-side (lhs) and a right-</span>
-<span class="sd">    hand-side (rhs) connected by the &quot;=&quot; operator (e.g. `p*V = n*R*T`).</span>
-
-<span class="sd">    Explanation</span>
-<span class="sd">    ===========</span>
-<span class="sd">    This class defines relations that all high school and college students</span>
-<span class="sd">    would recognize as mathematical equations. At present only the &quot;=&quot; relation</span>
-<span class="sd">    operator is recognized.</span>
-
-<span class="sd">    This class is intended to allow using the mathematical tools in SymPy to</span>
-<span class="sd">    rearrange equations and perform algebra in a stepwise fashion. In this</span>
-<span class="sd">    way more people can successfully perform algebraic rearrangements without</span>
-<span class="sd">    stumbling over missed details such as a negative sign.</span>
-
-<span class="sd">    Create an equation with the call ``Equation(lhs,rhs)``, where ``lhs`` and</span>
-<span class="sd">    ``rhs`` are any valid Sympy expression. ``Eqn(...)`` is a synonym for</span>
-<span class="sd">    ``Equation(...)``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-<span class="sd">    lhs: sympy expression, ``class Expr``.</span>
-<span class="sd">    rhs: sympy expression, ``class Expr``.</span>
-<span class="sd">    kwargs:</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from algebra_with_sympy import *</span>
-<span class="sd">    &gt;&gt;&gt; a, b, c, x = var(&#39;a b c x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Equation(a,b/c)</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t=Eqn(a,b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t*c</span>
-<span class="sd">    Equation(a*c, b)</span>
-<span class="sd">    &gt;&gt;&gt; c*t</span>
-<span class="sd">    Equation(a*c, b)</span>
-<span class="sd">    &gt;&gt;&gt; exp(t)</span>
-<span class="sd">    Equation(exp(a), exp(b/c))</span>
-<span class="sd">    &gt;&gt;&gt; exp(log(t))</span>
-<span class="sd">    Equation(a, b/c)</span>
-
-<span class="sd">    Simplification and Expansion</span>
-<span class="sd">    &gt;&gt;&gt; f = Eqn(x**2 - 1, c)</span>
-<span class="sd">    &gt;&gt;&gt; f</span>
-<span class="sd">    Equation(x**2 - 1, c)</span>
-<span class="sd">    &gt;&gt;&gt; f/(x+1)</span>
-<span class="sd">    Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; (f/(x+1)).simplify()</span>
-<span class="sd">    Equation(x - 1, c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; simplify(f/(x+1))</span>
-<span class="sd">    Equation(x - 1, c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; (f/(x+1)).expand()</span>
-<span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; expand(f/(x+1))</span>
-<span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; factor(f)</span>
-<span class="sd">    Equation((x - 1)*(x + 1), c)</span>
-<span class="sd">    &gt;&gt;&gt; f.factor()</span>
-<span class="sd">    Equation((x - 1)*(x + 1), c)</span>
-<span class="sd">    &gt;&gt;&gt; f2 = f+a*x**2+b*x +c</span>
-<span class="sd">    &gt;&gt;&gt; f2</span>
-<span class="sd">    Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>
-<span class="sd">    &gt;&gt;&gt; collect(f2,x)</span>
-<span class="sd">    Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>
-
-<span class="sd">    Apply operation to only one side</span>
-<span class="sd">    &gt;&gt;&gt; poly = Eqn(a*x**2 + b*x + c*x**2, a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.applyrhs(factor,x)</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>
-<span class="sd">    &gt;&gt;&gt; poly.applylhs(factor)</span>
-<span class="sd">    Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.applylhs(collect,x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
-
-<span class="sd">    ``.apply...`` also works with user defined python functions</span>
-<span class="sd">    &gt;&gt;&gt; def addsquare(eqn):</span>
-<span class="sd">    ...     return eqn+eqn**2</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; t.apply(addsquare)</span>
-<span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.applyrhs(addsquare)</span>
-<span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.apply(addsquare, side = &#39;rhs&#39;)</span>
-<span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.applylhs(addsquare)</span>
-<span class="sd">    Equation(a**2 + a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; addsquare(t)</span>
-<span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
-
-<span class="sd">    Inaddition to ``.apply...`` there is also the less general ``.do``,</span>
-<span class="sd">    ``.dolhs``, ``.dorhs``, which only works for operations defined on the</span>
-<span class="sd">    ``Expr`` class (e.g.``.collect(), .factor(), .expand()``, etc...).</span>
-<span class="sd">    &gt;&gt;&gt; poly.dolhs.collect(x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.dorhs.collect(x)</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>
-<span class="sd">    &gt;&gt;&gt; poly.do.collect(x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>
-<span class="sd">    &gt;&gt;&gt; poly.dorhs.factor()</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>
-
-<span class="sd">    ``poly.do.exp()`` or other sympy math functions will raise an error.</span>
-
-<span class="sd">    Rearranging an equation (simple example made complicated as illustration)</span>
-<span class="sd">    &gt;&gt;&gt; p, V, n, R, T = var(&#39;p V n R T&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; eq1=Eqn(p*V,n*R*T)</span>
-<span class="sd">    &gt;&gt;&gt; eq1</span>
-<span class="sd">    Equation(V*p, R*T*n)</span>
-<span class="sd">    &gt;&gt;&gt; eq2 =eq1/V</span>
-<span class="sd">    &gt;&gt;&gt; eq2</span>
-<span class="sd">    Equation(p, R*T*n/V)</span>
-<span class="sd">    &gt;&gt;&gt; eq3 = eq2/R/T</span>
-<span class="sd">    &gt;&gt;&gt; eq3</span>
-<span class="sd">    Equation(p/(R*T), n/V)</span>
-<span class="sd">    &gt;&gt;&gt; eq4 = eq3*R/p</span>
-<span class="sd">    &gt;&gt;&gt; eq4</span>
-<span class="sd">    Equation(1/T, R*n/(V*p))</span>
-<span class="sd">    &gt;&gt;&gt; 1/eq4</span>
-<span class="sd">    Equation(T, V*p/(R*n))</span>
-<span class="sd">    &gt;&gt;&gt; eq5 = 1/eq4 - T</span>
-<span class="sd">    &gt;&gt;&gt; eq5</span>
-<span class="sd">    Equation(0, -T + V*p/(R*n))</span>
-
-<span class="sd">    Substitution (#&#39;s and units)</span>
-<span class="sd">    &gt;&gt;&gt; L, atm, mol, K = var(&#39;L atm mol K&#39;, positive=True, real=True) # units</span>
-<span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L})</span>
-<span class="sd">    Equation(p, 0.9334325*atm)</span>
-<span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L}).evalf(4)</span>
-<span class="sd">    Equation(p, 0.9334*atm)</span>
-
-<span class="sd">    Combining equations (Math with equations: lhs with lhs and rhs with rhs)</span>
-<span class="sd">    &gt;&gt;&gt; q = Eqn(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q</span>
-<span class="sd">    Equation(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; t</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; q+t</span>
-<span class="sd">    Equation(a*c + a, b/c + b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q/t</span>
-<span class="sd">    Equation(c, 1/c)</span>
-<span class="sd">    &gt;&gt;&gt; t**q</span>
-<span class="sd">    Equation(a**(a*c), (b/c)**(b/c**2))</span>
-
-<span class="sd">    Utility operations</span>
-<span class="sd">    &gt;&gt;&gt; t.reversed</span>
-<span class="sd">    Equation(b/c, a)</span>
-<span class="sd">    &gt;&gt;&gt; t.swap</span>
-<span class="sd">    Equation(b/c, a)</span>
-<span class="sd">    &gt;&gt;&gt; t.lhs</span>
-<span class="sd">    a</span>
-<span class="sd">    &gt;&gt;&gt; t.rhs</span>
-<span class="sd">    b/c</span>
-<span class="sd">    &gt;&gt;&gt; t.as_Boolean()</span>
-<span class="sd">    Eq(a, b/c)</span>
-
-<span class="sd">    `.check()` convenience method for `.as_Boolean().simplify()`</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import I, pi</span>
-<span class="sd">    &gt;&gt;&gt; Equation(pi*(I+2), pi*I+2*pi).check()</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; Eqn(a,a+1).check()</span>
-<span class="sd">    False</span>
-
-<span class="sd">    Differentiation</span>
-<span class="sd">    Differentiation is applied to both sides if the wrt variable appears on</span>
-<span class="sd">    both sides.</span>
-<span class="sd">    &gt;&gt;&gt; q=Eqn(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q</span>
-<span class="sd">    Equation(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,b)</span>
-<span class="sd">    Equation(Derivative(a*c, b), c**(-2))</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c)</span>
-<span class="sd">    Equation(a, -2*b/c**3)</span>
-<span class="sd">    &gt;&gt;&gt; diff(log(q),b)</span>
-<span class="sd">    Equation(Derivative(log(a*c), b), 1/b)</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c,2)</span>
-<span class="sd">    Equation(Derivative(a, c), 6*b/c**4)</span>
-
-<span class="sd">    If you specify multiple differentiation all at once the assumption</span>
-<span class="sd">    is order of differentiation matters and the lhs will not be</span>
-<span class="sd">    evaluated.</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c,b)</span>
-<span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
-
-<span class="sd">    To overcome this specify the order of operations.</span>
-<span class="sd">    &gt;&gt;&gt; diff(diff(q,c),b)</span>
-<span class="sd">    Equation(Derivative(a, b), -2/c**3)</span>
-
-<span class="sd">    But the reverse order returns an unevaulated lhs (a may depend on b).</span>
-<span class="sd">    &gt;&gt;&gt; diff(diff(q,b),c)</span>
-<span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
-
-<span class="sd">    Integration can only be performed on one side at a time.</span>
-<span class="sd">    &gt;&gt;&gt; q=Eqn(a*c,b/c)</span>
-<span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;rhs&#39;)</span>
-<span class="sd">    b**2/(2*c)</span>
-<span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;lhs&#39;)</span>
-<span class="sd">    a*b*c</span>
-
-<span class="sd">    Make a pretty statement of integration from an equation</span>
-<span class="sd">    &gt;&gt;&gt; Eqn(Integral(q.lhs,b),integrate(q,b,side=&#39;rhs&#39;))</span>
-<span class="sd">    Equation(Integral(a*c, b), b**2/(2*c))</span>
-
-<span class="sd">    Integration of each side with respect to different variables</span>
-<span class="sd">    &gt;&gt;&gt; q.dorhs.integrate(b).dolhs.integrate(a)</span>
-<span class="sd">    Equation(a**2*c/2, b**2/(2*c))</span>
-
-<span class="sd">    SymPy&#39;s solvers do not understand these equations. They expect an</span>
-<span class="sd">    expression that the solver assumes = 0. Thus to use the solver the</span>
-<span class="sd">    equation must be rearranged so that all non-zero symbols are on one side.</span>
-<span class="sd">    Then just the non-zero symbolic side is passed to ``solve()``.</span>
-<span class="sd">    &gt;&gt;&gt; t2 = t-t.rhs</span>
-<span class="sd">    &gt;&gt;&gt; t2</span>
-<span class="sd">    Equation(a - b/c, 0)</span>
-<span class="sd">    &gt;&gt;&gt; solve(t2.lhs,c)</span>
-<span class="sd">    [b/a]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">lhs</span><span class="p">)</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;lhs and rhs must be valid sympy expressions.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_get_eqn_name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">IPython</span> <span class="kn">import</span> <span class="n">get_ipython</span>
-        <span class="n">global_dict</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">get_ipython</span><span class="p">(),</span> <span class="s1">&#39;user_ns&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-        <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
-        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
-        <span class="k">if</span> <span class="n">global_dict</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">var_name</span> <span class="ow">in</span> <span class="n">global_dict</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">])</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">==</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> \
-                            <span class="n">var_name</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s1">&#39;_&#39;</span><span class="p">):</span>
-                        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
-                        <span class="k">return</span> <span class="n">var_name</span>
-        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span> <span class="o">=</span> <span class="n">human_text</span>
-        <span class="k">return</span> <span class="s1">&#39;&#39;</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">lhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the lhs of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the rhs of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Converts the equation to an Equality.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
-<span class="sd">        Parameters</span>
-<span class="sd">        ----------</span>
-<span class="sd">        kwargs any appropriate for `Equality`.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        -------</span>
-<span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">reversed</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Swaps the lhs and the rhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">swap</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Synonym for `.reversed`</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">reversed</span>
-
-    <span class="k">def</span> <span class="nf">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># Applies a function to an expression checking whether there</span>
-        <span class="c1"># is a specialized version associated with the particular type of</span>
-        <span class="c1"># expression. Errors will be raised if the function cannot be</span>
-        <span class="c1"># applied to an expression.</span>
-        <span class="n">funcname</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s1">&#39;__name__&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">funcname</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-            <span class="n">localfunc</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">funcname</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">localfunc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">localfunc</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply an operation/function/method to the equation returning the</span>
-<span class="sd">        resulting equation.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        func: object</span>
-<span class="sd">            object to apply usually a function</span>
-
-<span class="sd">        args: as necessary for the function</span>
-
-<span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
-<span class="sd">            Specifies which side of the equation the operation will be applied</span>
-<span class="sd">            to. Default is &#39;both&#39;.</span>
-
-<span class="sd">        kwargs: as necessary for the function</span>
-<span class="sd">         &quot;&quot;&quot;</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the lhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the rhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
-
-    <span class="k">class</span> <span class="nc">_sides</span><span class="p">:</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Helper class for the `.do.`, `.dolhs.`, `.dorhs.` syntax for applying</span>
-<span class="sd">        submethods of expressions.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eqn</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">eqn</span> <span class="o">=</span> <span class="n">eqn</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="o">=</span> <span class="n">side</span>
-
-        <span class="k">def</span> <span class="fm">__getattr__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="kc">None</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;rhs&#39;</span><span class="p">,</span> <span class="s1">&#39;both&#39;</span><span class="p">):</span>
-                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;Expressions in the equation have no &#39;</span>
-                                     <span class="s1">&#39;attribute `&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span>
-                    <span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;`. Try `.apply(&#39;</span>
-                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;, *args)` or &#39;</span>
-                                                   <span class="s1">&#39;pass the equation as a parameter to `&#39;</span>
-                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;()`.&#39;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">apply</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">side</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">do</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dolhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dorhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Overrides of binary math operations</span>
-    <span class="c1">#####</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_binary_op</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">opfunc_ab</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rtruediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rpow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__pow__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Operation helper functions</span>
-    <span class="c1">#####</span>
-    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># TODO: cancel out factors common to both sides.</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># TODO Find why diff and Derivative do not appear to pass through</span>
-        <span class="c1">#  kwargs to this. Since we cannot set evaluation of lhs manually</span>
-        <span class="c1">#  try to be intelligent about when to do it.</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Derivative</span>
-        <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">False</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)):</span>
-            <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">sym</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span>
-                        <span class="n">_sympify</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">):</span>
-                    <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">True</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="n">eval_lhs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">side</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s1">&#39;side&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>  <span class="c1"># Could not seem to pass values for</span>
-        <span class="c1"># `evaluate` through to here.</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;You must specify `side=&quot;lhs&quot;` or `side=&quot;rhs&quot;` &#39;</span>
-                             <span class="s1">&#39;when integrating an Equation&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;`side` must equal &quot;lhs&quot; or &quot;rhs&quot;.&#39;</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Output helper functions</span>
-    <span class="c1">#####</span>
-    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">repstr</span> <span class="o">=</span> <span class="s1">&#39;Equation(</span><span class="si">%s</span><span class="s1">, </span><span class="si">%s</span><span class="s1">)&#39;</span> <span class="o">%</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">())</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__str__</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">repstr</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span> <span class="ow">and</span> <span class="ow">not</span> \
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span><span class="s1">&#39;</span><span class="se">\\</span><span class="s1">text{code version: &#39;</span><span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">+</span><span class="s1">&#39;} </span><span class="se">\\</span><span class="s1">newline &#39;</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;=&#39;</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span><span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,&#39;</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;(</span><span class="se">\\</span><span class="s1">text{&#39;</span><span class="o">+</span><span class="n">namestr</span><span class="o">+</span><span class="s1">&#39;})&#39;</span>
-        <span class="k">return</span> <span class="n">tempstr</span>
-
-    <span class="k">def</span> <span class="fm">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span><span class="p">:</span>
-            <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;code version: &#39;</span><span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="o">+</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39; = &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span> <span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;          (&#39;</span> <span class="o">+</span> <span class="n">namestr</span> <span class="o">+</span> <span class="s1">&#39;)&#39;</span>
-        <span class="k">return</span> <span class="n">tempstr</span>
-</pre></div>
-
-        </details>
+                            <input id="Equation-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Equation</span><wbr>(<span class="base">sympy.core.basic.Basic</span>, <span class="base">sympy.core.evalf.EvalfMixin</span>):
+
+                <label class="view-source-button" for="Equation-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Equation"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation-122"><a href="#Equation-122"><span class="linenos">122</span></a><span class="k">class</span> <span class="nc">Equation</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
+</span><span id="Equation-123"><a href="#Equation-123"><span class="linenos">123</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-124"><a href="#Equation-124"><span class="linenos">124</span></a><span class="sd">    This class defines an equation with a left-hand-side (lhs) and a right-</span>
+</span><span id="Equation-125"><a href="#Equation-125"><span class="linenos">125</span></a><span class="sd">    hand-side (rhs) connected by the &quot;=&quot; operator (e.g. `p*V = n*R*T`).</span>
+</span><span id="Equation-126"><a href="#Equation-126"><span class="linenos">126</span></a>
+</span><span id="Equation-127"><a href="#Equation-127"><span class="linenos">127</span></a><span class="sd">    Explanation</span>
+</span><span id="Equation-128"><a href="#Equation-128"><span class="linenos">128</span></a><span class="sd">    ===========</span>
+</span><span id="Equation-129"><a href="#Equation-129"><span class="linenos">129</span></a><span class="sd">    This class defines relations that all high school and college students</span>
+</span><span id="Equation-130"><a href="#Equation-130"><span class="linenos">130</span></a><span class="sd">    would recognize as mathematical equations. At present only the &quot;=&quot; relation</span>
+</span><span id="Equation-131"><a href="#Equation-131"><span class="linenos">131</span></a><span class="sd">    operator is recognized.</span>
+</span><span id="Equation-132"><a href="#Equation-132"><span class="linenos">132</span></a>
+</span><span id="Equation-133"><a href="#Equation-133"><span class="linenos">133</span></a><span class="sd">    This class is intended to allow using the mathematical tools in SymPy to</span>
+</span><span id="Equation-134"><a href="#Equation-134"><span class="linenos">134</span></a><span class="sd">    rearrange equations and perform algebra in a stepwise fashion. In this</span>
+</span><span id="Equation-135"><a href="#Equation-135"><span class="linenos">135</span></a><span class="sd">    way more people can successfully perform algebraic rearrangements without</span>
+</span><span id="Equation-136"><a href="#Equation-136"><span class="linenos">136</span></a><span class="sd">    stumbling over missed details such as a negative sign.</span>
+</span><span id="Equation-137"><a href="#Equation-137"><span class="linenos">137</span></a>
+</span><span id="Equation-138"><a href="#Equation-138"><span class="linenos">138</span></a><span class="sd">    Create an equation with the call ``Equation(lhs,rhs)``, where ``lhs`` and</span>
+</span><span id="Equation-139"><a href="#Equation-139"><span class="linenos">139</span></a><span class="sd">    ``rhs`` are any valid Sympy expression. ``Eqn(...)`` is a synonym for</span>
+</span><span id="Equation-140"><a href="#Equation-140"><span class="linenos">140</span></a><span class="sd">    ``Equation(...)``.</span>
+</span><span id="Equation-141"><a href="#Equation-141"><span class="linenos">141</span></a>
+</span><span id="Equation-142"><a href="#Equation-142"><span class="linenos">142</span></a><span class="sd">    Parameters</span>
+</span><span id="Equation-143"><a href="#Equation-143"><span class="linenos">143</span></a><span class="sd">    ==========</span>
+</span><span id="Equation-144"><a href="#Equation-144"><span class="linenos">144</span></a><span class="sd">    lhs: sympy expression, ``class Expr``.</span>
+</span><span id="Equation-145"><a href="#Equation-145"><span class="linenos">145</span></a><span class="sd">    rhs: sympy expression, ``class Expr``.</span>
+</span><span id="Equation-146"><a href="#Equation-146"><span class="linenos">146</span></a><span class="sd">    kwargs:</span>
+</span><span id="Equation-147"><a href="#Equation-147"><span class="linenos">147</span></a>
+</span><span id="Equation-148"><a href="#Equation-148"><span class="linenos">148</span></a><span class="sd">    Examples</span>
+</span><span id="Equation-149"><a href="#Equation-149"><span class="linenos">149</span></a><span class="sd">    ========</span>
+</span><span id="Equation-150"><a href="#Equation-150"><span class="linenos">150</span></a><span class="sd">    &gt;&gt;&gt; from algebra_with_sympy import *</span>
+</span><span id="Equation-151"><a href="#Equation-151"><span class="linenos">151</span></a><span class="sd">    &gt;&gt;&gt; a, b, c, x = var(&#39;a b c x&#39;)</span>
+</span><span id="Equation-152"><a href="#Equation-152"><span class="linenos">152</span></a><span class="sd">    &gt;&gt;&gt; Equation(a,b/c)</span>
+</span><span id="Equation-153"><a href="#Equation-153"><span class="linenos">153</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="Equation-154"><a href="#Equation-154"><span class="linenos">154</span></a><span class="sd">    &gt;&gt;&gt; t=Eqn(a,b/c)</span>
+</span><span id="Equation-155"><a href="#Equation-155"><span class="linenos">155</span></a><span class="sd">    &gt;&gt;&gt; t</span>
+</span><span id="Equation-156"><a href="#Equation-156"><span class="linenos">156</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="Equation-157"><a href="#Equation-157"><span class="linenos">157</span></a><span class="sd">    &gt;&gt;&gt; t*c</span>
+</span><span id="Equation-158"><a href="#Equation-158"><span class="linenos">158</span></a><span class="sd">    Equation(a*c, b)</span>
+</span><span id="Equation-159"><a href="#Equation-159"><span class="linenos">159</span></a><span class="sd">    &gt;&gt;&gt; c*t</span>
+</span><span id="Equation-160"><a href="#Equation-160"><span class="linenos">160</span></a><span class="sd">    Equation(a*c, b)</span>
+</span><span id="Equation-161"><a href="#Equation-161"><span class="linenos">161</span></a><span class="sd">    &gt;&gt;&gt; exp(t)</span>
+</span><span id="Equation-162"><a href="#Equation-162"><span class="linenos">162</span></a><span class="sd">    Equation(exp(a), exp(b/c))</span>
+</span><span id="Equation-163"><a href="#Equation-163"><span class="linenos">163</span></a><span class="sd">    &gt;&gt;&gt; exp(log(t))</span>
+</span><span id="Equation-164"><a href="#Equation-164"><span class="linenos">164</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="Equation-165"><a href="#Equation-165"><span class="linenos">165</span></a>
+</span><span id="Equation-166"><a href="#Equation-166"><span class="linenos">166</span></a><span class="sd">    Simplification and Expansion</span>
+</span><span id="Equation-167"><a href="#Equation-167"><span class="linenos">167</span></a><span class="sd">    &gt;&gt;&gt; f = Eqn(x**2 - 1, c)</span>
+</span><span id="Equation-168"><a href="#Equation-168"><span class="linenos">168</span></a><span class="sd">    &gt;&gt;&gt; f</span>
+</span><span id="Equation-169"><a href="#Equation-169"><span class="linenos">169</span></a><span class="sd">    Equation(x**2 - 1, c)</span>
+</span><span id="Equation-170"><a href="#Equation-170"><span class="linenos">170</span></a><span class="sd">    &gt;&gt;&gt; f/(x+1)</span>
+</span><span id="Equation-171"><a href="#Equation-171"><span class="linenos">171</span></a><span class="sd">    Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>
+</span><span id="Equation-172"><a href="#Equation-172"><span class="linenos">172</span></a><span class="sd">    &gt;&gt;&gt; (f/(x+1)).simplify()</span>
+</span><span id="Equation-173"><a href="#Equation-173"><span class="linenos">173</span></a><span class="sd">    Equation(x - 1, c/(x + 1))</span>
+</span><span id="Equation-174"><a href="#Equation-174"><span class="linenos">174</span></a><span class="sd">    &gt;&gt;&gt; simplify(f/(x+1))</span>
+</span><span id="Equation-175"><a href="#Equation-175"><span class="linenos">175</span></a><span class="sd">    Equation(x - 1, c/(x + 1))</span>
+</span><span id="Equation-176"><a href="#Equation-176"><span class="linenos">176</span></a><span class="sd">    &gt;&gt;&gt; (f/(x+1)).expand()</span>
+</span><span id="Equation-177"><a href="#Equation-177"><span class="linenos">177</span></a><span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
+</span><span id="Equation-178"><a href="#Equation-178"><span class="linenos">178</span></a><span class="sd">    &gt;&gt;&gt; expand(f/(x+1))</span>
+</span><span id="Equation-179"><a href="#Equation-179"><span class="linenos">179</span></a><span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
+</span><span id="Equation-180"><a href="#Equation-180"><span class="linenos">180</span></a><span class="sd">    &gt;&gt;&gt; factor(f)</span>
+</span><span id="Equation-181"><a href="#Equation-181"><span class="linenos">181</span></a><span class="sd">    Equation((x - 1)*(x + 1), c)</span>
+</span><span id="Equation-182"><a href="#Equation-182"><span class="linenos">182</span></a><span class="sd">    &gt;&gt;&gt; f.factor()</span>
+</span><span id="Equation-183"><a href="#Equation-183"><span class="linenos">183</span></a><span class="sd">    Equation((x - 1)*(x + 1), c)</span>
+</span><span id="Equation-184"><a href="#Equation-184"><span class="linenos">184</span></a><span class="sd">    &gt;&gt;&gt; f2 = f+a*x**2+b*x +c</span>
+</span><span id="Equation-185"><a href="#Equation-185"><span class="linenos">185</span></a><span class="sd">    &gt;&gt;&gt; f2</span>
+</span><span id="Equation-186"><a href="#Equation-186"><span class="linenos">186</span></a><span class="sd">    Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>
+</span><span id="Equation-187"><a href="#Equation-187"><span class="linenos">187</span></a><span class="sd">    &gt;&gt;&gt; collect(f2,x)</span>
+</span><span id="Equation-188"><a href="#Equation-188"><span class="linenos">188</span></a><span class="sd">    Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>
+</span><span id="Equation-189"><a href="#Equation-189"><span class="linenos">189</span></a>
+</span><span id="Equation-190"><a href="#Equation-190"><span class="linenos">190</span></a><span class="sd">    Apply operation to only one side</span>
+</span><span id="Equation-191"><a href="#Equation-191"><span class="linenos">191</span></a><span class="sd">    &gt;&gt;&gt; poly = Eqn(a*x**2 + b*x + c*x**2, a*x**3 + b*x**3 + c*x)</span>
+</span><span id="Equation-192"><a href="#Equation-192"><span class="linenos">192</span></a><span class="sd">    &gt;&gt;&gt; poly.applyrhs(factor,x)</span>
+</span><span id="Equation-193"><a href="#Equation-193"><span class="linenos">193</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>
+</span><span id="Equation-194"><a href="#Equation-194"><span class="linenos">194</span></a><span class="sd">    &gt;&gt;&gt; poly.applylhs(factor)</span>
+</span><span id="Equation-195"><a href="#Equation-195"><span class="linenos">195</span></a><span class="sd">    Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="Equation-196"><a href="#Equation-196"><span class="linenos">196</span></a><span class="sd">    &gt;&gt;&gt; poly.applylhs(collect,x)</span>
+</span><span id="Equation-197"><a href="#Equation-197"><span class="linenos">197</span></a><span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="Equation-198"><a href="#Equation-198"><span class="linenos">198</span></a>
+</span><span id="Equation-199"><a href="#Equation-199"><span class="linenos">199</span></a><span class="sd">    ``.apply...`` also works with user defined python functions</span>
+</span><span id="Equation-200"><a href="#Equation-200"><span class="linenos">200</span></a><span class="sd">    &gt;&gt;&gt; def addsquare(eqn):</span>
+</span><span id="Equation-201"><a href="#Equation-201"><span class="linenos">201</span></a><span class="sd">    ...     return eqn+eqn**2</span>
+</span><span id="Equation-202"><a href="#Equation-202"><span class="linenos">202</span></a><span class="sd">    ...</span>
+</span><span id="Equation-203"><a href="#Equation-203"><span class="linenos">203</span></a><span class="sd">    &gt;&gt;&gt; t.apply(addsquare)</span>
+</span><span id="Equation-204"><a href="#Equation-204"><span class="linenos">204</span></a><span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
+</span><span id="Equation-205"><a href="#Equation-205"><span class="linenos">205</span></a><span class="sd">    &gt;&gt;&gt; t.applyrhs(addsquare)</span>
+</span><span id="Equation-206"><a href="#Equation-206"><span class="linenos">206</span></a><span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
+</span><span id="Equation-207"><a href="#Equation-207"><span class="linenos">207</span></a><span class="sd">    &gt;&gt;&gt; t.apply(addsquare, side = &#39;rhs&#39;)</span>
+</span><span id="Equation-208"><a href="#Equation-208"><span class="linenos">208</span></a><span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
+</span><span id="Equation-209"><a href="#Equation-209"><span class="linenos">209</span></a><span class="sd">    &gt;&gt;&gt; t.applylhs(addsquare)</span>
+</span><span id="Equation-210"><a href="#Equation-210"><span class="linenos">210</span></a><span class="sd">    Equation(a**2 + a, b/c)</span>
+</span><span id="Equation-211"><a href="#Equation-211"><span class="linenos">211</span></a><span class="sd">    &gt;&gt;&gt; addsquare(t)</span>
+</span><span id="Equation-212"><a href="#Equation-212"><span class="linenos">212</span></a><span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
+</span><span id="Equation-213"><a href="#Equation-213"><span class="linenos">213</span></a>
+</span><span id="Equation-214"><a href="#Equation-214"><span class="linenos">214</span></a><span class="sd">    Inaddition to ``.apply...`` there is also the less general ``.do``,</span>
+</span><span id="Equation-215"><a href="#Equation-215"><span class="linenos">215</span></a><span class="sd">    ``.dolhs``, ``.dorhs``, which only works for operations defined on the</span>
+</span><span id="Equation-216"><a href="#Equation-216"><span class="linenos">216</span></a><span class="sd">    ``Expr`` class (e.g.``.collect(), .factor(), .expand()``, etc...).</span>
+</span><span id="Equation-217"><a href="#Equation-217"><span class="linenos">217</span></a><span class="sd">    &gt;&gt;&gt; poly.dolhs.collect(x)</span>
+</span><span id="Equation-218"><a href="#Equation-218"><span class="linenos">218</span></a><span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="Equation-219"><a href="#Equation-219"><span class="linenos">219</span></a><span class="sd">    &gt;&gt;&gt; poly.dorhs.collect(x)</span>
+</span><span id="Equation-220"><a href="#Equation-220"><span class="linenos">220</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>
+</span><span id="Equation-221"><a href="#Equation-221"><span class="linenos">221</span></a><span class="sd">    &gt;&gt;&gt; poly.do.collect(x)</span>
+</span><span id="Equation-222"><a href="#Equation-222"><span class="linenos">222</span></a><span class="sd">    Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>
+</span><span id="Equation-223"><a href="#Equation-223"><span class="linenos">223</span></a><span class="sd">    &gt;&gt;&gt; poly.dorhs.factor()</span>
+</span><span id="Equation-224"><a href="#Equation-224"><span class="linenos">224</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>
+</span><span id="Equation-225"><a href="#Equation-225"><span class="linenos">225</span></a>
+</span><span id="Equation-226"><a href="#Equation-226"><span class="linenos">226</span></a><span class="sd">    ``poly.do.exp()`` or other sympy math functions will raise an error.</span>
+</span><span id="Equation-227"><a href="#Equation-227"><span class="linenos">227</span></a>
+</span><span id="Equation-228"><a href="#Equation-228"><span class="linenos">228</span></a><span class="sd">    Rearranging an equation (simple example made complicated as illustration)</span>
+</span><span id="Equation-229"><a href="#Equation-229"><span class="linenos">229</span></a><span class="sd">    &gt;&gt;&gt; p, V, n, R, T = var(&#39;p V n R T&#39;)</span>
+</span><span id="Equation-230"><a href="#Equation-230"><span class="linenos">230</span></a><span class="sd">    &gt;&gt;&gt; eq1=Eqn(p*V,n*R*T)</span>
+</span><span id="Equation-231"><a href="#Equation-231"><span class="linenos">231</span></a><span class="sd">    &gt;&gt;&gt; eq1</span>
+</span><span id="Equation-232"><a href="#Equation-232"><span class="linenos">232</span></a><span class="sd">    Equation(V*p, R*T*n)</span>
+</span><span id="Equation-233"><a href="#Equation-233"><span class="linenos">233</span></a><span class="sd">    &gt;&gt;&gt; eq2 =eq1/V</span>
+</span><span id="Equation-234"><a href="#Equation-234"><span class="linenos">234</span></a><span class="sd">    &gt;&gt;&gt; eq2</span>
+</span><span id="Equation-235"><a href="#Equation-235"><span class="linenos">235</span></a><span class="sd">    Equation(p, R*T*n/V)</span>
+</span><span id="Equation-236"><a href="#Equation-236"><span class="linenos">236</span></a><span class="sd">    &gt;&gt;&gt; eq3 = eq2/R/T</span>
+</span><span id="Equation-237"><a href="#Equation-237"><span class="linenos">237</span></a><span class="sd">    &gt;&gt;&gt; eq3</span>
+</span><span id="Equation-238"><a href="#Equation-238"><span class="linenos">238</span></a><span class="sd">    Equation(p/(R*T), n/V)</span>
+</span><span id="Equation-239"><a href="#Equation-239"><span class="linenos">239</span></a><span class="sd">    &gt;&gt;&gt; eq4 = eq3*R/p</span>
+</span><span id="Equation-240"><a href="#Equation-240"><span class="linenos">240</span></a><span class="sd">    &gt;&gt;&gt; eq4</span>
+</span><span id="Equation-241"><a href="#Equation-241"><span class="linenos">241</span></a><span class="sd">    Equation(1/T, R*n/(V*p))</span>
+</span><span id="Equation-242"><a href="#Equation-242"><span class="linenos">242</span></a><span class="sd">    &gt;&gt;&gt; 1/eq4</span>
+</span><span id="Equation-243"><a href="#Equation-243"><span class="linenos">243</span></a><span class="sd">    Equation(T, V*p/(R*n))</span>
+</span><span id="Equation-244"><a href="#Equation-244"><span class="linenos">244</span></a><span class="sd">    &gt;&gt;&gt; eq5 = 1/eq4 - T</span>
+</span><span id="Equation-245"><a href="#Equation-245"><span class="linenos">245</span></a><span class="sd">    &gt;&gt;&gt; eq5</span>
+</span><span id="Equation-246"><a href="#Equation-246"><span class="linenos">246</span></a><span class="sd">    Equation(0, -T + V*p/(R*n))</span>
+</span><span id="Equation-247"><a href="#Equation-247"><span class="linenos">247</span></a>
+</span><span id="Equation-248"><a href="#Equation-248"><span class="linenos">248</span></a><span class="sd">    Substitution (#&#39;s and units)</span>
+</span><span id="Equation-249"><a href="#Equation-249"><span class="linenos">249</span></a><span class="sd">    &gt;&gt;&gt; L, atm, mol, K = var(&#39;L atm mol K&#39;, positive=True, real=True) # units</span>
+</span><span id="Equation-250"><a href="#Equation-250"><span class="linenos">250</span></a><span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L})</span>
+</span><span id="Equation-251"><a href="#Equation-251"><span class="linenos">251</span></a><span class="sd">    Equation(p, 0.9334325*atm)</span>
+</span><span id="Equation-252"><a href="#Equation-252"><span class="linenos">252</span></a><span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L}).evalf(4)</span>
+</span><span id="Equation-253"><a href="#Equation-253"><span class="linenos">253</span></a><span class="sd">    Equation(p, 0.9334*atm)</span>
+</span><span id="Equation-254"><a href="#Equation-254"><span class="linenos">254</span></a>
+</span><span id="Equation-255"><a href="#Equation-255"><span class="linenos">255</span></a><span class="sd">    Combining equations (Math with equations: lhs with lhs and rhs with rhs)</span>
+</span><span id="Equation-256"><a href="#Equation-256"><span class="linenos">256</span></a><span class="sd">    &gt;&gt;&gt; q = Eqn(a*c, b/c**2)</span>
+</span><span id="Equation-257"><a href="#Equation-257"><span class="linenos">257</span></a><span class="sd">    &gt;&gt;&gt; q</span>
+</span><span id="Equation-258"><a href="#Equation-258"><span class="linenos">258</span></a><span class="sd">    Equation(a*c, b/c**2)</span>
+</span><span id="Equation-259"><a href="#Equation-259"><span class="linenos">259</span></a><span class="sd">    &gt;&gt;&gt; t</span>
+</span><span id="Equation-260"><a href="#Equation-260"><span class="linenos">260</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="Equation-261"><a href="#Equation-261"><span class="linenos">261</span></a><span class="sd">    &gt;&gt;&gt; q+t</span>
+</span><span id="Equation-262"><a href="#Equation-262"><span class="linenos">262</span></a><span class="sd">    Equation(a*c + a, b/c + b/c**2)</span>
+</span><span id="Equation-263"><a href="#Equation-263"><span class="linenos">263</span></a><span class="sd">    &gt;&gt;&gt; q/t</span>
+</span><span id="Equation-264"><a href="#Equation-264"><span class="linenos">264</span></a><span class="sd">    Equation(c, 1/c)</span>
+</span><span id="Equation-265"><a href="#Equation-265"><span class="linenos">265</span></a><span class="sd">    &gt;&gt;&gt; t**q</span>
+</span><span id="Equation-266"><a href="#Equation-266"><span class="linenos">266</span></a><span class="sd">    Equation(a**(a*c), (b/c)**(b/c**2))</span>
+</span><span id="Equation-267"><a href="#Equation-267"><span class="linenos">267</span></a>
+</span><span id="Equation-268"><a href="#Equation-268"><span class="linenos">268</span></a><span class="sd">    Utility operations</span>
+</span><span id="Equation-269"><a href="#Equation-269"><span class="linenos">269</span></a><span class="sd">    &gt;&gt;&gt; t.reversed</span>
+</span><span id="Equation-270"><a href="#Equation-270"><span class="linenos">270</span></a><span class="sd">    Equation(b/c, a)</span>
+</span><span id="Equation-271"><a href="#Equation-271"><span class="linenos">271</span></a><span class="sd">    &gt;&gt;&gt; t.swap</span>
+</span><span id="Equation-272"><a href="#Equation-272"><span class="linenos">272</span></a><span class="sd">    Equation(b/c, a)</span>
+</span><span id="Equation-273"><a href="#Equation-273"><span class="linenos">273</span></a><span class="sd">    &gt;&gt;&gt; t.lhs</span>
+</span><span id="Equation-274"><a href="#Equation-274"><span class="linenos">274</span></a><span class="sd">    a</span>
+</span><span id="Equation-275"><a href="#Equation-275"><span class="linenos">275</span></a><span class="sd">    &gt;&gt;&gt; t.rhs</span>
+</span><span id="Equation-276"><a href="#Equation-276"><span class="linenos">276</span></a><span class="sd">    b/c</span>
+</span><span id="Equation-277"><a href="#Equation-277"><span class="linenos">277</span></a><span class="sd">    &gt;&gt;&gt; t.as_Boolean()</span>
+</span><span id="Equation-278"><a href="#Equation-278"><span class="linenos">278</span></a><span class="sd">    Eq(a, b/c)</span>
+</span><span id="Equation-279"><a href="#Equation-279"><span class="linenos">279</span></a>
+</span><span id="Equation-280"><a href="#Equation-280"><span class="linenos">280</span></a><span class="sd">    `.check()` convenience method for `.as_Boolean().simplify()`</span>
+</span><span id="Equation-281"><a href="#Equation-281"><span class="linenos">281</span></a><span class="sd">    &gt;&gt;&gt; from sympy import I, pi</span>
+</span><span id="Equation-282"><a href="#Equation-282"><span class="linenos">282</span></a><span class="sd">    &gt;&gt;&gt; Equation(pi*(I+2), pi*I+2*pi).check()</span>
+</span><span id="Equation-283"><a href="#Equation-283"><span class="linenos">283</span></a><span class="sd">    True</span>
+</span><span id="Equation-284"><a href="#Equation-284"><span class="linenos">284</span></a><span class="sd">    &gt;&gt;&gt; Eqn(a,a+1).check()</span>
+</span><span id="Equation-285"><a href="#Equation-285"><span class="linenos">285</span></a><span class="sd">    False</span>
+</span><span id="Equation-286"><a href="#Equation-286"><span class="linenos">286</span></a>
+</span><span id="Equation-287"><a href="#Equation-287"><span class="linenos">287</span></a><span class="sd">    Differentiation</span>
+</span><span id="Equation-288"><a href="#Equation-288"><span class="linenos">288</span></a><span class="sd">    Differentiation is applied to both sides if the wrt variable appears on</span>
+</span><span id="Equation-289"><a href="#Equation-289"><span class="linenos">289</span></a><span class="sd">    both sides.</span>
+</span><span id="Equation-290"><a href="#Equation-290"><span class="linenos">290</span></a><span class="sd">    &gt;&gt;&gt; q=Eqn(a*c, b/c**2)</span>
+</span><span id="Equation-291"><a href="#Equation-291"><span class="linenos">291</span></a><span class="sd">    &gt;&gt;&gt; q</span>
+</span><span id="Equation-292"><a href="#Equation-292"><span class="linenos">292</span></a><span class="sd">    Equation(a*c, b/c**2)</span>
+</span><span id="Equation-293"><a href="#Equation-293"><span class="linenos">293</span></a><span class="sd">    &gt;&gt;&gt; diff(q,b)</span>
+</span><span id="Equation-294"><a href="#Equation-294"><span class="linenos">294</span></a><span class="sd">    Equation(Derivative(a*c, b), c**(-2))</span>
+</span><span id="Equation-295"><a href="#Equation-295"><span class="linenos">295</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c)</span>
+</span><span id="Equation-296"><a href="#Equation-296"><span class="linenos">296</span></a><span class="sd">    Equation(a, -2*b/c**3)</span>
+</span><span id="Equation-297"><a href="#Equation-297"><span class="linenos">297</span></a><span class="sd">    &gt;&gt;&gt; diff(log(q),b)</span>
+</span><span id="Equation-298"><a href="#Equation-298"><span class="linenos">298</span></a><span class="sd">    Equation(Derivative(log(a*c), b), 1/b)</span>
+</span><span id="Equation-299"><a href="#Equation-299"><span class="linenos">299</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c,2)</span>
+</span><span id="Equation-300"><a href="#Equation-300"><span class="linenos">300</span></a><span class="sd">    Equation(Derivative(a, c), 6*b/c**4)</span>
+</span><span id="Equation-301"><a href="#Equation-301"><span class="linenos">301</span></a>
+</span><span id="Equation-302"><a href="#Equation-302"><span class="linenos">302</span></a><span class="sd">    If you specify multiple differentiation all at once the assumption</span>
+</span><span id="Equation-303"><a href="#Equation-303"><span class="linenos">303</span></a><span class="sd">    is order of differentiation matters and the lhs will not be</span>
+</span><span id="Equation-304"><a href="#Equation-304"><span class="linenos">304</span></a><span class="sd">    evaluated.</span>
+</span><span id="Equation-305"><a href="#Equation-305"><span class="linenos">305</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c,b)</span>
+</span><span id="Equation-306"><a href="#Equation-306"><span class="linenos">306</span></a><span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
+</span><span id="Equation-307"><a href="#Equation-307"><span class="linenos">307</span></a>
+</span><span id="Equation-308"><a href="#Equation-308"><span class="linenos">308</span></a><span class="sd">    To overcome this specify the order of operations.</span>
+</span><span id="Equation-309"><a href="#Equation-309"><span class="linenos">309</span></a><span class="sd">    &gt;&gt;&gt; diff(diff(q,c),b)</span>
+</span><span id="Equation-310"><a href="#Equation-310"><span class="linenos">310</span></a><span class="sd">    Equation(Derivative(a, b), -2/c**3)</span>
+</span><span id="Equation-311"><a href="#Equation-311"><span class="linenos">311</span></a>
+</span><span id="Equation-312"><a href="#Equation-312"><span class="linenos">312</span></a><span class="sd">    But the reverse order returns an unevaulated lhs (a may depend on b).</span>
+</span><span id="Equation-313"><a href="#Equation-313"><span class="linenos">313</span></a><span class="sd">    &gt;&gt;&gt; diff(diff(q,b),c)</span>
+</span><span id="Equation-314"><a href="#Equation-314"><span class="linenos">314</span></a><span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
+</span><span id="Equation-315"><a href="#Equation-315"><span class="linenos">315</span></a>
+</span><span id="Equation-316"><a href="#Equation-316"><span class="linenos">316</span></a><span class="sd">    Integration can only be performed on one side at a time.</span>
+</span><span id="Equation-317"><a href="#Equation-317"><span class="linenos">317</span></a><span class="sd">    &gt;&gt;&gt; q=Eqn(a*c,b/c)</span>
+</span><span id="Equation-318"><a href="#Equation-318"><span class="linenos">318</span></a><span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;rhs&#39;)</span>
+</span><span id="Equation-319"><a href="#Equation-319"><span class="linenos">319</span></a><span class="sd">    b**2/(2*c)</span>
+</span><span id="Equation-320"><a href="#Equation-320"><span class="linenos">320</span></a><span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;lhs&#39;)</span>
+</span><span id="Equation-321"><a href="#Equation-321"><span class="linenos">321</span></a><span class="sd">    a*b*c</span>
+</span><span id="Equation-322"><a href="#Equation-322"><span class="linenos">322</span></a>
+</span><span id="Equation-323"><a href="#Equation-323"><span class="linenos">323</span></a><span class="sd">    Make a pretty statement of integration from an equation</span>
+</span><span id="Equation-324"><a href="#Equation-324"><span class="linenos">324</span></a><span class="sd">    &gt;&gt;&gt; Eqn(Integral(q.lhs,b),integrate(q,b,side=&#39;rhs&#39;))</span>
+</span><span id="Equation-325"><a href="#Equation-325"><span class="linenos">325</span></a><span class="sd">    Equation(Integral(a*c, b), b**2/(2*c))</span>
+</span><span id="Equation-326"><a href="#Equation-326"><span class="linenos">326</span></a>
+</span><span id="Equation-327"><a href="#Equation-327"><span class="linenos">327</span></a><span class="sd">    Integration of each side with respect to different variables</span>
+</span><span id="Equation-328"><a href="#Equation-328"><span class="linenos">328</span></a><span class="sd">    &gt;&gt;&gt; q.dorhs.integrate(b).dolhs.integrate(a)</span>
+</span><span id="Equation-329"><a href="#Equation-329"><span class="linenos">329</span></a><span class="sd">    Equation(a**2*c/2, b**2/(2*c))</span>
+</span><span id="Equation-330"><a href="#Equation-330"><span class="linenos">330</span></a>
+</span><span id="Equation-331"><a href="#Equation-331"><span class="linenos">331</span></a><span class="sd">    SymPy&#39;s solvers do not understand these equations. They expect an</span>
+</span><span id="Equation-332"><a href="#Equation-332"><span class="linenos">332</span></a><span class="sd">    expression that the solver assumes = 0. Thus to use the solver the</span>
+</span><span id="Equation-333"><a href="#Equation-333"><span class="linenos">333</span></a><span class="sd">    equation must be rearranged so that all non-zero symbols are on one side.</span>
+</span><span id="Equation-334"><a href="#Equation-334"><span class="linenos">334</span></a><span class="sd">    Then just the non-zero symbolic side is passed to ``solve()``.</span>
+</span><span id="Equation-335"><a href="#Equation-335"><span class="linenos">335</span></a><span class="sd">    &gt;&gt;&gt; t2 = t-t.rhs</span>
+</span><span id="Equation-336"><a href="#Equation-336"><span class="linenos">336</span></a><span class="sd">    &gt;&gt;&gt; t2</span>
+</span><span id="Equation-337"><a href="#Equation-337"><span class="linenos">337</span></a><span class="sd">    Equation(a - b/c, 0)</span>
+</span><span id="Equation-338"><a href="#Equation-338"><span class="linenos">338</span></a><span class="sd">    &gt;&gt;&gt; solve(t2.lhs,c)</span>
+</span><span id="Equation-339"><a href="#Equation-339"><span class="linenos">339</span></a><span class="sd">    [b/a]</span>
+</span><span id="Equation-340"><a href="#Equation-340"><span class="linenos">340</span></a>
+</span><span id="Equation-341"><a href="#Equation-341"><span class="linenos">341</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="Equation-342"><a href="#Equation-342"><span class="linenos">342</span></a>
+</span><span id="Equation-343"><a href="#Equation-343"><span class="linenos">343</span></a>    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-344"><a href="#Equation-344"><span class="linenos">344</span></a>        <span class="n">lhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">lhs</span><span class="p">)</span>
+</span><span id="Equation-345"><a href="#Equation-345"><span class="linenos">345</span></a>        <span class="n">rhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="Equation-346"><a href="#Equation-346"><span class="linenos">346</span></a>        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
+</span><span id="Equation-347"><a href="#Equation-347"><span class="linenos">347</span></a>            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;lhs and rhs must be valid sympy expressions.&#39;</span><span class="p">)</span>
+</span><span id="Equation-348"><a href="#Equation-348"><span class="linenos">348</span></a>        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
+</span><span id="Equation-349"><a href="#Equation-349"><span class="linenos">349</span></a>
+</span><span id="Equation-350"><a href="#Equation-350"><span class="linenos">350</span></a>    <span class="k">def</span> <span class="nf">_get_eqn_name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-351"><a href="#Equation-351"><span class="linenos">351</span></a>        <span class="kn">from</span> <span class="nn">IPython</span> <span class="kn">import</span> <span class="n">get_ipython</span>
+</span><span id="Equation-352"><a href="#Equation-352"><span class="linenos">352</span></a>        <span class="n">global_dict</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">get_ipython</span><span class="p">(),</span> <span class="s1">&#39;user_ns&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Equation-353"><a href="#Equation-353"><span class="linenos">353</span></a>        <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="Equation-354"><a href="#Equation-354"><span class="linenos">354</span></a>        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
+</span><span id="Equation-355"><a href="#Equation-355"><span class="linenos">355</span></a>        <span class="k">if</span> <span class="n">global_dict</span><span class="p">:</span>
+</span><span id="Equation-356"><a href="#Equation-356"><span class="linenos">356</span></a>            <span class="k">for</span> <span class="n">var_name</span> <span class="ow">in</span> <span class="n">global_dict</span><span class="p">:</span>
+</span><span id="Equation-357"><a href="#Equation-357"><span class="linenos">357</span></a>                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="Equation-358"><a href="#Equation-358"><span class="linenos">358</span></a>                    <span class="k">if</span> <span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">])</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">==</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> \
+</span><span id="Equation-359"><a href="#Equation-359"><span class="linenos">359</span></a>                            <span class="n">var_name</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s1">&#39;_&#39;</span><span class="p">):</span>
+</span><span id="Equation-360"><a href="#Equation-360"><span class="linenos">360</span></a>                        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
+</span><span id="Equation-361"><a href="#Equation-361"><span class="linenos">361</span></a>                        <span class="k">return</span> <span class="n">var_name</span>
+</span><span id="Equation-362"><a href="#Equation-362"><span class="linenos">362</span></a>        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span> <span class="o">=</span> <span class="n">human_text</span>
+</span><span id="Equation-363"><a href="#Equation-363"><span class="linenos">363</span></a>        <span class="k">return</span> <span class="s1">&#39;&#39;</span>
+</span><span id="Equation-364"><a href="#Equation-364"><span class="linenos">364</span></a>
+</span><span id="Equation-365"><a href="#Equation-365"><span class="linenos">365</span></a>    <span class="nd">@property</span>
+</span><span id="Equation-366"><a href="#Equation-366"><span class="linenos">366</span></a>    <span class="k">def</span> <span class="nf">lhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-367"><a href="#Equation-367"><span class="linenos">367</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-368"><a href="#Equation-368"><span class="linenos">368</span></a><span class="sd">        Returns the lhs of the equation.</span>
+</span><span id="Equation-369"><a href="#Equation-369"><span class="linenos">369</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-370"><a href="#Equation-370"><span class="linenos">370</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+</span><span id="Equation-371"><a href="#Equation-371"><span class="linenos">371</span></a>
+</span><span id="Equation-372"><a href="#Equation-372"><span class="linenos">372</span></a>    <span class="nd">@property</span>
+</span><span id="Equation-373"><a href="#Equation-373"><span class="linenos">373</span></a>    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-374"><a href="#Equation-374"><span class="linenos">374</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-375"><a href="#Equation-375"><span class="linenos">375</span></a><span class="sd">        Returns the rhs of the equation.</span>
+</span><span id="Equation-376"><a href="#Equation-376"><span class="linenos">376</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-377"><a href="#Equation-377"><span class="linenos">377</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span><span id="Equation-378"><a href="#Equation-378"><span class="linenos">378</span></a>
+</span><span id="Equation-379"><a href="#Equation-379"><span class="linenos">379</span></a>    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-380"><a href="#Equation-380"><span class="linenos">380</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-381"><a href="#Equation-381"><span class="linenos">381</span></a><span class="sd">        Converts the equation to an Equality.</span>
+</span><span id="Equation-382"><a href="#Equation-382"><span class="linenos">382</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-383"><a href="#Equation-383"><span class="linenos">383</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="Equation-384"><a href="#Equation-384"><span class="linenos">384</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="Equation-385"><a href="#Equation-385"><span class="linenos">385</span></a>
+</span><span id="Equation-386"><a href="#Equation-386"><span class="linenos">386</span></a>    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-387"><a href="#Equation-387"><span class="linenos">387</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-388"><a href="#Equation-388"><span class="linenos">388</span></a><span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
+</span><span id="Equation-389"><a href="#Equation-389"><span class="linenos">389</span></a><span class="sd">        Parameters</span>
+</span><span id="Equation-390"><a href="#Equation-390"><span class="linenos">390</span></a><span class="sd">        ----------</span>
+</span><span id="Equation-391"><a href="#Equation-391"><span class="linenos">391</span></a><span class="sd">        kwargs any appropriate for `Equality`.</span>
+</span><span id="Equation-392"><a href="#Equation-392"><span class="linenos">392</span></a>
+</span><span id="Equation-393"><a href="#Equation-393"><span class="linenos">393</span></a><span class="sd">        Returns</span>
+</span><span id="Equation-394"><a href="#Equation-394"><span class="linenos">394</span></a><span class="sd">        -------</span>
+</span><span id="Equation-395"><a href="#Equation-395"><span class="linenos">395</span></a><span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
+</span><span id="Equation-396"><a href="#Equation-396"><span class="linenos">396</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-397"><a href="#Equation-397"><span class="linenos">397</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="Equation-398"><a href="#Equation-398"><span class="linenos">398</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
+</span><span id="Equation-399"><a href="#Equation-399"><span class="linenos">399</span></a>
+</span><span id="Equation-400"><a href="#Equation-400"><span class="linenos">400</span></a>    <span class="nd">@property</span>
+</span><span id="Equation-401"><a href="#Equation-401"><span class="linenos">401</span></a>    <span class="k">def</span> <span class="nf">reversed</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-402"><a href="#Equation-402"><span class="linenos">402</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-403"><a href="#Equation-403"><span class="linenos">403</span></a><span class="sd">        Swaps the lhs and the rhs.</span>
+</span><span id="Equation-404"><a href="#Equation-404"><span class="linenos">404</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-405"><a href="#Equation-405"><span class="linenos">405</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
+</span><span id="Equation-406"><a href="#Equation-406"><span class="linenos">406</span></a>
+</span><span id="Equation-407"><a href="#Equation-407"><span class="linenos">407</span></a>    <span class="nd">@property</span>
+</span><span id="Equation-408"><a href="#Equation-408"><span class="linenos">408</span></a>    <span class="k">def</span> <span class="nf">swap</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-409"><a href="#Equation-409"><span class="linenos">409</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-410"><a href="#Equation-410"><span class="linenos">410</span></a><span class="sd">        Synonym for `.reversed`</span>
+</span><span id="Equation-411"><a href="#Equation-411"><span class="linenos">411</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-412"><a href="#Equation-412"><span class="linenos">412</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">reversed</span>
+</span><span id="Equation-413"><a href="#Equation-413"><span class="linenos">413</span></a>
+</span><span id="Equation-414"><a href="#Equation-414"><span class="linenos">414</span></a>    <span class="k">def</span> <span class="nf">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-415"><a href="#Equation-415"><span class="linenos">415</span></a>        <span class="c1"># Applies a function to an expression checking whether there</span>
+</span><span id="Equation-416"><a href="#Equation-416"><span class="linenos">416</span></a>        <span class="c1"># is a specialized version associated with the particular type of</span>
+</span><span id="Equation-417"><a href="#Equation-417"><span class="linenos">417</span></a>        <span class="c1"># expression. Errors will be raised if the function cannot be</span>
+</span><span id="Equation-418"><a href="#Equation-418"><span class="linenos">418</span></a>        <span class="c1"># applied to an expression.</span>
+</span><span id="Equation-419"><a href="#Equation-419"><span class="linenos">419</span></a>        <span class="n">funcname</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s1">&#39;__name__&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Equation-420"><a href="#Equation-420"><span class="linenos">420</span></a>        <span class="k">if</span> <span class="n">funcname</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="Equation-421"><a href="#Equation-421"><span class="linenos">421</span></a>            <span class="n">localfunc</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">funcname</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Equation-422"><a href="#Equation-422"><span class="linenos">422</span></a>            <span class="k">if</span> <span class="n">localfunc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="Equation-423"><a href="#Equation-423"><span class="linenos">423</span></a>                <span class="k">return</span> <span class="n">localfunc</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-424"><a href="#Equation-424"><span class="linenos">424</span></a>        <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-425"><a href="#Equation-425"><span class="linenos">425</span></a>
+</span><span id="Equation-426"><a href="#Equation-426"><span class="linenos">426</span></a>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-427"><a href="#Equation-427"><span class="linenos">427</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-428"><a href="#Equation-428"><span class="linenos">428</span></a><span class="sd">        Apply an operation/function/method to the equation returning the</span>
+</span><span id="Equation-429"><a href="#Equation-429"><span class="linenos">429</span></a><span class="sd">        resulting equation.</span>
+</span><span id="Equation-430"><a href="#Equation-430"><span class="linenos">430</span></a>
+</span><span id="Equation-431"><a href="#Equation-431"><span class="linenos">431</span></a><span class="sd">        Parameters</span>
+</span><span id="Equation-432"><a href="#Equation-432"><span class="linenos">432</span></a><span class="sd">        ==========</span>
+</span><span id="Equation-433"><a href="#Equation-433"><span class="linenos">433</span></a>
+</span><span id="Equation-434"><a href="#Equation-434"><span class="linenos">434</span></a><span class="sd">        func: object</span>
+</span><span id="Equation-435"><a href="#Equation-435"><span class="linenos">435</span></a><span class="sd">            object to apply usually a function</span>
+</span><span id="Equation-436"><a href="#Equation-436"><span class="linenos">436</span></a>
+</span><span id="Equation-437"><a href="#Equation-437"><span class="linenos">437</span></a><span class="sd">        args: as necessary for the function</span>
+</span><span id="Equation-438"><a href="#Equation-438"><span class="linenos">438</span></a>
+</span><span id="Equation-439"><a href="#Equation-439"><span class="linenos">439</span></a><span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
+</span><span id="Equation-440"><a href="#Equation-440"><span class="linenos">440</span></a><span class="sd">            Specifies which side of the equation the operation will be applied</span>
+</span><span id="Equation-441"><a href="#Equation-441"><span class="linenos">441</span></a><span class="sd">            to. Default is &#39;both&#39;.</span>
+</span><span id="Equation-442"><a href="#Equation-442"><span class="linenos">442</span></a>
+</span><span id="Equation-443"><a href="#Equation-443"><span class="linenos">443</span></a><span class="sd">        kwargs: as necessary for the function</span>
+</span><span id="Equation-444"><a href="#Equation-444"><span class="linenos">444</span></a><span class="sd">         &quot;&quot;&quot;</span>
+</span><span id="Equation-445"><a href="#Equation-445"><span class="linenos">445</span></a>        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
+</span><span id="Equation-446"><a href="#Equation-446"><span class="linenos">446</span></a>        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
+</span><span id="Equation-447"><a href="#Equation-447"><span class="linenos">447</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
+</span><span id="Equation-448"><a href="#Equation-448"><span class="linenos">448</span></a>            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-449"><a href="#Equation-449"><span class="linenos">449</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
+</span><span id="Equation-450"><a href="#Equation-450"><span class="linenos">450</span></a>            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-451"><a href="#Equation-451"><span class="linenos">451</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
+</span><span id="Equation-452"><a href="#Equation-452"><span class="linenos">452</span></a>
+</span><span id="Equation-453"><a href="#Equation-453"><span class="linenos">453</span></a>    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-454"><a href="#Equation-454"><span class="linenos">454</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-455"><a href="#Equation-455"><span class="linenos">455</span></a><span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="Equation-456"><a href="#Equation-456"><span class="linenos">456</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="Equation-457"><a href="#Equation-457"><span class="linenos">457</span></a><span class="sd">        The operation is applied to only the lhs.</span>
+</span><span id="Equation-458"><a href="#Equation-458"><span class="linenos">458</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-459"><a href="#Equation-459"><span class="linenos">459</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
+</span><span id="Equation-460"><a href="#Equation-460"><span class="linenos">460</span></a>
+</span><span id="Equation-461"><a href="#Equation-461"><span class="linenos">461</span></a>    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-462"><a href="#Equation-462"><span class="linenos">462</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-463"><a href="#Equation-463"><span class="linenos">463</span></a><span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="Equation-464"><a href="#Equation-464"><span class="linenos">464</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="Equation-465"><a href="#Equation-465"><span class="linenos">465</span></a><span class="sd">        The operation is applied to only the rhs.</span>
+</span><span id="Equation-466"><a href="#Equation-466"><span class="linenos">466</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-467"><a href="#Equation-467"><span class="linenos">467</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
+</span><span id="Equation-468"><a href="#Equation-468"><span class="linenos">468</span></a>
+</span><span id="Equation-469"><a href="#Equation-469"><span class="linenos">469</span></a>    <span class="k">class</span> <span class="nc">_sides</span><span class="p">:</span>
+</span><span id="Equation-470"><a href="#Equation-470"><span class="linenos">470</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation-471"><a href="#Equation-471"><span class="linenos">471</span></a><span class="sd">        Helper class for the `.do.`, `.dolhs.`, `.dorhs.` syntax for applying</span>
+</span><span id="Equation-472"><a href="#Equation-472"><span class="linenos">472</span></a><span class="sd">        submethods of expressions.</span>
+</span><span id="Equation-473"><a href="#Equation-473"><span class="linenos">473</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation-474"><a href="#Equation-474"><span class="linenos">474</span></a>
+</span><span id="Equation-475"><a href="#Equation-475"><span class="linenos">475</span></a>        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eqn</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">):</span>
+</span><span id="Equation-476"><a href="#Equation-476"><span class="linenos">476</span></a>            <span class="bp">self</span><span class="o">.</span><span class="n">eqn</span> <span class="o">=</span> <span class="n">eqn</span>
+</span><span id="Equation-477"><a href="#Equation-477"><span class="linenos">477</span></a>            <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="o">=</span> <span class="n">side</span>
+</span><span id="Equation-478"><a href="#Equation-478"><span class="linenos">478</span></a>
+</span><span id="Equation-479"><a href="#Equation-479"><span class="linenos">479</span></a>        <span class="k">def</span> <span class="fm">__getattr__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
+</span><span id="Equation-480"><a href="#Equation-480"><span class="linenos">480</span></a>            <span class="n">func</span> <span class="o">=</span> <span class="kc">None</span>
+</span><span id="Equation-481"><a href="#Equation-481"><span class="linenos">481</span></a>            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;rhs&#39;</span><span class="p">,</span> <span class="s1">&#39;both&#39;</span><span class="p">):</span>
+</span><span id="Equation-482"><a href="#Equation-482"><span class="linenos">482</span></a>                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Equation-483"><a href="#Equation-483"><span class="linenos">483</span></a>            <span class="k">else</span><span class="p">:</span>
+</span><span id="Equation-484"><a href="#Equation-484"><span class="linenos">484</span></a>                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Equation-485"><a href="#Equation-485"><span class="linenos">485</span></a>            <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="Equation-486"><a href="#Equation-486"><span class="linenos">486</span></a>                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;Expressions in the equation have no &#39;</span>
+</span><span id="Equation-487"><a href="#Equation-487"><span class="linenos">487</span></a>                                     <span class="s1">&#39;attribute `&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span>
+</span><span id="Equation-488"><a href="#Equation-488"><span class="linenos">488</span></a>                    <span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;`. Try `.apply(&#39;</span>
+</span><span id="Equation-489"><a href="#Equation-489"><span class="linenos">489</span></a>                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;, *args)` or &#39;</span>
+</span><span id="Equation-490"><a href="#Equation-490"><span class="linenos">490</span></a>                                                   <span class="s1">&#39;pass the equation as a parameter to `&#39;</span>
+</span><span id="Equation-491"><a href="#Equation-491"><span class="linenos">491</span></a>                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;()`.&#39;</span><span class="p">)</span>
+</span><span id="Equation-492"><a href="#Equation-492"><span class="linenos">492</span></a>            <span class="k">return</span> <span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">apply</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">side</span><span class="p">)</span>
+</span><span id="Equation-493"><a href="#Equation-493"><span class="linenos">493</span></a>
+</span><span id="Equation-494"><a href="#Equation-494"><span class="linenos">494</span></a>    <span class="nd">@property</span>
+</span><span id="Equation-495"><a href="#Equation-495"><span class="linenos">495</span></a>    <span class="k">def</span> <span class="nf">do</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-496"><a href="#Equation-496"><span class="linenos">496</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">)</span>
+</span><span id="Equation-497"><a href="#Equation-497"><span class="linenos">497</span></a>
+</span><span id="Equation-498"><a href="#Equation-498"><span class="linenos">498</span></a>    <span class="nd">@property</span>
+</span><span id="Equation-499"><a href="#Equation-499"><span class="linenos">499</span></a>    <span class="k">def</span> <span class="nf">dolhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-500"><a href="#Equation-500"><span class="linenos">500</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
+</span><span id="Equation-501"><a href="#Equation-501"><span class="linenos">501</span></a>
+</span><span id="Equation-502"><a href="#Equation-502"><span class="linenos">502</span></a>    <span class="nd">@property</span>
+</span><span id="Equation-503"><a href="#Equation-503"><span class="linenos">503</span></a>    <span class="k">def</span> <span class="nf">dorhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-504"><a href="#Equation-504"><span class="linenos">504</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
+</span><span id="Equation-505"><a href="#Equation-505"><span class="linenos">505</span></a>
+</span><span id="Equation-506"><a href="#Equation-506"><span class="linenos">506</span></a>    <span class="c1">#####</span>
+</span><span id="Equation-507"><a href="#Equation-507"><span class="linenos">507</span></a>    <span class="c1"># Overrides of binary math operations</span>
+</span><span id="Equation-508"><a href="#Equation-508"><span class="linenos">508</span></a>    <span class="c1">#####</span>
+</span><span id="Equation-509"><a href="#Equation-509"><span class="linenos">509</span></a>
+</span><span id="Equation-510"><a href="#Equation-510"><span class="linenos">510</span></a>    <span class="nd">@classmethod</span>
+</span><span id="Equation-511"><a href="#Equation-511"><span class="linenos">511</span></a>    <span class="k">def</span> <span class="nf">_binary_op</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">opfunc_ab</span><span class="p">):</span>
+</span><span id="Equation-512"><a href="#Equation-512"><span class="linenos">512</span></a>        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="Equation-513"><a href="#Equation-513"><span class="linenos">513</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
+</span><span id="Equation-514"><a href="#Equation-514"><span class="linenos">514</span></a>        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="Equation-515"><a href="#Equation-515"><span class="linenos">515</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
+</span><span id="Equation-516"><a href="#Equation-516"><span class="linenos">516</span></a>        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="Equation-517"><a href="#Equation-517"><span class="linenos">517</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
+</span><span id="Equation-518"><a href="#Equation-518"><span class="linenos">518</span></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="Equation-519"><a href="#Equation-519"><span class="linenos">519</span></a>            <span class="k">return</span> <span class="bp">NotImplemented</span>
+</span><span id="Equation-520"><a href="#Equation-520"><span class="linenos">520</span></a>
+</span><span id="Equation-521"><a href="#Equation-521"><span class="linenos">521</span></a>    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-522"><a href="#Equation-522"><span class="linenos">522</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-523"><a href="#Equation-523"><span class="linenos">523</span></a>
+</span><span id="Equation-524"><a href="#Equation-524"><span class="linenos">524</span></a>    <span class="k">def</span> <span class="fm">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-525"><a href="#Equation-525"><span class="linenos">525</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-526"><a href="#Equation-526"><span class="linenos">526</span></a>
+</span><span id="Equation-527"><a href="#Equation-527"><span class="linenos">527</span></a>    <span class="k">def</span> <span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-528"><a href="#Equation-528"><span class="linenos">528</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-529"><a href="#Equation-529"><span class="linenos">529</span></a>
+</span><span id="Equation-530"><a href="#Equation-530"><span class="linenos">530</span></a>    <span class="k">def</span> <span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-531"><a href="#Equation-531"><span class="linenos">531</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-532"><a href="#Equation-532"><span class="linenos">532</span></a>
+</span><span id="Equation-533"><a href="#Equation-533"><span class="linenos">533</span></a>    <span class="k">def</span> <span class="fm">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-534"><a href="#Equation-534"><span class="linenos">534</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-535"><a href="#Equation-535"><span class="linenos">535</span></a>
+</span><span id="Equation-536"><a href="#Equation-536"><span class="linenos">536</span></a>    <span class="k">def</span> <span class="fm">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-537"><a href="#Equation-537"><span class="linenos">537</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-538"><a href="#Equation-538"><span class="linenos">538</span></a>
+</span><span id="Equation-539"><a href="#Equation-539"><span class="linenos">539</span></a>    <span class="k">def</span> <span class="fm">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-540"><a href="#Equation-540"><span class="linenos">540</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-541"><a href="#Equation-541"><span class="linenos">541</span></a>
+</span><span id="Equation-542"><a href="#Equation-542"><span class="linenos">542</span></a>    <span class="k">def</span> <span class="fm">__rtruediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-543"><a href="#Equation-543"><span class="linenos">543</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-544"><a href="#Equation-544"><span class="linenos">544</span></a>
+</span><span id="Equation-545"><a href="#Equation-545"><span class="linenos">545</span></a>    <span class="k">def</span> <span class="fm">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-546"><a href="#Equation-546"><span class="linenos">546</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-547"><a href="#Equation-547"><span class="linenos">547</span></a>
+</span><span id="Equation-548"><a href="#Equation-548"><span class="linenos">548</span></a>    <span class="k">def</span> <span class="fm">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-549"><a href="#Equation-549"><span class="linenos">549</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-550"><a href="#Equation-550"><span class="linenos">550</span></a>
+</span><span id="Equation-551"><a href="#Equation-551"><span class="linenos">551</span></a>    <span class="k">def</span> <span class="fm">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-552"><a href="#Equation-552"><span class="linenos">552</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-553"><a href="#Equation-553"><span class="linenos">553</span></a>
+</span><span id="Equation-554"><a href="#Equation-554"><span class="linenos">554</span></a>    <span class="k">def</span> <span class="fm">__rpow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-555"><a href="#Equation-555"><span class="linenos">555</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Equation-556"><a href="#Equation-556"><span class="linenos">556</span></a>
+</span><span id="Equation-557"><a href="#Equation-557"><span class="linenos">557</span></a>    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Equation-558"><a href="#Equation-558"><span class="linenos">558</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__pow__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
+</span><span id="Equation-559"><a href="#Equation-559"><span class="linenos">559</span></a>
+</span><span id="Equation-560"><a href="#Equation-560"><span class="linenos">560</span></a>    <span class="c1">#####</span>
+</span><span id="Equation-561"><a href="#Equation-561"><span class="linenos">561</span></a>    <span class="c1"># Operation helper functions</span>
+</span><span id="Equation-562"><a href="#Equation-562"><span class="linenos">562</span></a>    <span class="c1">#####</span>
+</span><span id="Equation-563"><a href="#Equation-563"><span class="linenos">563</span></a>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-564"><a href="#Equation-564"><span class="linenos">564</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
+</span><span id="Equation-565"><a href="#Equation-565"><span class="linenos">565</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Equation-566"><a href="#Equation-566"><span class="linenos">566</span></a>
+</span><span id="Equation-567"><a href="#Equation-567"><span class="linenos">567</span></a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-568"><a href="#Equation-568"><span class="linenos">568</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-569"><a href="#Equation-569"><span class="linenos">569</span></a>
+</span><span id="Equation-570"><a href="#Equation-570"><span class="linenos">570</span></a>    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-571"><a href="#Equation-571"><span class="linenos">571</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span>
+</span><span id="Equation-572"><a href="#Equation-572"><span class="linenos">572</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Equation-573"><a href="#Equation-573"><span class="linenos">573</span></a>
+</span><span id="Equation-574"><a href="#Equation-574"><span class="linenos">574</span></a>    <span class="k">def</span> <span class="nf">_eval_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-575"><a href="#Equation-575"><span class="linenos">575</span></a>        <span class="c1"># TODO: cancel out factors common to both sides.</span>
+</span><span id="Equation-576"><a href="#Equation-576"><span class="linenos">576</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span>
+</span><span id="Equation-577"><a href="#Equation-577"><span class="linenos">577</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Equation-578"><a href="#Equation-578"><span class="linenos">578</span></a>
+</span><span id="Equation-579"><a href="#Equation-579"><span class="linenos">579</span></a>    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-580"><a href="#Equation-580"><span class="linenos">580</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-581"><a href="#Equation-581"><span class="linenos">581</span></a>
+</span><span id="Equation-582"><a href="#Equation-582"><span class="linenos">582</span></a>    <span class="k">def</span> <span class="nf">_eval_collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-583"><a href="#Equation-583"><span class="linenos">583</span></a>        <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
+</span><span id="Equation-584"><a href="#Equation-584"><span class="linenos">584</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="Equation-585"><a href="#Equation-585"><span class="linenos">585</span></a>                        <span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Equation-586"><a href="#Equation-586"><span class="linenos">586</span></a>
+</span><span id="Equation-587"><a href="#Equation-587"><span class="linenos">587</span></a>    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-588"><a href="#Equation-588"><span class="linenos">588</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-589"><a href="#Equation-589"><span class="linenos">589</span></a>
+</span><span id="Equation-590"><a href="#Equation-590"><span class="linenos">590</span></a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-591"><a href="#Equation-591"><span class="linenos">591</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="Equation-592"><a href="#Equation-592"><span class="linenos">592</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Equation-593"><a href="#Equation-593"><span class="linenos">593</span></a>
+</span><span id="Equation-594"><a href="#Equation-594"><span class="linenos">594</span></a>    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
+</span><span id="Equation-595"><a href="#Equation-595"><span class="linenos">595</span></a>
+</span><span id="Equation-596"><a href="#Equation-596"><span class="linenos">596</span></a>    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-597"><a href="#Equation-597"><span class="linenos">597</span></a>        <span class="c1"># TODO Find why diff and Derivative do not appear to pass through</span>
+</span><span id="Equation-598"><a href="#Equation-598"><span class="linenos">598</span></a>        <span class="c1">#  kwargs to this. Since we cannot set evaluation of lhs manually</span>
+</span><span id="Equation-599"><a href="#Equation-599"><span class="linenos">599</span></a>        <span class="c1">#  try to be intelligent about when to do it.</span>
+</span><span id="Equation-600"><a href="#Equation-600"><span class="linenos">600</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Derivative</span>
+</span><span id="Equation-601"><a href="#Equation-601"><span class="linenos">601</span></a>        <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">False</span>
+</span><span id="Equation-602"><a href="#Equation-602"><span class="linenos">602</span></a>        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)):</span>
+</span><span id="Equation-603"><a href="#Equation-603"><span class="linenos">603</span></a>            <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
+</span><span id="Equation-604"><a href="#Equation-604"><span class="linenos">604</span></a>                <span class="k">if</span> <span class="n">sym</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span>
+</span><span id="Equation-605"><a href="#Equation-605"><span class="linenos">605</span></a>                        <span class="n">_sympify</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">):</span>
+</span><span id="Equation-606"><a href="#Equation-606"><span class="linenos">606</span></a>                    <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="Equation-607"><a href="#Equation-607"><span class="linenos">607</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="n">eval_lhs</span><span class="p">),</span>
+</span><span id="Equation-608"><a href="#Equation-608"><span class="linenos">608</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Equation-609"><a href="#Equation-609"><span class="linenos">609</span></a>
+</span><span id="Equation-610"><a href="#Equation-610"><span class="linenos">610</span></a>    <span class="k">def</span> <span class="nf">_eval_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-611"><a href="#Equation-611"><span class="linenos">611</span></a>        <span class="n">side</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s1">&#39;side&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>  <span class="c1"># Could not seem to pass values for</span>
+</span><span id="Equation-612"><a href="#Equation-612"><span class="linenos">612</span></a>        <span class="c1"># `evaluate` through to here.</span>
+</span><span id="Equation-613"><a href="#Equation-613"><span class="linenos">613</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="Equation-614"><a href="#Equation-614"><span class="linenos">614</span></a>            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;You must specify `side=&quot;lhs&quot;` or `side=&quot;rhs&quot;` &#39;</span>
+</span><span id="Equation-615"><a href="#Equation-615"><span class="linenos">615</span></a>                             <span class="s1">&#39;when integrating an Equation&#39;</span><span class="p">)</span>
+</span><span id="Equation-616"><a href="#Equation-616"><span class="linenos">616</span></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="Equation-617"><a href="#Equation-617"><span class="linenos">617</span></a>            <span class="k">try</span><span class="p">:</span>
+</span><span id="Equation-618"><a href="#Equation-618"><span class="linenos">618</span></a>                <span class="k">return</span> <span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Equation-619"><a href="#Equation-619"><span class="linenos">619</span></a>            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
+</span><span id="Equation-620"><a href="#Equation-620"><span class="linenos">620</span></a>                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;`side` must equal &quot;lhs&quot; or &quot;rhs&quot;.&#39;</span><span class="p">)</span>
+</span><span id="Equation-621"><a href="#Equation-621"><span class="linenos">621</span></a>
+</span><span id="Equation-622"><a href="#Equation-622"><span class="linenos">622</span></a>    <span class="c1">#####</span>
+</span><span id="Equation-623"><a href="#Equation-623"><span class="linenos">623</span></a>    <span class="c1"># Output helper functions</span>
+</span><span id="Equation-624"><a href="#Equation-624"><span class="linenos">624</span></a>    <span class="c1">#####</span>
+</span><span id="Equation-625"><a href="#Equation-625"><span class="linenos">625</span></a>    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-626"><a href="#Equation-626"><span class="linenos">626</span></a>        <span class="n">repstr</span> <span class="o">=</span> <span class="s1">&#39;Equation(</span><span class="si">%s</span><span class="s1">, </span><span class="si">%s</span><span class="s1">)&#39;</span> <span class="o">%</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">())</span>
+</span><span id="Equation-627"><a href="#Equation-627"><span class="linenos">627</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
+</span><span id="Equation-628"><a href="#Equation-628"><span class="linenos">628</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__str__</span><span class="p">()</span>
+</span><span id="Equation-629"><a href="#Equation-629"><span class="linenos">629</span></a>        <span class="k">return</span> <span class="n">repstr</span>
+</span><span id="Equation-630"><a href="#Equation-630"><span class="linenos">630</span></a>
+</span><span id="Equation-631"><a href="#Equation-631"><span class="linenos">631</span></a>    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation-632"><a href="#Equation-632"><span class="linenos">632</span></a>        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
+</span><span id="Equation-633"><a href="#Equation-633"><span class="linenos">633</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span> <span class="ow">and</span> <span class="ow">not</span> \
+</span><span id="Equation-634"><a href="#Equation-634"><span class="linenos">634</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
+</span><span id="Equation-635"><a href="#Equation-635"><span class="linenos">635</span></a>            <span class="n">tempstr</span> <span class="o">+=</span><span class="s1">&#39;</span><span class="se">\\</span><span class="s1">text{code version: &#39;</span><span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">+</span><span class="s1">&#39;} </span><span class="se">\\</span><span class="s1">newline &#39;</span>
+</span><span id="Equation-636"><a href="#Equation-636"><span class="linenos">636</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-637"><a href="#Equation-637"><span class="linenos">637</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;=&#39;</span>
+</span><span id="Equation-638"><a href="#Equation-638"><span class="linenos">638</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation-639"><a href="#Equation-639"><span class="linenos">639</span></a>        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
+</span><span id="Equation-640"><a href="#Equation-640"><span class="linenos">640</span></a>        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span><span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
+</span><span id="Equation-641"><a href="#Equation-641"><span class="linenos">641</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,&#39;</span>
+</span><span id="Equation-642"><a href="#Equation-642"><span class="linenos">642</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;(</span><span class="se">\\</span><span class="s1">text{&#39;</span><span class="o">+</span><span class="n">namestr</span><span class="o">+</span><span class="s1">&#39;})&#39;</span>
+</span><span id="Equation-643"><a href="#Equation-643"><span class="linenos">643</span></a>        <span class="k">return</span> <span class="n">tempstr</span>
+</span><span id="Equation-644"><a href="#Equation-644"><span class="linenos">644</span></a>
+</span><span id="Equation-645"><a href="#Equation-645"><span class="linenos">645</span></a>    <span class="k">def</span> <span class="fm">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation-646"><a href="#Equation-646"><span class="linenos">646</span></a>        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
+</span><span id="Equation-647"><a href="#Equation-647"><span class="linenos">647</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span><span class="p">:</span>
+</span><span id="Equation-648"><a href="#Equation-648"><span class="linenos">648</span></a>            <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="Equation-649"><a href="#Equation-649"><span class="linenos">649</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
+</span><span id="Equation-650"><a href="#Equation-650"><span class="linenos">650</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;code version: &#39;</span><span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="o">+</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
+</span><span id="Equation-651"><a href="#Equation-651"><span class="linenos">651</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
+</span><span id="Equation-652"><a href="#Equation-652"><span class="linenos">652</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39; = &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="Equation-653"><a href="#Equation-653"><span class="linenos">653</span></a>        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
+</span><span id="Equation-654"><a href="#Equation-654"><span class="linenos">654</span></a>        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span> <span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
+</span><span id="Equation-655"><a href="#Equation-655"><span class="linenos">655</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;          (&#39;</span> <span class="o">+</span> <span class="n">namestr</span> <span class="o">+</span> <span class="s1">&#39;)&#39;</span>
+</span><span id="Equation-656"><a href="#Equation-656"><span class="linenos">656</span></a>        <span class="k">return</span> <span class="n">tempstr</span>
+</span></pre></div>
+
 
             <div class="docstring"><p>This class defines an equation with a left-hand-side (lhs) and a right-
 hand-side (rhs) connected by the "=" operator (e.g. <code>p*V = n*R*T</code>).</p>
@@ -4174,22 +4226,24 @@ <h1 id="examples">Examples</h1>
 
 
                             <div id="Equation.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Equation</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Equation</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Equation.__init__"></a>
     
     
 
                             </div>
                             <div id="Equation.lhs" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Equation.lhs">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">lhs</span>
 
-        <span class="name">lhs</span>
+        
     </div>
-
+    <a class="headerlink" href="#Equation.lhs"></a>
     
             <div class="docstring"><p>Returns the lhs of the equation.</p>
 </div>
@@ -4197,11 +4251,12 @@ <h1 id="examples">Examples</h1>
 
                             </div>
                             <div id="Equation.rhs" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Equation.rhs">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">rhs</span>
 
-        <span class="name">rhs</span>
+        
     </div>
-
+    <a class="headerlink" href="#Equation.rhs"></a>
     
             <div class="docstring"><p>Returns the rhs of the equation.</p>
 </div>
@@ -4209,24 +4264,24 @@ <h1 id="examples">Examples</h1>
 
                             </div>
                             <div id="Equation.as_Boolean" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.as_Boolean">#&nbsp;&nbsp</a>
+                                        <input id="Equation.as_Boolean-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">as_Boolean</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">as_Boolean</span><span class="signature">(self)</span>:
-    </div>
+                <label class="view-source-button" for="Equation.as_Boolean-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Converts the equation to an Equality.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Equation.as_Boolean"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.as_Boolean-379"><a href="#Equation.as_Boolean-379"><span class="linenos">379</span></a>    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Equation.as_Boolean-380"><a href="#Equation.as_Boolean-380"><span class="linenos">380</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation.as_Boolean-381"><a href="#Equation.as_Boolean-381"><span class="linenos">381</span></a><span class="sd">        Converts the equation to an Equality.</span>
+</span><span id="Equation.as_Boolean-382"><a href="#Equation.as_Boolean-382"><span class="linenos">382</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation.as_Boolean-383"><a href="#Equation.as_Boolean-383"><span class="linenos">383</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="Equation.as_Boolean-384"><a href="#Equation.as_Boolean-384"><span class="linenos">384</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>Converts the equation to an Equality.</p>
 </div>
@@ -4234,31 +4289,31 @@ <h1 id="examples">Examples</h1>
 
                             </div>
                             <div id="Equation.check" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.check">#&nbsp;&nbsp</a>
+                                        <input id="Equation.check-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">check</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
+
+                <label class="view-source-button" for="Equation.check-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Equation.check"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.check-386"><a href="#Equation.check-386"><span class="linenos">386</span></a>    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.check-387"><a href="#Equation.check-387"><span class="linenos">387</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation.check-388"><a href="#Equation.check-388"><span class="linenos">388</span></a><span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
+</span><span id="Equation.check-389"><a href="#Equation.check-389"><span class="linenos">389</span></a><span class="sd">        Parameters</span>
+</span><span id="Equation.check-390"><a href="#Equation.check-390"><span class="linenos">390</span></a><span class="sd">        ----------</span>
+</span><span id="Equation.check-391"><a href="#Equation.check-391"><span class="linenos">391</span></a><span class="sd">        kwargs any appropriate for `Equality`.</span>
+</span><span id="Equation.check-392"><a href="#Equation.check-392"><span class="linenos">392</span></a>
+</span><span id="Equation.check-393"><a href="#Equation.check-393"><span class="linenos">393</span></a><span class="sd">        Returns</span>
+</span><span id="Equation.check-394"><a href="#Equation.check-394"><span class="linenos">394</span></a><span class="sd">        -------</span>
+</span><span id="Equation.check-395"><a href="#Equation.check-395"><span class="linenos">395</span></a><span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
+</span><span id="Equation.check-396"><a href="#Equation.check-396"><span class="linenos">396</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation.check-397"><a href="#Equation.check-397"><span class="linenos">397</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="Equation.check-398"><a href="#Equation.check-398"><span class="linenos">398</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
+</span></pre></div>
 
-        
-            <span class="def">def</span>
-            <span class="name">check</span><span class="signature">(self, **kwargs)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
-<span class="sd">        Parameters</span>
-<span class="sd">        ----------</span>
-<span class="sd">        kwargs any appropriate for `Equality`.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        -------</span>
-<span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-</pre></div>
-
-        </details>
 
             <div class="docstring"><p>Forces simplification and casts as <code>Equality</code> to check validity.</p>
 
@@ -4274,11 +4329,12 @@ <h2 id="returns">Returns</h2>
 
                             </div>
                             <div id="Equation.reversed" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Equation.reversed">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">reversed</span>
 
-        <span class="name">reversed</span>
+        
     </div>
-
+    <a class="headerlink" href="#Equation.reversed"></a>
     
             <div class="docstring"><p>Swaps the lhs and the rhs.</p>
 </div>
@@ -4286,11 +4342,12 @@ <h2 id="returns">Returns</h2>
 
                             </div>
                             <div id="Equation.swap" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Equation.swap">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">swap</span>
 
-        <span class="name">swap</span>
+        
     </div>
-
+    <a class="headerlink" href="#Equation.swap"></a>
     
             <div class="docstring"><p>Synonym for <code>.reversed</code></p>
 </div>
@@ -4298,44 +4355,44 @@ <h2 id="returns">Returns</h2>
 
                             </div>
                             <div id="Equation.apply" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.apply">#&nbsp;&nbsp</a>
-
-        
-            <span class="def">def</span>
-            <span class="name">apply</span><span class="signature">(self, func, *args, side=&#39;both&#39;, **kwargs)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply an operation/function/method to the equation returning the</span>
-<span class="sd">        resulting equation.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        func: object</span>
-<span class="sd">            object to apply usually a function</span>
-
-<span class="sd">        args: as necessary for the function</span>
-
-<span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
-<span class="sd">            Specifies which side of the equation the operation will be applied</span>
-<span class="sd">            to. Default is &#39;both&#39;.</span>
+                                        <input id="Equation.apply-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">apply</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="n">func</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
+
+                <label class="view-source-button" for="Equation.apply-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Equation.apply"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.apply-426"><a href="#Equation.apply-426"><span class="linenos">426</span></a>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.apply-427"><a href="#Equation.apply-427"><span class="linenos">427</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation.apply-428"><a href="#Equation.apply-428"><span class="linenos">428</span></a><span class="sd">        Apply an operation/function/method to the equation returning the</span>
+</span><span id="Equation.apply-429"><a href="#Equation.apply-429"><span class="linenos">429</span></a><span class="sd">        resulting equation.</span>
+</span><span id="Equation.apply-430"><a href="#Equation.apply-430"><span class="linenos">430</span></a>
+</span><span id="Equation.apply-431"><a href="#Equation.apply-431"><span class="linenos">431</span></a><span class="sd">        Parameters</span>
+</span><span id="Equation.apply-432"><a href="#Equation.apply-432"><span class="linenos">432</span></a><span class="sd">        ==========</span>
+</span><span id="Equation.apply-433"><a href="#Equation.apply-433"><span class="linenos">433</span></a>
+</span><span id="Equation.apply-434"><a href="#Equation.apply-434"><span class="linenos">434</span></a><span class="sd">        func: object</span>
+</span><span id="Equation.apply-435"><a href="#Equation.apply-435"><span class="linenos">435</span></a><span class="sd">            object to apply usually a function</span>
+</span><span id="Equation.apply-436"><a href="#Equation.apply-436"><span class="linenos">436</span></a>
+</span><span id="Equation.apply-437"><a href="#Equation.apply-437"><span class="linenos">437</span></a><span class="sd">        args: as necessary for the function</span>
+</span><span id="Equation.apply-438"><a href="#Equation.apply-438"><span class="linenos">438</span></a>
+</span><span id="Equation.apply-439"><a href="#Equation.apply-439"><span class="linenos">439</span></a><span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
+</span><span id="Equation.apply-440"><a href="#Equation.apply-440"><span class="linenos">440</span></a><span class="sd">            Specifies which side of the equation the operation will be applied</span>
+</span><span id="Equation.apply-441"><a href="#Equation.apply-441"><span class="linenos">441</span></a><span class="sd">            to. Default is &#39;both&#39;.</span>
+</span><span id="Equation.apply-442"><a href="#Equation.apply-442"><span class="linenos">442</span></a>
+</span><span id="Equation.apply-443"><a href="#Equation.apply-443"><span class="linenos">443</span></a><span class="sd">        kwargs: as necessary for the function</span>
+</span><span id="Equation.apply-444"><a href="#Equation.apply-444"><span class="linenos">444</span></a><span class="sd">         &quot;&quot;&quot;</span>
+</span><span id="Equation.apply-445"><a href="#Equation.apply-445"><span class="linenos">445</span></a>        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
+</span><span id="Equation.apply-446"><a href="#Equation.apply-446"><span class="linenos">446</span></a>        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
+</span><span id="Equation.apply-447"><a href="#Equation.apply-447"><span class="linenos">447</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
+</span><span id="Equation.apply-448"><a href="#Equation.apply-448"><span class="linenos">448</span></a>            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation.apply-449"><a href="#Equation.apply-449"><span class="linenos">449</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
+</span><span id="Equation.apply-450"><a href="#Equation.apply-450"><span class="linenos">450</span></a>            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Equation.apply-451"><a href="#Equation.apply-451"><span class="linenos">451</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
+</span></pre></div>
 
-<span class="sd">        kwargs: as necessary for the function</span>
-<span class="sd">         &quot;&quot;&quot;</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-</pre></div>
-
-        </details>
 
             <div class="docstring"><p>Apply an operation/function/method to the equation returning the
 resulting equation.</p>
@@ -4357,25 +4414,25 @@ <h1 id="parameters">Parameters</h1>
 
                             </div>
                             <div id="Equation.applylhs" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.applylhs">#&nbsp;&nbsp</a>
+                                        <input id="Equation.applylhs-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">applylhs</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="n">func</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">applylhs</span><span class="signature">(self, func, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Equation.applylhs-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the lhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Equation.applylhs"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.applylhs-453"><a href="#Equation.applylhs-453"><span class="linenos">453</span></a>    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.applylhs-454"><a href="#Equation.applylhs-454"><span class="linenos">454</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation.applylhs-455"><a href="#Equation.applylhs-455"><span class="linenos">455</span></a><span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="Equation.applylhs-456"><a href="#Equation.applylhs-456"><span class="linenos">456</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="Equation.applylhs-457"><a href="#Equation.applylhs-457"><span class="linenos">457</span></a><span class="sd">        The operation is applied to only the lhs.</span>
+</span><span id="Equation.applylhs-458"><a href="#Equation.applylhs-458"><span class="linenos">458</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation.applylhs-459"><a href="#Equation.applylhs-459"><span class="linenos">459</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>If lhs side of the equation has a defined subfunction (attribute) of
 name <code><a href="#Equation.func">func</a></code>, that will be applied instead of the global function.
@@ -4385,25 +4442,25 @@ <h1 id="parameters">Parameters</h1>
 
                             </div>
                             <div id="Equation.applyrhs" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.applyrhs">#&nbsp;&nbsp</a>
+                                        <input id="Equation.applyrhs-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">applyrhs</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="n">func</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">applyrhs</span><span class="signature">(self, func, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Equation.applyrhs-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the rhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Equation.applyrhs"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.applyrhs-461"><a href="#Equation.applyrhs-461"><span class="linenos">461</span></a>    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.applyrhs-462"><a href="#Equation.applyrhs-462"><span class="linenos">462</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Equation.applyrhs-463"><a href="#Equation.applyrhs-463"><span class="linenos">463</span></a><span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="Equation.applyrhs-464"><a href="#Equation.applyrhs-464"><span class="linenos">464</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="Equation.applyrhs-465"><a href="#Equation.applyrhs-465"><span class="linenos">465</span></a><span class="sd">        The operation is applied to only the rhs.</span>
+</span><span id="Equation.applyrhs-466"><a href="#Equation.applyrhs-466"><span class="linenos">466</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Equation.applyrhs-467"><a href="#Equation.applyrhs-467"><span class="linenos">467</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>If rhs side of the equation has a defined subfunction (attribute) of
 name <code><a href="#Equation.func">func</a></code>, that will be applied instead of the global function.
@@ -4413,70 +4470,73 @@ <h1 id="parameters">Parameters</h1>
 
                             </div>
                             <div id="Equation.do" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Equation.do">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">do</span>
 
-        <span class="name">do</span>
+        
     </div>
-
+    <a class="headerlink" href="#Equation.do"></a>
     
     
 
                             </div>
                             <div id="Equation.dolhs" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Equation.dolhs">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">dolhs</span>
 
-        <span class="name">dolhs</span>
+        
     </div>
-
+    <a class="headerlink" href="#Equation.dolhs"></a>
     
     
 
                             </div>
                             <div id="Equation.dorhs" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Equation.dorhs">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">dorhs</span>
 
-        <span class="name">dorhs</span>
+        
     </div>
-
+    <a class="headerlink" href="#Equation.dorhs"></a>
     
     
 
                             </div>
                             <div id="Equation.expand" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.expand">#&nbsp;&nbsp</a>
+                                        <input id="Equation.expand-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">expand</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">expand</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Equation.expand-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Equation.expand"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.expand-563"><a href="#Equation.expand-563"><span class="linenos">563</span></a>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.expand-564"><a href="#Equation.expand-564"><span class="linenos">564</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
+</span><span id="Equation.expand-565"><a href="#Equation.expand-565"><span class="linenos">565</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span></pre></div>
 
-        </details>
 
     
 
                             </div>
                             <div id="Equation.simplify" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.simplify">#&nbsp;&nbsp</a>
+                                        <input id="Equation.simplify-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">simplify</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">simplify</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Equation.simplify-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Equation.simplify"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.simplify-567"><a href="#Equation.simplify-567"><span class="linenos">567</span></a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.simplify-568"><a href="#Equation.simplify-568"><span class="linenos">568</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>See the simplify function in sympy.simplify</p>
 </div>
@@ -4484,143 +4544,143 @@ <h1 id="parameters">Parameters</h1>
 
                             </div>
                             <div id="Equation.factor" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.factor">#&nbsp;&nbsp</a>
+                                        <input id="Equation.factor-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">factor</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">factor</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Equation.factor-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Equation.factor"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.factor-579"><a href="#Equation.factor-579"><span class="linenos">579</span></a>    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.factor-580"><a href="#Equation.factor-580"><span class="linenos">580</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
     
 
                             </div>
                             <div id="Equation.collect" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.collect">#&nbsp;&nbsp</a>
+                                        <input id="Equation.collect-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">collect</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">collect</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Equation.collect-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Equation.collect"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.collect-587"><a href="#Equation.collect-587"><span class="linenos">587</span></a>    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.collect-588"><a href="#Equation.collect-588"><span class="linenos">588</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
     
 
                             </div>
                             <div id="Equation.evalf" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.evalf">#&nbsp;&nbsp</a>
+                                        <input id="Equation.evalf-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">evalf</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
+
+                <label class="view-source-button" for="Equation.evalf-view-source"><span>View Source</span></label>
 
-        
-            <span class="def">def</span>
-            <span class="name">evalf</span><span class="signature">(self, *args, **kwargs)</span>:
     </div>
+    <a class="headerlink" href="#Equation.evalf"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.evalf-590"><a href="#Equation.evalf-590"><span class="linenos">590</span></a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.evalf-591"><a href="#Equation.evalf-591"><span class="linenos">591</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="Equation.evalf-592"><a href="#Equation.evalf-592"><span class="linenos">592</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span></pre></div>
+
+
+            <div class="docstring"><p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>
+
+<h1 id="parameters">Parameters</h1>
+
+<p>subs : dict, optional
+    Substitute numerical values for symbols, e.g.
+    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a
+    dictionary.</p>
+
+<p>maxn : int, optional
+    Allow a maximum temporary working precision of maxn digits.</p>
+
+<p>chop : bool or number, optional
+    Specifies how to replace tiny real or imaginary parts in
+    subresults by exact zeros.</p>
+
+<pre><code>When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+&gt;&gt;&gt; from sympy import N
+&gt;&gt;&gt; x = 1e-4
+&gt;&gt;&gt; N(x, chop=True)
+0.000100000000000000
+&gt;&gt;&gt; N(x, chop=1e-5)
+0.000100000000000000
+&gt;&gt;&gt; N(x, chop=1e-4)
+0
+</code></pre>
+
+<p>strict : bool, optional
+    Raise <code>PrecisionExhausted</code> if any subresult fails to
+    evaluate to full accuracy, given the available maxprec.</p>
+
+<p>quad : str, optional
+    Choose algorithm for numerical quadrature. By default,
+    tanh-sinh quadrature is used. For oscillatory
+    integrals on an infinite interval, try <code>quad='osc'</code>.</p>
+
+<p>verbose : bool, optional
+    Print debug information.</p>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-</pre></div>
+<h1 id="notes">Notes</h1>
+
+<p>When Floats are naively substituted into an expression,
+precision errors may adversely affect the result. For example,
+adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is
+then subtracted, the result will be 0.
+That is exactly what happens in the following:</p>
+
+<div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">values</span> <span class="o">=</span> <span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="mf">1e16</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">:</span> <span class="mf">1e16</span><span class="p">}</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">values</span><span class="p">)</span>
+<span class="go">0</span>
+</code></pre></div>
+
+<p>Using the subs argument for evalf is the accurate way to
+evaluate such an expression:</p>
+
+<div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">subs</span><span class="o">=</span><span class="n">values</span><span class="p">)</span>
+<span class="go">1.00000000000000</span>
+</code></pre></div>
+</div>
+
+
+                            </div>
+                            <div id="Equation.n" class="classattr">
+                                        <input id="Equation.n-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">n</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
+
+                <label class="view-source-button" for="Equation.n-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Equation.n"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Equation.n-590"><a href="#Equation.n-590"><span class="linenos">590</span></a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Equation.n-591"><a href="#Equation.n-591"><span class="linenos">591</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="Equation.n-592"><a href="#Equation.n-592"><span class="linenos">592</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span></pre></div>
 
-        </details>
-
-            <div class="docstring"><p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>
-
-<h1 id="parameters">Parameters</h1>
-
-<p>subs : dict, optional
-    Substitute numerical values for symbols, e.g.
-    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a
-    dictionary.</p>
-
-<p>maxn : int, optional
-    Allow a maximum temporary working precision of maxn digits.</p>
-
-<p>chop : bool or number, optional
-    Specifies how to replace tiny real or imaginary parts in
-    subresults by exact zeros.</p>
-
-<pre><code>When ``True`` the chop value defaults to standard precision.
-
-Otherwise the chop value is used to determine the
-magnitude of "small" for purposes of chopping.
-
-&gt;&gt;&gt; from sympy import N
-&gt;&gt;&gt; x = 1e-4
-&gt;&gt;&gt; N(x, chop=True)
-0.000100000000000000
-&gt;&gt;&gt; N(x, chop=1e-5)
-0.000100000000000000
-&gt;&gt;&gt; N(x, chop=1e-4)
-0
-</code></pre>
-
-<p>strict : bool, optional
-    Raise <code>PrecisionExhausted</code> if any subresult fails to
-    evaluate to full accuracy, given the available maxprec.</p>
-
-<p>quad : str, optional
-    Choose algorithm for numerical quadrature. By default,
-    tanh-sinh quadrature is used. For oscillatory
-    integrals on an infinite interval, try <code>quad='osc'</code>.</p>
-
-<p>verbose : bool, optional
-    Print debug information.</p>
-
-<h1 id="notes">Notes</h1>
-
-<p>When Floats are naively substituted into an expression,
-precision errors may adversely affect the result. For example,
-adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is
-then subtracted, the result will be 0.
-That is exactly what happens in the following:</p>
-
-<div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">values</span> <span class="o">=</span> <span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="mf">1e16</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">:</span> <span class="mf">1e16</span><span class="p">}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">values</span><span class="p">)</span>
-<span class="go">0</span>
-</code></pre></div>
-
-<p>Using the subs argument for evalf is the accurate way to
-evaluate such an expression:</p>
-
-<div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">subs</span><span class="o">=</span><span class="n">values</span><span class="p">)</span>
-<span class="go">1.00000000000000</span>
-</code></pre></div>
-</div>
-
-
-                            </div>
-                            <div id="Equation.n" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Equation.n">#&nbsp;&nbsp</a>
-
-        
-            <span class="def">def</span>
-            <span class="name">n</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-</pre></div>
-
-        </details>
 
             <div class="docstring"><p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>
 
@@ -4690,11 +4750,12 @@ <h1 id="notes">Notes</h1>
 
                             </div>
                             <div id="Equation.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Equation.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Equation.default_assumptions"></a>
     
     
 
@@ -4766,591 +4827,590 @@ <h5>Inherited Members</h5>
                 <dd id="Equation.refine" class="function">refine</dd>
                 <dd id="Equation.rewrite" class="function">rewrite</dd>
                 <dd id="Equation.could_extract_minus_sign" class="function">could_extract_minus_sign</dd>
-                <dd id="Equation.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Equation.is_integer" class="variable">is_integer</dd>
-                <dd id="Equation.is_complex" class="variable">is_complex</dd>
-                <dd id="Equation.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Equation.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Equation.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Equation.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Equation.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Equation.is_positive" class="variable">is_positive</dd>
+                <dd id="Equation.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Equation.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Equation.is_negative" class="variable">is_negative</dd>
+                <dd id="Equation.is_composite" class="variable">is_composite</dd>
+                <dd id="Equation.is_finite" class="variable">is_finite</dd>
                 <dd id="Equation.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Equation.is_zero" class="variable">is_zero</dd>
                 <dd id="Equation.is_even" class="variable">is_even</dd>
-                <dd id="Equation.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Equation.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Equation.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Equation.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Equation.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Equation.is_real" class="variable">is_real</dd>
-                <dd id="Equation.is_positive" class="variable">is_positive</dd>
-                <dd id="Equation.is_finite" class="variable">is_finite</dd>
-                <dd id="Equation.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Equation.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Equation.is_rational" class="variable">is_rational</dd>
-                <dd id="Equation.is_composite" class="variable">is_composite</dd>
-                <dd id="Equation.is_negative" class="variable">is_negative</dd>
-                <dd id="Equation.is_polar" class="variable">is_polar</dd>
+                <dd id="Equation.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Equation.is_odd" class="variable">is_odd</dd>
-                <dd id="Equation.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Equation.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Equation.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Equation.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Equation.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Equation.is_integer" class="variable">is_integer</dd>
+                <dd id="Equation.is_rational" class="variable">is_rational</dd>
+                <dd id="Equation.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Equation.is_complex" class="variable">is_complex</dd>
                 <dd id="Equation.is_prime" class="variable">is_prime</dd>
+                <dd id="Equation.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Equation.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Equation.is_real" class="variable">is_real</dd>
+                <dd id="Equation.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Equation.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Equation.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Equation.is_polar" class="variable">is_polar</dd>
+                <dd id="Equation.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Equation.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Equation.is_zero" class="variable">is_zero</dd>
 
             </div>
                                 </dl>
                             </div>
                 </section>
                 <section id="Eqn">
-                                <div class="attr class">
-        <a class="headerlink" href="#Eqn">#&nbsp;&nbsp</a>
-
-        
-        <span class="def">class</span>
-        <span class="name">Eqn</span><wbr>(<span class="base">sympy.core.basic.Basic</span>, <span class="base">sympy.core.evalf.EvalfMixin</span>):
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="k">class</span> <span class="nc">Equation</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class defines an equation with a left-hand-side (lhs) and a right-</span>
-<span class="sd">    hand-side (rhs) connected by the &quot;=&quot; operator (e.g. `p*V = n*R*T`).</span>
-
-<span class="sd">    Explanation</span>
-<span class="sd">    ===========</span>
-<span class="sd">    This class defines relations that all high school and college students</span>
-<span class="sd">    would recognize as mathematical equations. At present only the &quot;=&quot; relation</span>
-<span class="sd">    operator is recognized.</span>
-
-<span class="sd">    This class is intended to allow using the mathematical tools in SymPy to</span>
-<span class="sd">    rearrange equations and perform algebra in a stepwise fashion. In this</span>
-<span class="sd">    way more people can successfully perform algebraic rearrangements without</span>
-<span class="sd">    stumbling over missed details such as a negative sign.</span>
-
-<span class="sd">    Create an equation with the call ``Equation(lhs,rhs)``, where ``lhs`` and</span>
-<span class="sd">    ``rhs`` are any valid Sympy expression. ``Eqn(...)`` is a synonym for</span>
-<span class="sd">    ``Equation(...)``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-<span class="sd">    lhs: sympy expression, ``class Expr``.</span>
-<span class="sd">    rhs: sympy expression, ``class Expr``.</span>
-<span class="sd">    kwargs:</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from algebra_with_sympy import *</span>
-<span class="sd">    &gt;&gt;&gt; a, b, c, x = var(&#39;a b c x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Equation(a,b/c)</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t=Eqn(a,b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t*c</span>
-<span class="sd">    Equation(a*c, b)</span>
-<span class="sd">    &gt;&gt;&gt; c*t</span>
-<span class="sd">    Equation(a*c, b)</span>
-<span class="sd">    &gt;&gt;&gt; exp(t)</span>
-<span class="sd">    Equation(exp(a), exp(b/c))</span>
-<span class="sd">    &gt;&gt;&gt; exp(log(t))</span>
-<span class="sd">    Equation(a, b/c)</span>
-
-<span class="sd">    Simplification and Expansion</span>
-<span class="sd">    &gt;&gt;&gt; f = Eqn(x**2 - 1, c)</span>
-<span class="sd">    &gt;&gt;&gt; f</span>
-<span class="sd">    Equation(x**2 - 1, c)</span>
-<span class="sd">    &gt;&gt;&gt; f/(x+1)</span>
-<span class="sd">    Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; (f/(x+1)).simplify()</span>
-<span class="sd">    Equation(x - 1, c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; simplify(f/(x+1))</span>
-<span class="sd">    Equation(x - 1, c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; (f/(x+1)).expand()</span>
-<span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; expand(f/(x+1))</span>
-<span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
-<span class="sd">    &gt;&gt;&gt; factor(f)</span>
-<span class="sd">    Equation((x - 1)*(x + 1), c)</span>
-<span class="sd">    &gt;&gt;&gt; f.factor()</span>
-<span class="sd">    Equation((x - 1)*(x + 1), c)</span>
-<span class="sd">    &gt;&gt;&gt; f2 = f+a*x**2+b*x +c</span>
-<span class="sd">    &gt;&gt;&gt; f2</span>
-<span class="sd">    Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>
-<span class="sd">    &gt;&gt;&gt; collect(f2,x)</span>
-<span class="sd">    Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>
-
-<span class="sd">    Apply operation to only one side</span>
-<span class="sd">    &gt;&gt;&gt; poly = Eqn(a*x**2 + b*x + c*x**2, a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.applyrhs(factor,x)</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>
-<span class="sd">    &gt;&gt;&gt; poly.applylhs(factor)</span>
-<span class="sd">    Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.applylhs(collect,x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
-
-<span class="sd">    ``.apply...`` also works with user defined python functions</span>
-<span class="sd">    &gt;&gt;&gt; def addsquare(eqn):</span>
-<span class="sd">    ...     return eqn+eqn**2</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; t.apply(addsquare)</span>
-<span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.applyrhs(addsquare)</span>
-<span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.apply(addsquare, side = &#39;rhs&#39;)</span>
-<span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
-<span class="sd">    &gt;&gt;&gt; t.applylhs(addsquare)</span>
-<span class="sd">    Equation(a**2 + a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; addsquare(t)</span>
-<span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
-
-<span class="sd">    Inaddition to ``.apply...`` there is also the less general ``.do``,</span>
-<span class="sd">    ``.dolhs``, ``.dorhs``, which only works for operations defined on the</span>
-<span class="sd">    ``Expr`` class (e.g.``.collect(), .factor(), .expand()``, etc...).</span>
-<span class="sd">    &gt;&gt;&gt; poly.dolhs.collect(x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
-<span class="sd">    &gt;&gt;&gt; poly.dorhs.collect(x)</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>
-<span class="sd">    &gt;&gt;&gt; poly.do.collect(x)</span>
-<span class="sd">    Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>
-<span class="sd">    &gt;&gt;&gt; poly.dorhs.factor()</span>
-<span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>
-
-<span class="sd">    ``poly.do.exp()`` or other sympy math functions will raise an error.</span>
-
-<span class="sd">    Rearranging an equation (simple example made complicated as illustration)</span>
-<span class="sd">    &gt;&gt;&gt; p, V, n, R, T = var(&#39;p V n R T&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; eq1=Eqn(p*V,n*R*T)</span>
-<span class="sd">    &gt;&gt;&gt; eq1</span>
-<span class="sd">    Equation(V*p, R*T*n)</span>
-<span class="sd">    &gt;&gt;&gt; eq2 =eq1/V</span>
-<span class="sd">    &gt;&gt;&gt; eq2</span>
-<span class="sd">    Equation(p, R*T*n/V)</span>
-<span class="sd">    &gt;&gt;&gt; eq3 = eq2/R/T</span>
-<span class="sd">    &gt;&gt;&gt; eq3</span>
-<span class="sd">    Equation(p/(R*T), n/V)</span>
-<span class="sd">    &gt;&gt;&gt; eq4 = eq3*R/p</span>
-<span class="sd">    &gt;&gt;&gt; eq4</span>
-<span class="sd">    Equation(1/T, R*n/(V*p))</span>
-<span class="sd">    &gt;&gt;&gt; 1/eq4</span>
-<span class="sd">    Equation(T, V*p/(R*n))</span>
-<span class="sd">    &gt;&gt;&gt; eq5 = 1/eq4 - T</span>
-<span class="sd">    &gt;&gt;&gt; eq5</span>
-<span class="sd">    Equation(0, -T + V*p/(R*n))</span>
-
-<span class="sd">    Substitution (#&#39;s and units)</span>
-<span class="sd">    &gt;&gt;&gt; L, atm, mol, K = var(&#39;L atm mol K&#39;, positive=True, real=True) # units</span>
-<span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L})</span>
-<span class="sd">    Equation(p, 0.9334325*atm)</span>
-<span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L}).evalf(4)</span>
-<span class="sd">    Equation(p, 0.9334*atm)</span>
-
-<span class="sd">    Combining equations (Math with equations: lhs with lhs and rhs with rhs)</span>
-<span class="sd">    &gt;&gt;&gt; q = Eqn(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q</span>
-<span class="sd">    Equation(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; t</span>
-<span class="sd">    Equation(a, b/c)</span>
-<span class="sd">    &gt;&gt;&gt; q+t</span>
-<span class="sd">    Equation(a*c + a, b/c + b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q/t</span>
-<span class="sd">    Equation(c, 1/c)</span>
-<span class="sd">    &gt;&gt;&gt; t**q</span>
-<span class="sd">    Equation(a**(a*c), (b/c)**(b/c**2))</span>
-
-<span class="sd">    Utility operations</span>
-<span class="sd">    &gt;&gt;&gt; t.reversed</span>
-<span class="sd">    Equation(b/c, a)</span>
-<span class="sd">    &gt;&gt;&gt; t.swap</span>
-<span class="sd">    Equation(b/c, a)</span>
-<span class="sd">    &gt;&gt;&gt; t.lhs</span>
-<span class="sd">    a</span>
-<span class="sd">    &gt;&gt;&gt; t.rhs</span>
-<span class="sd">    b/c</span>
-<span class="sd">    &gt;&gt;&gt; t.as_Boolean()</span>
-<span class="sd">    Eq(a, b/c)</span>
-
-<span class="sd">    `.check()` convenience method for `.as_Boolean().simplify()`</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import I, pi</span>
-<span class="sd">    &gt;&gt;&gt; Equation(pi*(I+2), pi*I+2*pi).check()</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; Eqn(a,a+1).check()</span>
-<span class="sd">    False</span>
-
-<span class="sd">    Differentiation</span>
-<span class="sd">    Differentiation is applied to both sides if the wrt variable appears on</span>
-<span class="sd">    both sides.</span>
-<span class="sd">    &gt;&gt;&gt; q=Eqn(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; q</span>
-<span class="sd">    Equation(a*c, b/c**2)</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,b)</span>
-<span class="sd">    Equation(Derivative(a*c, b), c**(-2))</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c)</span>
-<span class="sd">    Equation(a, -2*b/c**3)</span>
-<span class="sd">    &gt;&gt;&gt; diff(log(q),b)</span>
-<span class="sd">    Equation(Derivative(log(a*c), b), 1/b)</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c,2)</span>
-<span class="sd">    Equation(Derivative(a, c), 6*b/c**4)</span>
-
-<span class="sd">    If you specify multiple differentiation all at once the assumption</span>
-<span class="sd">    is order of differentiation matters and the lhs will not be</span>
-<span class="sd">    evaluated.</span>
-<span class="sd">    &gt;&gt;&gt; diff(q,c,b)</span>
-<span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
-
-<span class="sd">    To overcome this specify the order of operations.</span>
-<span class="sd">    &gt;&gt;&gt; diff(diff(q,c),b)</span>
-<span class="sd">    Equation(Derivative(a, b), -2/c**3)</span>
-
-<span class="sd">    But the reverse order returns an unevaulated lhs (a may depend on b).</span>
-<span class="sd">    &gt;&gt;&gt; diff(diff(q,b),c)</span>
-<span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
-
-<span class="sd">    Integration can only be performed on one side at a time.</span>
-<span class="sd">    &gt;&gt;&gt; q=Eqn(a*c,b/c)</span>
-<span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;rhs&#39;)</span>
-<span class="sd">    b**2/(2*c)</span>
-<span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;lhs&#39;)</span>
-<span class="sd">    a*b*c</span>
-
-<span class="sd">    Make a pretty statement of integration from an equation</span>
-<span class="sd">    &gt;&gt;&gt; Eqn(Integral(q.lhs,b),integrate(q,b,side=&#39;rhs&#39;))</span>
-<span class="sd">    Equation(Integral(a*c, b), b**2/(2*c))</span>
-
-<span class="sd">    Integration of each side with respect to different variables</span>
-<span class="sd">    &gt;&gt;&gt; q.dorhs.integrate(b).dolhs.integrate(a)</span>
-<span class="sd">    Equation(a**2*c/2, b**2/(2*c))</span>
-
-<span class="sd">    SymPy&#39;s solvers do not understand these equations. They expect an</span>
-<span class="sd">    expression that the solver assumes = 0. Thus to use the solver the</span>
-<span class="sd">    equation must be rearranged so that all non-zero symbols are on one side.</span>
-<span class="sd">    Then just the non-zero symbolic side is passed to ``solve()``.</span>
-<span class="sd">    &gt;&gt;&gt; t2 = t-t.rhs</span>
-<span class="sd">    &gt;&gt;&gt; t2</span>
-<span class="sd">    Equation(a - b/c, 0)</span>
-<span class="sd">    &gt;&gt;&gt; solve(t2.lhs,c)</span>
-<span class="sd">    [b/a]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">lhs</span><span class="p">)</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;lhs and rhs must be valid sympy expressions.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_get_eqn_name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">IPython</span> <span class="kn">import</span> <span class="n">get_ipython</span>
-        <span class="n">global_dict</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">get_ipython</span><span class="p">(),</span> <span class="s1">&#39;user_ns&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-        <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
-        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
-        <span class="k">if</span> <span class="n">global_dict</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">var_name</span> <span class="ow">in</span> <span class="n">global_dict</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">])</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">==</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> \
-                            <span class="n">var_name</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s1">&#39;_&#39;</span><span class="p">):</span>
-                        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
-                        <span class="k">return</span> <span class="n">var_name</span>
-        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span> <span class="o">=</span> <span class="n">human_text</span>
-        <span class="k">return</span> <span class="s1">&#39;&#39;</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">lhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the lhs of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the rhs of the equation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Converts the equation to an Equality.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
-<span class="sd">        Parameters</span>
-<span class="sd">        ----------</span>
-<span class="sd">        kwargs any appropriate for `Equality`.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        -------</span>
-<span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">reversed</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Swaps the lhs and the rhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">swap</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Synonym for `.reversed`</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">reversed</span>
-
-    <span class="k">def</span> <span class="nf">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># Applies a function to an expression checking whether there</span>
-        <span class="c1"># is a specialized version associated with the particular type of</span>
-        <span class="c1"># expression. Errors will be raised if the function cannot be</span>
-        <span class="c1"># applied to an expression.</span>
-        <span class="n">funcname</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s1">&#39;__name__&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">funcname</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-            <span class="n">localfunc</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">funcname</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">localfunc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">localfunc</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply an operation/function/method to the equation returning the</span>
-<span class="sd">        resulting equation.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        func: object</span>
-<span class="sd">            object to apply usually a function</span>
-
-<span class="sd">        args: as necessary for the function</span>
-
-<span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
-<span class="sd">            Specifies which side of the equation the operation will be applied</span>
-<span class="sd">            to. Default is &#39;both&#39;.</span>
-
-<span class="sd">        kwargs: as necessary for the function</span>
-<span class="sd">         &quot;&quot;&quot;</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the lhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the rhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
-
-    <span class="k">class</span> <span class="nc">_sides</span><span class="p">:</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Helper class for the `.do.`, `.dolhs.`, `.dorhs.` syntax for applying</span>
-<span class="sd">        submethods of expressions.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eqn</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">eqn</span> <span class="o">=</span> <span class="n">eqn</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="o">=</span> <span class="n">side</span>
-
-        <span class="k">def</span> <span class="fm">__getattr__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="kc">None</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;rhs&#39;</span><span class="p">,</span> <span class="s1">&#39;both&#39;</span><span class="p">):</span>
-                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;Expressions in the equation have no &#39;</span>
-                                     <span class="s1">&#39;attribute `&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span>
-                    <span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;`. Try `.apply(&#39;</span>
-                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;, *args)` or &#39;</span>
-                                                   <span class="s1">&#39;pass the equation as a parameter to `&#39;</span>
-                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;()`.&#39;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">apply</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">side</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">do</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dolhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dorhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Overrides of binary math operations</span>
-    <span class="c1">#####</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_binary_op</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">opfunc_ab</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rtruediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="fm">__rpow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__pow__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Operation helper functions</span>
-    <span class="c1">#####</span>
-    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># TODO: cancel out factors common to both sides.</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1"># TODO Find why diff and Derivative do not appear to pass through</span>
-        <span class="c1">#  kwargs to this. Since we cannot set evaluation of lhs manually</span>
-        <span class="c1">#  try to be intelligent about when to do it.</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Derivative</span>
-        <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">False</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)):</span>
-            <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">sym</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span>
-                        <span class="n">_sympify</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">):</span>
-                    <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">True</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="n">eval_lhs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">side</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s1">&#39;side&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>  <span class="c1"># Could not seem to pass values for</span>
-        <span class="c1"># `evaluate` through to here.</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;You must specify `side=&quot;lhs&quot;` or `side=&quot;rhs&quot;` &#39;</span>
-                             <span class="s1">&#39;when integrating an Equation&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;`side` must equal &quot;lhs&quot; or &quot;rhs&quot;.&#39;</span><span class="p">)</span>
-
-    <span class="c1">#####</span>
-    <span class="c1"># Output helper functions</span>
-    <span class="c1">#####</span>
-    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">repstr</span> <span class="o">=</span> <span class="s1">&#39;Equation(</span><span class="si">%s</span><span class="s1">, </span><span class="si">%s</span><span class="s1">)&#39;</span> <span class="o">%</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">())</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__str__</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">repstr</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span> <span class="ow">and</span> <span class="ow">not</span> \
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span><span class="s1">&#39;</span><span class="se">\\</span><span class="s1">text{code version: &#39;</span><span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">+</span><span class="s1">&#39;} </span><span class="se">\\</span><span class="s1">newline &#39;</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;=&#39;</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span><span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,&#39;</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;(</span><span class="se">\\</span><span class="s1">text{&#39;</span><span class="o">+</span><span class="n">namestr</span><span class="o">+</span><span class="s1">&#39;})&#39;</span>
-        <span class="k">return</span> <span class="n">tempstr</span>
-
-    <span class="k">def</span> <span class="fm">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
-        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span><span class="p">:</span>
-            <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;code version: &#39;</span><span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="o">+</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
-            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
-        <span class="n">tempstr</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39; = &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span> <span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
-            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;          (&#39;</span> <span class="o">+</span> <span class="n">namestr</span> <span class="o">+</span> <span class="s1">&#39;)&#39;</span>
-        <span class="k">return</span> <span class="n">tempstr</span>
-</pre></div>
-
-        </details>
+                            <input id="Eqn-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Eqn</span><wbr>(<span class="base">sympy.core.basic.Basic</span>, <span class="base">sympy.core.evalf.EvalfMixin</span>):
+
+                <label class="view-source-button" for="Eqn-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Eqn"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn-122"><a href="#Eqn-122"><span class="linenos">122</span></a><span class="k">class</span> <span class="nc">Equation</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
+</span><span id="Eqn-123"><a href="#Eqn-123"><span class="linenos">123</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-124"><a href="#Eqn-124"><span class="linenos">124</span></a><span class="sd">    This class defines an equation with a left-hand-side (lhs) and a right-</span>
+</span><span id="Eqn-125"><a href="#Eqn-125"><span class="linenos">125</span></a><span class="sd">    hand-side (rhs) connected by the &quot;=&quot; operator (e.g. `p*V = n*R*T`).</span>
+</span><span id="Eqn-126"><a href="#Eqn-126"><span class="linenos">126</span></a>
+</span><span id="Eqn-127"><a href="#Eqn-127"><span class="linenos">127</span></a><span class="sd">    Explanation</span>
+</span><span id="Eqn-128"><a href="#Eqn-128"><span class="linenos">128</span></a><span class="sd">    ===========</span>
+</span><span id="Eqn-129"><a href="#Eqn-129"><span class="linenos">129</span></a><span class="sd">    This class defines relations that all high school and college students</span>
+</span><span id="Eqn-130"><a href="#Eqn-130"><span class="linenos">130</span></a><span class="sd">    would recognize as mathematical equations. At present only the &quot;=&quot; relation</span>
+</span><span id="Eqn-131"><a href="#Eqn-131"><span class="linenos">131</span></a><span class="sd">    operator is recognized.</span>
+</span><span id="Eqn-132"><a href="#Eqn-132"><span class="linenos">132</span></a>
+</span><span id="Eqn-133"><a href="#Eqn-133"><span class="linenos">133</span></a><span class="sd">    This class is intended to allow using the mathematical tools in SymPy to</span>
+</span><span id="Eqn-134"><a href="#Eqn-134"><span class="linenos">134</span></a><span class="sd">    rearrange equations and perform algebra in a stepwise fashion. In this</span>
+</span><span id="Eqn-135"><a href="#Eqn-135"><span class="linenos">135</span></a><span class="sd">    way more people can successfully perform algebraic rearrangements without</span>
+</span><span id="Eqn-136"><a href="#Eqn-136"><span class="linenos">136</span></a><span class="sd">    stumbling over missed details such as a negative sign.</span>
+</span><span id="Eqn-137"><a href="#Eqn-137"><span class="linenos">137</span></a>
+</span><span id="Eqn-138"><a href="#Eqn-138"><span class="linenos">138</span></a><span class="sd">    Create an equation with the call ``Equation(lhs,rhs)``, where ``lhs`` and</span>
+</span><span id="Eqn-139"><a href="#Eqn-139"><span class="linenos">139</span></a><span class="sd">    ``rhs`` are any valid Sympy expression. ``Eqn(...)`` is a synonym for</span>
+</span><span id="Eqn-140"><a href="#Eqn-140"><span class="linenos">140</span></a><span class="sd">    ``Equation(...)``.</span>
+</span><span id="Eqn-141"><a href="#Eqn-141"><span class="linenos">141</span></a>
+</span><span id="Eqn-142"><a href="#Eqn-142"><span class="linenos">142</span></a><span class="sd">    Parameters</span>
+</span><span id="Eqn-143"><a href="#Eqn-143"><span class="linenos">143</span></a><span class="sd">    ==========</span>
+</span><span id="Eqn-144"><a href="#Eqn-144"><span class="linenos">144</span></a><span class="sd">    lhs: sympy expression, ``class Expr``.</span>
+</span><span id="Eqn-145"><a href="#Eqn-145"><span class="linenos">145</span></a><span class="sd">    rhs: sympy expression, ``class Expr``.</span>
+</span><span id="Eqn-146"><a href="#Eqn-146"><span class="linenos">146</span></a><span class="sd">    kwargs:</span>
+</span><span id="Eqn-147"><a href="#Eqn-147"><span class="linenos">147</span></a>
+</span><span id="Eqn-148"><a href="#Eqn-148"><span class="linenos">148</span></a><span class="sd">    Examples</span>
+</span><span id="Eqn-149"><a href="#Eqn-149"><span class="linenos">149</span></a><span class="sd">    ========</span>
+</span><span id="Eqn-150"><a href="#Eqn-150"><span class="linenos">150</span></a><span class="sd">    &gt;&gt;&gt; from algebra_with_sympy import *</span>
+</span><span id="Eqn-151"><a href="#Eqn-151"><span class="linenos">151</span></a><span class="sd">    &gt;&gt;&gt; a, b, c, x = var(&#39;a b c x&#39;)</span>
+</span><span id="Eqn-152"><a href="#Eqn-152"><span class="linenos">152</span></a><span class="sd">    &gt;&gt;&gt; Equation(a,b/c)</span>
+</span><span id="Eqn-153"><a href="#Eqn-153"><span class="linenos">153</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="Eqn-154"><a href="#Eqn-154"><span class="linenos">154</span></a><span class="sd">    &gt;&gt;&gt; t=Eqn(a,b/c)</span>
+</span><span id="Eqn-155"><a href="#Eqn-155"><span class="linenos">155</span></a><span class="sd">    &gt;&gt;&gt; t</span>
+</span><span id="Eqn-156"><a href="#Eqn-156"><span class="linenos">156</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="Eqn-157"><a href="#Eqn-157"><span class="linenos">157</span></a><span class="sd">    &gt;&gt;&gt; t*c</span>
+</span><span id="Eqn-158"><a href="#Eqn-158"><span class="linenos">158</span></a><span class="sd">    Equation(a*c, b)</span>
+</span><span id="Eqn-159"><a href="#Eqn-159"><span class="linenos">159</span></a><span class="sd">    &gt;&gt;&gt; c*t</span>
+</span><span id="Eqn-160"><a href="#Eqn-160"><span class="linenos">160</span></a><span class="sd">    Equation(a*c, b)</span>
+</span><span id="Eqn-161"><a href="#Eqn-161"><span class="linenos">161</span></a><span class="sd">    &gt;&gt;&gt; exp(t)</span>
+</span><span id="Eqn-162"><a href="#Eqn-162"><span class="linenos">162</span></a><span class="sd">    Equation(exp(a), exp(b/c))</span>
+</span><span id="Eqn-163"><a href="#Eqn-163"><span class="linenos">163</span></a><span class="sd">    &gt;&gt;&gt; exp(log(t))</span>
+</span><span id="Eqn-164"><a href="#Eqn-164"><span class="linenos">164</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="Eqn-165"><a href="#Eqn-165"><span class="linenos">165</span></a>
+</span><span id="Eqn-166"><a href="#Eqn-166"><span class="linenos">166</span></a><span class="sd">    Simplification and Expansion</span>
+</span><span id="Eqn-167"><a href="#Eqn-167"><span class="linenos">167</span></a><span class="sd">    &gt;&gt;&gt; f = Eqn(x**2 - 1, c)</span>
+</span><span id="Eqn-168"><a href="#Eqn-168"><span class="linenos">168</span></a><span class="sd">    &gt;&gt;&gt; f</span>
+</span><span id="Eqn-169"><a href="#Eqn-169"><span class="linenos">169</span></a><span class="sd">    Equation(x**2 - 1, c)</span>
+</span><span id="Eqn-170"><a href="#Eqn-170"><span class="linenos">170</span></a><span class="sd">    &gt;&gt;&gt; f/(x+1)</span>
+</span><span id="Eqn-171"><a href="#Eqn-171"><span class="linenos">171</span></a><span class="sd">    Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>
+</span><span id="Eqn-172"><a href="#Eqn-172"><span class="linenos">172</span></a><span class="sd">    &gt;&gt;&gt; (f/(x+1)).simplify()</span>
+</span><span id="Eqn-173"><a href="#Eqn-173"><span class="linenos">173</span></a><span class="sd">    Equation(x - 1, c/(x + 1))</span>
+</span><span id="Eqn-174"><a href="#Eqn-174"><span class="linenos">174</span></a><span class="sd">    &gt;&gt;&gt; simplify(f/(x+1))</span>
+</span><span id="Eqn-175"><a href="#Eqn-175"><span class="linenos">175</span></a><span class="sd">    Equation(x - 1, c/(x + 1))</span>
+</span><span id="Eqn-176"><a href="#Eqn-176"><span class="linenos">176</span></a><span class="sd">    &gt;&gt;&gt; (f/(x+1)).expand()</span>
+</span><span id="Eqn-177"><a href="#Eqn-177"><span class="linenos">177</span></a><span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
+</span><span id="Eqn-178"><a href="#Eqn-178"><span class="linenos">178</span></a><span class="sd">    &gt;&gt;&gt; expand(f/(x+1))</span>
+</span><span id="Eqn-179"><a href="#Eqn-179"><span class="linenos">179</span></a><span class="sd">    Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>
+</span><span id="Eqn-180"><a href="#Eqn-180"><span class="linenos">180</span></a><span class="sd">    &gt;&gt;&gt; factor(f)</span>
+</span><span id="Eqn-181"><a href="#Eqn-181"><span class="linenos">181</span></a><span class="sd">    Equation((x - 1)*(x + 1), c)</span>
+</span><span id="Eqn-182"><a href="#Eqn-182"><span class="linenos">182</span></a><span class="sd">    &gt;&gt;&gt; f.factor()</span>
+</span><span id="Eqn-183"><a href="#Eqn-183"><span class="linenos">183</span></a><span class="sd">    Equation((x - 1)*(x + 1), c)</span>
+</span><span id="Eqn-184"><a href="#Eqn-184"><span class="linenos">184</span></a><span class="sd">    &gt;&gt;&gt; f2 = f+a*x**2+b*x +c</span>
+</span><span id="Eqn-185"><a href="#Eqn-185"><span class="linenos">185</span></a><span class="sd">    &gt;&gt;&gt; f2</span>
+</span><span id="Eqn-186"><a href="#Eqn-186"><span class="linenos">186</span></a><span class="sd">    Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>
+</span><span id="Eqn-187"><a href="#Eqn-187"><span class="linenos">187</span></a><span class="sd">    &gt;&gt;&gt; collect(f2,x)</span>
+</span><span id="Eqn-188"><a href="#Eqn-188"><span class="linenos">188</span></a><span class="sd">    Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>
+</span><span id="Eqn-189"><a href="#Eqn-189"><span class="linenos">189</span></a>
+</span><span id="Eqn-190"><a href="#Eqn-190"><span class="linenos">190</span></a><span class="sd">    Apply operation to only one side</span>
+</span><span id="Eqn-191"><a href="#Eqn-191"><span class="linenos">191</span></a><span class="sd">    &gt;&gt;&gt; poly = Eqn(a*x**2 + b*x + c*x**2, a*x**3 + b*x**3 + c*x)</span>
+</span><span id="Eqn-192"><a href="#Eqn-192"><span class="linenos">192</span></a><span class="sd">    &gt;&gt;&gt; poly.applyrhs(factor,x)</span>
+</span><span id="Eqn-193"><a href="#Eqn-193"><span class="linenos">193</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>
+</span><span id="Eqn-194"><a href="#Eqn-194"><span class="linenos">194</span></a><span class="sd">    &gt;&gt;&gt; poly.applylhs(factor)</span>
+</span><span id="Eqn-195"><a href="#Eqn-195"><span class="linenos">195</span></a><span class="sd">    Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="Eqn-196"><a href="#Eqn-196"><span class="linenos">196</span></a><span class="sd">    &gt;&gt;&gt; poly.applylhs(collect,x)</span>
+</span><span id="Eqn-197"><a href="#Eqn-197"><span class="linenos">197</span></a><span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="Eqn-198"><a href="#Eqn-198"><span class="linenos">198</span></a>
+</span><span id="Eqn-199"><a href="#Eqn-199"><span class="linenos">199</span></a><span class="sd">    ``.apply...`` also works with user defined python functions</span>
+</span><span id="Eqn-200"><a href="#Eqn-200"><span class="linenos">200</span></a><span class="sd">    &gt;&gt;&gt; def addsquare(eqn):</span>
+</span><span id="Eqn-201"><a href="#Eqn-201"><span class="linenos">201</span></a><span class="sd">    ...     return eqn+eqn**2</span>
+</span><span id="Eqn-202"><a href="#Eqn-202"><span class="linenos">202</span></a><span class="sd">    ...</span>
+</span><span id="Eqn-203"><a href="#Eqn-203"><span class="linenos">203</span></a><span class="sd">    &gt;&gt;&gt; t.apply(addsquare)</span>
+</span><span id="Eqn-204"><a href="#Eqn-204"><span class="linenos">204</span></a><span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
+</span><span id="Eqn-205"><a href="#Eqn-205"><span class="linenos">205</span></a><span class="sd">    &gt;&gt;&gt; t.applyrhs(addsquare)</span>
+</span><span id="Eqn-206"><a href="#Eqn-206"><span class="linenos">206</span></a><span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
+</span><span id="Eqn-207"><a href="#Eqn-207"><span class="linenos">207</span></a><span class="sd">    &gt;&gt;&gt; t.apply(addsquare, side = &#39;rhs&#39;)</span>
+</span><span id="Eqn-208"><a href="#Eqn-208"><span class="linenos">208</span></a><span class="sd">    Equation(a, b**2/c**2 + b/c)</span>
+</span><span id="Eqn-209"><a href="#Eqn-209"><span class="linenos">209</span></a><span class="sd">    &gt;&gt;&gt; t.applylhs(addsquare)</span>
+</span><span id="Eqn-210"><a href="#Eqn-210"><span class="linenos">210</span></a><span class="sd">    Equation(a**2 + a, b/c)</span>
+</span><span id="Eqn-211"><a href="#Eqn-211"><span class="linenos">211</span></a><span class="sd">    &gt;&gt;&gt; addsquare(t)</span>
+</span><span id="Eqn-212"><a href="#Eqn-212"><span class="linenos">212</span></a><span class="sd">    Equation(a**2 + a, b**2/c**2 + b/c)</span>
+</span><span id="Eqn-213"><a href="#Eqn-213"><span class="linenos">213</span></a>
+</span><span id="Eqn-214"><a href="#Eqn-214"><span class="linenos">214</span></a><span class="sd">    Inaddition to ``.apply...`` there is also the less general ``.do``,</span>
+</span><span id="Eqn-215"><a href="#Eqn-215"><span class="linenos">215</span></a><span class="sd">    ``.dolhs``, ``.dorhs``, which only works for operations defined on the</span>
+</span><span id="Eqn-216"><a href="#Eqn-216"><span class="linenos">216</span></a><span class="sd">    ``Expr`` class (e.g.``.collect(), .factor(), .expand()``, etc...).</span>
+</span><span id="Eqn-217"><a href="#Eqn-217"><span class="linenos">217</span></a><span class="sd">    &gt;&gt;&gt; poly.dolhs.collect(x)</span>
+</span><span id="Eqn-218"><a href="#Eqn-218"><span class="linenos">218</span></a><span class="sd">    Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>
+</span><span id="Eqn-219"><a href="#Eqn-219"><span class="linenos">219</span></a><span class="sd">    &gt;&gt;&gt; poly.dorhs.collect(x)</span>
+</span><span id="Eqn-220"><a href="#Eqn-220"><span class="linenos">220</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>
+</span><span id="Eqn-221"><a href="#Eqn-221"><span class="linenos">221</span></a><span class="sd">    &gt;&gt;&gt; poly.do.collect(x)</span>
+</span><span id="Eqn-222"><a href="#Eqn-222"><span class="linenos">222</span></a><span class="sd">    Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>
+</span><span id="Eqn-223"><a href="#Eqn-223"><span class="linenos">223</span></a><span class="sd">    &gt;&gt;&gt; poly.dorhs.factor()</span>
+</span><span id="Eqn-224"><a href="#Eqn-224"><span class="linenos">224</span></a><span class="sd">    Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>
+</span><span id="Eqn-225"><a href="#Eqn-225"><span class="linenos">225</span></a>
+</span><span id="Eqn-226"><a href="#Eqn-226"><span class="linenos">226</span></a><span class="sd">    ``poly.do.exp()`` or other sympy math functions will raise an error.</span>
+</span><span id="Eqn-227"><a href="#Eqn-227"><span class="linenos">227</span></a>
+</span><span id="Eqn-228"><a href="#Eqn-228"><span class="linenos">228</span></a><span class="sd">    Rearranging an equation (simple example made complicated as illustration)</span>
+</span><span id="Eqn-229"><a href="#Eqn-229"><span class="linenos">229</span></a><span class="sd">    &gt;&gt;&gt; p, V, n, R, T = var(&#39;p V n R T&#39;)</span>
+</span><span id="Eqn-230"><a href="#Eqn-230"><span class="linenos">230</span></a><span class="sd">    &gt;&gt;&gt; eq1=Eqn(p*V,n*R*T)</span>
+</span><span id="Eqn-231"><a href="#Eqn-231"><span class="linenos">231</span></a><span class="sd">    &gt;&gt;&gt; eq1</span>
+</span><span id="Eqn-232"><a href="#Eqn-232"><span class="linenos">232</span></a><span class="sd">    Equation(V*p, R*T*n)</span>
+</span><span id="Eqn-233"><a href="#Eqn-233"><span class="linenos">233</span></a><span class="sd">    &gt;&gt;&gt; eq2 =eq1/V</span>
+</span><span id="Eqn-234"><a href="#Eqn-234"><span class="linenos">234</span></a><span class="sd">    &gt;&gt;&gt; eq2</span>
+</span><span id="Eqn-235"><a href="#Eqn-235"><span class="linenos">235</span></a><span class="sd">    Equation(p, R*T*n/V)</span>
+</span><span id="Eqn-236"><a href="#Eqn-236"><span class="linenos">236</span></a><span class="sd">    &gt;&gt;&gt; eq3 = eq2/R/T</span>
+</span><span id="Eqn-237"><a href="#Eqn-237"><span class="linenos">237</span></a><span class="sd">    &gt;&gt;&gt; eq3</span>
+</span><span id="Eqn-238"><a href="#Eqn-238"><span class="linenos">238</span></a><span class="sd">    Equation(p/(R*T), n/V)</span>
+</span><span id="Eqn-239"><a href="#Eqn-239"><span class="linenos">239</span></a><span class="sd">    &gt;&gt;&gt; eq4 = eq3*R/p</span>
+</span><span id="Eqn-240"><a href="#Eqn-240"><span class="linenos">240</span></a><span class="sd">    &gt;&gt;&gt; eq4</span>
+</span><span id="Eqn-241"><a href="#Eqn-241"><span class="linenos">241</span></a><span class="sd">    Equation(1/T, R*n/(V*p))</span>
+</span><span id="Eqn-242"><a href="#Eqn-242"><span class="linenos">242</span></a><span class="sd">    &gt;&gt;&gt; 1/eq4</span>
+</span><span id="Eqn-243"><a href="#Eqn-243"><span class="linenos">243</span></a><span class="sd">    Equation(T, V*p/(R*n))</span>
+</span><span id="Eqn-244"><a href="#Eqn-244"><span class="linenos">244</span></a><span class="sd">    &gt;&gt;&gt; eq5 = 1/eq4 - T</span>
+</span><span id="Eqn-245"><a href="#Eqn-245"><span class="linenos">245</span></a><span class="sd">    &gt;&gt;&gt; eq5</span>
+</span><span id="Eqn-246"><a href="#Eqn-246"><span class="linenos">246</span></a><span class="sd">    Equation(0, -T + V*p/(R*n))</span>
+</span><span id="Eqn-247"><a href="#Eqn-247"><span class="linenos">247</span></a>
+</span><span id="Eqn-248"><a href="#Eqn-248"><span class="linenos">248</span></a><span class="sd">    Substitution (#&#39;s and units)</span>
+</span><span id="Eqn-249"><a href="#Eqn-249"><span class="linenos">249</span></a><span class="sd">    &gt;&gt;&gt; L, atm, mol, K = var(&#39;L atm mol K&#39;, positive=True, real=True) # units</span>
+</span><span id="Eqn-250"><a href="#Eqn-250"><span class="linenos">250</span></a><span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L})</span>
+</span><span id="Eqn-251"><a href="#Eqn-251"><span class="linenos">251</span></a><span class="sd">    Equation(p, 0.9334325*atm)</span>
+</span><span id="Eqn-252"><a href="#Eqn-252"><span class="linenos">252</span></a><span class="sd">    &gt;&gt;&gt; eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L}).evalf(4)</span>
+</span><span id="Eqn-253"><a href="#Eqn-253"><span class="linenos">253</span></a><span class="sd">    Equation(p, 0.9334*atm)</span>
+</span><span id="Eqn-254"><a href="#Eqn-254"><span class="linenos">254</span></a>
+</span><span id="Eqn-255"><a href="#Eqn-255"><span class="linenos">255</span></a><span class="sd">    Combining equations (Math with equations: lhs with lhs and rhs with rhs)</span>
+</span><span id="Eqn-256"><a href="#Eqn-256"><span class="linenos">256</span></a><span class="sd">    &gt;&gt;&gt; q = Eqn(a*c, b/c**2)</span>
+</span><span id="Eqn-257"><a href="#Eqn-257"><span class="linenos">257</span></a><span class="sd">    &gt;&gt;&gt; q</span>
+</span><span id="Eqn-258"><a href="#Eqn-258"><span class="linenos">258</span></a><span class="sd">    Equation(a*c, b/c**2)</span>
+</span><span id="Eqn-259"><a href="#Eqn-259"><span class="linenos">259</span></a><span class="sd">    &gt;&gt;&gt; t</span>
+</span><span id="Eqn-260"><a href="#Eqn-260"><span class="linenos">260</span></a><span class="sd">    Equation(a, b/c)</span>
+</span><span id="Eqn-261"><a href="#Eqn-261"><span class="linenos">261</span></a><span class="sd">    &gt;&gt;&gt; q+t</span>
+</span><span id="Eqn-262"><a href="#Eqn-262"><span class="linenos">262</span></a><span class="sd">    Equation(a*c + a, b/c + b/c**2)</span>
+</span><span id="Eqn-263"><a href="#Eqn-263"><span class="linenos">263</span></a><span class="sd">    &gt;&gt;&gt; q/t</span>
+</span><span id="Eqn-264"><a href="#Eqn-264"><span class="linenos">264</span></a><span class="sd">    Equation(c, 1/c)</span>
+</span><span id="Eqn-265"><a href="#Eqn-265"><span class="linenos">265</span></a><span class="sd">    &gt;&gt;&gt; t**q</span>
+</span><span id="Eqn-266"><a href="#Eqn-266"><span class="linenos">266</span></a><span class="sd">    Equation(a**(a*c), (b/c)**(b/c**2))</span>
+</span><span id="Eqn-267"><a href="#Eqn-267"><span class="linenos">267</span></a>
+</span><span id="Eqn-268"><a href="#Eqn-268"><span class="linenos">268</span></a><span class="sd">    Utility operations</span>
+</span><span id="Eqn-269"><a href="#Eqn-269"><span class="linenos">269</span></a><span class="sd">    &gt;&gt;&gt; t.reversed</span>
+</span><span id="Eqn-270"><a href="#Eqn-270"><span class="linenos">270</span></a><span class="sd">    Equation(b/c, a)</span>
+</span><span id="Eqn-271"><a href="#Eqn-271"><span class="linenos">271</span></a><span class="sd">    &gt;&gt;&gt; t.swap</span>
+</span><span id="Eqn-272"><a href="#Eqn-272"><span class="linenos">272</span></a><span class="sd">    Equation(b/c, a)</span>
+</span><span id="Eqn-273"><a href="#Eqn-273"><span class="linenos">273</span></a><span class="sd">    &gt;&gt;&gt; t.lhs</span>
+</span><span id="Eqn-274"><a href="#Eqn-274"><span class="linenos">274</span></a><span class="sd">    a</span>
+</span><span id="Eqn-275"><a href="#Eqn-275"><span class="linenos">275</span></a><span class="sd">    &gt;&gt;&gt; t.rhs</span>
+</span><span id="Eqn-276"><a href="#Eqn-276"><span class="linenos">276</span></a><span class="sd">    b/c</span>
+</span><span id="Eqn-277"><a href="#Eqn-277"><span class="linenos">277</span></a><span class="sd">    &gt;&gt;&gt; t.as_Boolean()</span>
+</span><span id="Eqn-278"><a href="#Eqn-278"><span class="linenos">278</span></a><span class="sd">    Eq(a, b/c)</span>
+</span><span id="Eqn-279"><a href="#Eqn-279"><span class="linenos">279</span></a>
+</span><span id="Eqn-280"><a href="#Eqn-280"><span class="linenos">280</span></a><span class="sd">    `.check()` convenience method for `.as_Boolean().simplify()`</span>
+</span><span id="Eqn-281"><a href="#Eqn-281"><span class="linenos">281</span></a><span class="sd">    &gt;&gt;&gt; from sympy import I, pi</span>
+</span><span id="Eqn-282"><a href="#Eqn-282"><span class="linenos">282</span></a><span class="sd">    &gt;&gt;&gt; Equation(pi*(I+2), pi*I+2*pi).check()</span>
+</span><span id="Eqn-283"><a href="#Eqn-283"><span class="linenos">283</span></a><span class="sd">    True</span>
+</span><span id="Eqn-284"><a href="#Eqn-284"><span class="linenos">284</span></a><span class="sd">    &gt;&gt;&gt; Eqn(a,a+1).check()</span>
+</span><span id="Eqn-285"><a href="#Eqn-285"><span class="linenos">285</span></a><span class="sd">    False</span>
+</span><span id="Eqn-286"><a href="#Eqn-286"><span class="linenos">286</span></a>
+</span><span id="Eqn-287"><a href="#Eqn-287"><span class="linenos">287</span></a><span class="sd">    Differentiation</span>
+</span><span id="Eqn-288"><a href="#Eqn-288"><span class="linenos">288</span></a><span class="sd">    Differentiation is applied to both sides if the wrt variable appears on</span>
+</span><span id="Eqn-289"><a href="#Eqn-289"><span class="linenos">289</span></a><span class="sd">    both sides.</span>
+</span><span id="Eqn-290"><a href="#Eqn-290"><span class="linenos">290</span></a><span class="sd">    &gt;&gt;&gt; q=Eqn(a*c, b/c**2)</span>
+</span><span id="Eqn-291"><a href="#Eqn-291"><span class="linenos">291</span></a><span class="sd">    &gt;&gt;&gt; q</span>
+</span><span id="Eqn-292"><a href="#Eqn-292"><span class="linenos">292</span></a><span class="sd">    Equation(a*c, b/c**2)</span>
+</span><span id="Eqn-293"><a href="#Eqn-293"><span class="linenos">293</span></a><span class="sd">    &gt;&gt;&gt; diff(q,b)</span>
+</span><span id="Eqn-294"><a href="#Eqn-294"><span class="linenos">294</span></a><span class="sd">    Equation(Derivative(a*c, b), c**(-2))</span>
+</span><span id="Eqn-295"><a href="#Eqn-295"><span class="linenos">295</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c)</span>
+</span><span id="Eqn-296"><a href="#Eqn-296"><span class="linenos">296</span></a><span class="sd">    Equation(a, -2*b/c**3)</span>
+</span><span id="Eqn-297"><a href="#Eqn-297"><span class="linenos">297</span></a><span class="sd">    &gt;&gt;&gt; diff(log(q),b)</span>
+</span><span id="Eqn-298"><a href="#Eqn-298"><span class="linenos">298</span></a><span class="sd">    Equation(Derivative(log(a*c), b), 1/b)</span>
+</span><span id="Eqn-299"><a href="#Eqn-299"><span class="linenos">299</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c,2)</span>
+</span><span id="Eqn-300"><a href="#Eqn-300"><span class="linenos">300</span></a><span class="sd">    Equation(Derivative(a, c), 6*b/c**4)</span>
+</span><span id="Eqn-301"><a href="#Eqn-301"><span class="linenos">301</span></a>
+</span><span id="Eqn-302"><a href="#Eqn-302"><span class="linenos">302</span></a><span class="sd">    If you specify multiple differentiation all at once the assumption</span>
+</span><span id="Eqn-303"><a href="#Eqn-303"><span class="linenos">303</span></a><span class="sd">    is order of differentiation matters and the lhs will not be</span>
+</span><span id="Eqn-304"><a href="#Eqn-304"><span class="linenos">304</span></a><span class="sd">    evaluated.</span>
+</span><span id="Eqn-305"><a href="#Eqn-305"><span class="linenos">305</span></a><span class="sd">    &gt;&gt;&gt; diff(q,c,b)</span>
+</span><span id="Eqn-306"><a href="#Eqn-306"><span class="linenos">306</span></a><span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
+</span><span id="Eqn-307"><a href="#Eqn-307"><span class="linenos">307</span></a>
+</span><span id="Eqn-308"><a href="#Eqn-308"><span class="linenos">308</span></a><span class="sd">    To overcome this specify the order of operations.</span>
+</span><span id="Eqn-309"><a href="#Eqn-309"><span class="linenos">309</span></a><span class="sd">    &gt;&gt;&gt; diff(diff(q,c),b)</span>
+</span><span id="Eqn-310"><a href="#Eqn-310"><span class="linenos">310</span></a><span class="sd">    Equation(Derivative(a, b), -2/c**3)</span>
+</span><span id="Eqn-311"><a href="#Eqn-311"><span class="linenos">311</span></a>
+</span><span id="Eqn-312"><a href="#Eqn-312"><span class="linenos">312</span></a><span class="sd">    But the reverse order returns an unevaulated lhs (a may depend on b).</span>
+</span><span id="Eqn-313"><a href="#Eqn-313"><span class="linenos">313</span></a><span class="sd">    &gt;&gt;&gt; diff(diff(q,b),c)</span>
+</span><span id="Eqn-314"><a href="#Eqn-314"><span class="linenos">314</span></a><span class="sd">    Equation(Derivative(a*c, b, c), -2/c**3)</span>
+</span><span id="Eqn-315"><a href="#Eqn-315"><span class="linenos">315</span></a>
+</span><span id="Eqn-316"><a href="#Eqn-316"><span class="linenos">316</span></a><span class="sd">    Integration can only be performed on one side at a time.</span>
+</span><span id="Eqn-317"><a href="#Eqn-317"><span class="linenos">317</span></a><span class="sd">    &gt;&gt;&gt; q=Eqn(a*c,b/c)</span>
+</span><span id="Eqn-318"><a href="#Eqn-318"><span class="linenos">318</span></a><span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;rhs&#39;)</span>
+</span><span id="Eqn-319"><a href="#Eqn-319"><span class="linenos">319</span></a><span class="sd">    b**2/(2*c)</span>
+</span><span id="Eqn-320"><a href="#Eqn-320"><span class="linenos">320</span></a><span class="sd">    &gt;&gt;&gt; integrate(q,b,side=&#39;lhs&#39;)</span>
+</span><span id="Eqn-321"><a href="#Eqn-321"><span class="linenos">321</span></a><span class="sd">    a*b*c</span>
+</span><span id="Eqn-322"><a href="#Eqn-322"><span class="linenos">322</span></a>
+</span><span id="Eqn-323"><a href="#Eqn-323"><span class="linenos">323</span></a><span class="sd">    Make a pretty statement of integration from an equation</span>
+</span><span id="Eqn-324"><a href="#Eqn-324"><span class="linenos">324</span></a><span class="sd">    &gt;&gt;&gt; Eqn(Integral(q.lhs,b),integrate(q,b,side=&#39;rhs&#39;))</span>
+</span><span id="Eqn-325"><a href="#Eqn-325"><span class="linenos">325</span></a><span class="sd">    Equation(Integral(a*c, b), b**2/(2*c))</span>
+</span><span id="Eqn-326"><a href="#Eqn-326"><span class="linenos">326</span></a>
+</span><span id="Eqn-327"><a href="#Eqn-327"><span class="linenos">327</span></a><span class="sd">    Integration of each side with respect to different variables</span>
+</span><span id="Eqn-328"><a href="#Eqn-328"><span class="linenos">328</span></a><span class="sd">    &gt;&gt;&gt; q.dorhs.integrate(b).dolhs.integrate(a)</span>
+</span><span id="Eqn-329"><a href="#Eqn-329"><span class="linenos">329</span></a><span class="sd">    Equation(a**2*c/2, b**2/(2*c))</span>
+</span><span id="Eqn-330"><a href="#Eqn-330"><span class="linenos">330</span></a>
+</span><span id="Eqn-331"><a href="#Eqn-331"><span class="linenos">331</span></a><span class="sd">    SymPy&#39;s solvers do not understand these equations. They expect an</span>
+</span><span id="Eqn-332"><a href="#Eqn-332"><span class="linenos">332</span></a><span class="sd">    expression that the solver assumes = 0. Thus to use the solver the</span>
+</span><span id="Eqn-333"><a href="#Eqn-333"><span class="linenos">333</span></a><span class="sd">    equation must be rearranged so that all non-zero symbols are on one side.</span>
+</span><span id="Eqn-334"><a href="#Eqn-334"><span class="linenos">334</span></a><span class="sd">    Then just the non-zero symbolic side is passed to ``solve()``.</span>
+</span><span id="Eqn-335"><a href="#Eqn-335"><span class="linenos">335</span></a><span class="sd">    &gt;&gt;&gt; t2 = t-t.rhs</span>
+</span><span id="Eqn-336"><a href="#Eqn-336"><span class="linenos">336</span></a><span class="sd">    &gt;&gt;&gt; t2</span>
+</span><span id="Eqn-337"><a href="#Eqn-337"><span class="linenos">337</span></a><span class="sd">    Equation(a - b/c, 0)</span>
+</span><span id="Eqn-338"><a href="#Eqn-338"><span class="linenos">338</span></a><span class="sd">    &gt;&gt;&gt; solve(t2.lhs,c)</span>
+</span><span id="Eqn-339"><a href="#Eqn-339"><span class="linenos">339</span></a><span class="sd">    [b/a]</span>
+</span><span id="Eqn-340"><a href="#Eqn-340"><span class="linenos">340</span></a>
+</span><span id="Eqn-341"><a href="#Eqn-341"><span class="linenos">341</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="Eqn-342"><a href="#Eqn-342"><span class="linenos">342</span></a>
+</span><span id="Eqn-343"><a href="#Eqn-343"><span class="linenos">343</span></a>    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-344"><a href="#Eqn-344"><span class="linenos">344</span></a>        <span class="n">lhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">lhs</span><span class="p">)</span>
+</span><span id="Eqn-345"><a href="#Eqn-345"><span class="linenos">345</span></a>        <span class="n">rhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="Eqn-346"><a href="#Eqn-346"><span class="linenos">346</span></a>        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
+</span><span id="Eqn-347"><a href="#Eqn-347"><span class="linenos">347</span></a>            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;lhs and rhs must be valid sympy expressions.&#39;</span><span class="p">)</span>
+</span><span id="Eqn-348"><a href="#Eqn-348"><span class="linenos">348</span></a>        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
+</span><span id="Eqn-349"><a href="#Eqn-349"><span class="linenos">349</span></a>
+</span><span id="Eqn-350"><a href="#Eqn-350"><span class="linenos">350</span></a>    <span class="k">def</span> <span class="nf">_get_eqn_name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-351"><a href="#Eqn-351"><span class="linenos">351</span></a>        <span class="kn">from</span> <span class="nn">IPython</span> <span class="kn">import</span> <span class="n">get_ipython</span>
+</span><span id="Eqn-352"><a href="#Eqn-352"><span class="linenos">352</span></a>        <span class="n">global_dict</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">get_ipython</span><span class="p">(),</span> <span class="s1">&#39;user_ns&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Eqn-353"><a href="#Eqn-353"><span class="linenos">353</span></a>        <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="Eqn-354"><a href="#Eqn-354"><span class="linenos">354</span></a>        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
+</span><span id="Eqn-355"><a href="#Eqn-355"><span class="linenos">355</span></a>        <span class="k">if</span> <span class="n">global_dict</span><span class="p">:</span>
+</span><span id="Eqn-356"><a href="#Eqn-356"><span class="linenos">356</span></a>            <span class="k">for</span> <span class="n">var_name</span> <span class="ow">in</span> <span class="n">global_dict</span><span class="p">:</span>
+</span><span id="Eqn-357"><a href="#Eqn-357"><span class="linenos">357</span></a>                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="Eqn-358"><a href="#Eqn-358"><span class="linenos">358</span></a>                    <span class="k">if</span> <span class="p">(</span><span class="n">global_dict</span><span class="p">[</span><span class="n">var_name</span><span class="p">])</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">==</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> \
+</span><span id="Eqn-359"><a href="#Eqn-359"><span class="linenos">359</span></a>                            <span class="n">var_name</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s1">&#39;_&#39;</span><span class="p">):</span>
+</span><span id="Eqn-360"><a href="#Eqn-360"><span class="linenos">360</span></a>                        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
+</span><span id="Eqn-361"><a href="#Eqn-361"><span class="linenos">361</span></a>                        <span class="k">return</span> <span class="n">var_name</span>
+</span><span id="Eqn-362"><a href="#Eqn-362"><span class="linenos">362</span></a>        <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span> <span class="o">=</span> <span class="n">human_text</span>
+</span><span id="Eqn-363"><a href="#Eqn-363"><span class="linenos">363</span></a>        <span class="k">return</span> <span class="s1">&#39;&#39;</span>
+</span><span id="Eqn-364"><a href="#Eqn-364"><span class="linenos">364</span></a>
+</span><span id="Eqn-365"><a href="#Eqn-365"><span class="linenos">365</span></a>    <span class="nd">@property</span>
+</span><span id="Eqn-366"><a href="#Eqn-366"><span class="linenos">366</span></a>    <span class="k">def</span> <span class="nf">lhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-367"><a href="#Eqn-367"><span class="linenos">367</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-368"><a href="#Eqn-368"><span class="linenos">368</span></a><span class="sd">        Returns the lhs of the equation.</span>
+</span><span id="Eqn-369"><a href="#Eqn-369"><span class="linenos">369</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-370"><a href="#Eqn-370"><span class="linenos">370</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+</span><span id="Eqn-371"><a href="#Eqn-371"><span class="linenos">371</span></a>
+</span><span id="Eqn-372"><a href="#Eqn-372"><span class="linenos">372</span></a>    <span class="nd">@property</span>
+</span><span id="Eqn-373"><a href="#Eqn-373"><span class="linenos">373</span></a>    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-374"><a href="#Eqn-374"><span class="linenos">374</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-375"><a href="#Eqn-375"><span class="linenos">375</span></a><span class="sd">        Returns the rhs of the equation.</span>
+</span><span id="Eqn-376"><a href="#Eqn-376"><span class="linenos">376</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-377"><a href="#Eqn-377"><span class="linenos">377</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span><span id="Eqn-378"><a href="#Eqn-378"><span class="linenos">378</span></a>
+</span><span id="Eqn-379"><a href="#Eqn-379"><span class="linenos">379</span></a>    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-380"><a href="#Eqn-380"><span class="linenos">380</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-381"><a href="#Eqn-381"><span class="linenos">381</span></a><span class="sd">        Converts the equation to an Equality.</span>
+</span><span id="Eqn-382"><a href="#Eqn-382"><span class="linenos">382</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-383"><a href="#Eqn-383"><span class="linenos">383</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="Eqn-384"><a href="#Eqn-384"><span class="linenos">384</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="Eqn-385"><a href="#Eqn-385"><span class="linenos">385</span></a>
+</span><span id="Eqn-386"><a href="#Eqn-386"><span class="linenos">386</span></a>    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-387"><a href="#Eqn-387"><span class="linenos">387</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-388"><a href="#Eqn-388"><span class="linenos">388</span></a><span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
+</span><span id="Eqn-389"><a href="#Eqn-389"><span class="linenos">389</span></a><span class="sd">        Parameters</span>
+</span><span id="Eqn-390"><a href="#Eqn-390"><span class="linenos">390</span></a><span class="sd">        ----------</span>
+</span><span id="Eqn-391"><a href="#Eqn-391"><span class="linenos">391</span></a><span class="sd">        kwargs any appropriate for `Equality`.</span>
+</span><span id="Eqn-392"><a href="#Eqn-392"><span class="linenos">392</span></a>
+</span><span id="Eqn-393"><a href="#Eqn-393"><span class="linenos">393</span></a><span class="sd">        Returns</span>
+</span><span id="Eqn-394"><a href="#Eqn-394"><span class="linenos">394</span></a><span class="sd">        -------</span>
+</span><span id="Eqn-395"><a href="#Eqn-395"><span class="linenos">395</span></a><span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
+</span><span id="Eqn-396"><a href="#Eqn-396"><span class="linenos">396</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-397"><a href="#Eqn-397"><span class="linenos">397</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="Eqn-398"><a href="#Eqn-398"><span class="linenos">398</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
+</span><span id="Eqn-399"><a href="#Eqn-399"><span class="linenos">399</span></a>
+</span><span id="Eqn-400"><a href="#Eqn-400"><span class="linenos">400</span></a>    <span class="nd">@property</span>
+</span><span id="Eqn-401"><a href="#Eqn-401"><span class="linenos">401</span></a>    <span class="k">def</span> <span class="nf">reversed</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-402"><a href="#Eqn-402"><span class="linenos">402</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-403"><a href="#Eqn-403"><span class="linenos">403</span></a><span class="sd">        Swaps the lhs and the rhs.</span>
+</span><span id="Eqn-404"><a href="#Eqn-404"><span class="linenos">404</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-405"><a href="#Eqn-405"><span class="linenos">405</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
+</span><span id="Eqn-406"><a href="#Eqn-406"><span class="linenos">406</span></a>
+</span><span id="Eqn-407"><a href="#Eqn-407"><span class="linenos">407</span></a>    <span class="nd">@property</span>
+</span><span id="Eqn-408"><a href="#Eqn-408"><span class="linenos">408</span></a>    <span class="k">def</span> <span class="nf">swap</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-409"><a href="#Eqn-409"><span class="linenos">409</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-410"><a href="#Eqn-410"><span class="linenos">410</span></a><span class="sd">        Synonym for `.reversed`</span>
+</span><span id="Eqn-411"><a href="#Eqn-411"><span class="linenos">411</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-412"><a href="#Eqn-412"><span class="linenos">412</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">reversed</span>
+</span><span id="Eqn-413"><a href="#Eqn-413"><span class="linenos">413</span></a>
+</span><span id="Eqn-414"><a href="#Eqn-414"><span class="linenos">414</span></a>    <span class="k">def</span> <span class="nf">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-415"><a href="#Eqn-415"><span class="linenos">415</span></a>        <span class="c1"># Applies a function to an expression checking whether there</span>
+</span><span id="Eqn-416"><a href="#Eqn-416"><span class="linenos">416</span></a>        <span class="c1"># is a specialized version associated with the particular type of</span>
+</span><span id="Eqn-417"><a href="#Eqn-417"><span class="linenos">417</span></a>        <span class="c1"># expression. Errors will be raised if the function cannot be</span>
+</span><span id="Eqn-418"><a href="#Eqn-418"><span class="linenos">418</span></a>        <span class="c1"># applied to an expression.</span>
+</span><span id="Eqn-419"><a href="#Eqn-419"><span class="linenos">419</span></a>        <span class="n">funcname</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s1">&#39;__name__&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Eqn-420"><a href="#Eqn-420"><span class="linenos">420</span></a>        <span class="k">if</span> <span class="n">funcname</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="Eqn-421"><a href="#Eqn-421"><span class="linenos">421</span></a>            <span class="n">localfunc</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">funcname</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Eqn-422"><a href="#Eqn-422"><span class="linenos">422</span></a>            <span class="k">if</span> <span class="n">localfunc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="Eqn-423"><a href="#Eqn-423"><span class="linenos">423</span></a>                <span class="k">return</span> <span class="n">localfunc</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-424"><a href="#Eqn-424"><span class="linenos">424</span></a>        <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-425"><a href="#Eqn-425"><span class="linenos">425</span></a>
+</span><span id="Eqn-426"><a href="#Eqn-426"><span class="linenos">426</span></a>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-427"><a href="#Eqn-427"><span class="linenos">427</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-428"><a href="#Eqn-428"><span class="linenos">428</span></a><span class="sd">        Apply an operation/function/method to the equation returning the</span>
+</span><span id="Eqn-429"><a href="#Eqn-429"><span class="linenos">429</span></a><span class="sd">        resulting equation.</span>
+</span><span id="Eqn-430"><a href="#Eqn-430"><span class="linenos">430</span></a>
+</span><span id="Eqn-431"><a href="#Eqn-431"><span class="linenos">431</span></a><span class="sd">        Parameters</span>
+</span><span id="Eqn-432"><a href="#Eqn-432"><span class="linenos">432</span></a><span class="sd">        ==========</span>
+</span><span id="Eqn-433"><a href="#Eqn-433"><span class="linenos">433</span></a>
+</span><span id="Eqn-434"><a href="#Eqn-434"><span class="linenos">434</span></a><span class="sd">        func: object</span>
+</span><span id="Eqn-435"><a href="#Eqn-435"><span class="linenos">435</span></a><span class="sd">            object to apply usually a function</span>
+</span><span id="Eqn-436"><a href="#Eqn-436"><span class="linenos">436</span></a>
+</span><span id="Eqn-437"><a href="#Eqn-437"><span class="linenos">437</span></a><span class="sd">        args: as necessary for the function</span>
+</span><span id="Eqn-438"><a href="#Eqn-438"><span class="linenos">438</span></a>
+</span><span id="Eqn-439"><a href="#Eqn-439"><span class="linenos">439</span></a><span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
+</span><span id="Eqn-440"><a href="#Eqn-440"><span class="linenos">440</span></a><span class="sd">            Specifies which side of the equation the operation will be applied</span>
+</span><span id="Eqn-441"><a href="#Eqn-441"><span class="linenos">441</span></a><span class="sd">            to. Default is &#39;both&#39;.</span>
+</span><span id="Eqn-442"><a href="#Eqn-442"><span class="linenos">442</span></a>
+</span><span id="Eqn-443"><a href="#Eqn-443"><span class="linenos">443</span></a><span class="sd">        kwargs: as necessary for the function</span>
+</span><span id="Eqn-444"><a href="#Eqn-444"><span class="linenos">444</span></a><span class="sd">         &quot;&quot;&quot;</span>
+</span><span id="Eqn-445"><a href="#Eqn-445"><span class="linenos">445</span></a>        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
+</span><span id="Eqn-446"><a href="#Eqn-446"><span class="linenos">446</span></a>        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
+</span><span id="Eqn-447"><a href="#Eqn-447"><span class="linenos">447</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
+</span><span id="Eqn-448"><a href="#Eqn-448"><span class="linenos">448</span></a>            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-449"><a href="#Eqn-449"><span class="linenos">449</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
+</span><span id="Eqn-450"><a href="#Eqn-450"><span class="linenos">450</span></a>            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-451"><a href="#Eqn-451"><span class="linenos">451</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
+</span><span id="Eqn-452"><a href="#Eqn-452"><span class="linenos">452</span></a>
+</span><span id="Eqn-453"><a href="#Eqn-453"><span class="linenos">453</span></a>    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-454"><a href="#Eqn-454"><span class="linenos">454</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-455"><a href="#Eqn-455"><span class="linenos">455</span></a><span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="Eqn-456"><a href="#Eqn-456"><span class="linenos">456</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="Eqn-457"><a href="#Eqn-457"><span class="linenos">457</span></a><span class="sd">        The operation is applied to only the lhs.</span>
+</span><span id="Eqn-458"><a href="#Eqn-458"><span class="linenos">458</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-459"><a href="#Eqn-459"><span class="linenos">459</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
+</span><span id="Eqn-460"><a href="#Eqn-460"><span class="linenos">460</span></a>
+</span><span id="Eqn-461"><a href="#Eqn-461"><span class="linenos">461</span></a>    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-462"><a href="#Eqn-462"><span class="linenos">462</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-463"><a href="#Eqn-463"><span class="linenos">463</span></a><span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="Eqn-464"><a href="#Eqn-464"><span class="linenos">464</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="Eqn-465"><a href="#Eqn-465"><span class="linenos">465</span></a><span class="sd">        The operation is applied to only the rhs.</span>
+</span><span id="Eqn-466"><a href="#Eqn-466"><span class="linenos">466</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-467"><a href="#Eqn-467"><span class="linenos">467</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
+</span><span id="Eqn-468"><a href="#Eqn-468"><span class="linenos">468</span></a>
+</span><span id="Eqn-469"><a href="#Eqn-469"><span class="linenos">469</span></a>    <span class="k">class</span> <span class="nc">_sides</span><span class="p">:</span>
+</span><span id="Eqn-470"><a href="#Eqn-470"><span class="linenos">470</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn-471"><a href="#Eqn-471"><span class="linenos">471</span></a><span class="sd">        Helper class for the `.do.`, `.dolhs.`, `.dorhs.` syntax for applying</span>
+</span><span id="Eqn-472"><a href="#Eqn-472"><span class="linenos">472</span></a><span class="sd">        submethods of expressions.</span>
+</span><span id="Eqn-473"><a href="#Eqn-473"><span class="linenos">473</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn-474"><a href="#Eqn-474"><span class="linenos">474</span></a>
+</span><span id="Eqn-475"><a href="#Eqn-475"><span class="linenos">475</span></a>        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eqn</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">):</span>
+</span><span id="Eqn-476"><a href="#Eqn-476"><span class="linenos">476</span></a>            <span class="bp">self</span><span class="o">.</span><span class="n">eqn</span> <span class="o">=</span> <span class="n">eqn</span>
+</span><span id="Eqn-477"><a href="#Eqn-477"><span class="linenos">477</span></a>            <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="o">=</span> <span class="n">side</span>
+</span><span id="Eqn-478"><a href="#Eqn-478"><span class="linenos">478</span></a>
+</span><span id="Eqn-479"><a href="#Eqn-479"><span class="linenos">479</span></a>        <span class="k">def</span> <span class="fm">__getattr__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
+</span><span id="Eqn-480"><a href="#Eqn-480"><span class="linenos">480</span></a>            <span class="n">func</span> <span class="o">=</span> <span class="kc">None</span>
+</span><span id="Eqn-481"><a href="#Eqn-481"><span class="linenos">481</span></a>            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;rhs&#39;</span><span class="p">,</span> <span class="s1">&#39;both&#39;</span><span class="p">):</span>
+</span><span id="Eqn-482"><a href="#Eqn-482"><span class="linenos">482</span></a>                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Eqn-483"><a href="#Eqn-483"><span class="linenos">483</span></a>            <span class="k">else</span><span class="p">:</span>
+</span><span id="Eqn-484"><a href="#Eqn-484"><span class="linenos">484</span></a>                <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="Eqn-485"><a href="#Eqn-485"><span class="linenos">485</span></a>            <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="Eqn-486"><a href="#Eqn-486"><span class="linenos">486</span></a>                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;Expressions in the equation have no &#39;</span>
+</span><span id="Eqn-487"><a href="#Eqn-487"><span class="linenos">487</span></a>                                     <span class="s1">&#39;attribute `&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span>
+</span><span id="Eqn-488"><a href="#Eqn-488"><span class="linenos">488</span></a>                    <span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;`. Try `.apply(&#39;</span>
+</span><span id="Eqn-489"><a href="#Eqn-489"><span class="linenos">489</span></a>                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;, *args)` or &#39;</span>
+</span><span id="Eqn-490"><a href="#Eqn-490"><span class="linenos">490</span></a>                                                   <span class="s1">&#39;pass the equation as a parameter to `&#39;</span>
+</span><span id="Eqn-491"><a href="#Eqn-491"><span class="linenos">491</span></a>                                     <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;()`.&#39;</span><span class="p">)</span>
+</span><span id="Eqn-492"><a href="#Eqn-492"><span class="linenos">492</span></a>            <span class="k">return</span> <span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eqn</span><span class="o">.</span><span class="n">apply</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">side</span><span class="p">)</span>
+</span><span id="Eqn-493"><a href="#Eqn-493"><span class="linenos">493</span></a>
+</span><span id="Eqn-494"><a href="#Eqn-494"><span class="linenos">494</span></a>    <span class="nd">@property</span>
+</span><span id="Eqn-495"><a href="#Eqn-495"><span class="linenos">495</span></a>    <span class="k">def</span> <span class="nf">do</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-496"><a href="#Eqn-496"><span class="linenos">496</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">)</span>
+</span><span id="Eqn-497"><a href="#Eqn-497"><span class="linenos">497</span></a>
+</span><span id="Eqn-498"><a href="#Eqn-498"><span class="linenos">498</span></a>    <span class="nd">@property</span>
+</span><span id="Eqn-499"><a href="#Eqn-499"><span class="linenos">499</span></a>    <span class="k">def</span> <span class="nf">dolhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-500"><a href="#Eqn-500"><span class="linenos">500</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
+</span><span id="Eqn-501"><a href="#Eqn-501"><span class="linenos">501</span></a>
+</span><span id="Eqn-502"><a href="#Eqn-502"><span class="linenos">502</span></a>    <span class="nd">@property</span>
+</span><span id="Eqn-503"><a href="#Eqn-503"><span class="linenos">503</span></a>    <span class="k">def</span> <span class="nf">dorhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-504"><a href="#Eqn-504"><span class="linenos">504</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sides</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
+</span><span id="Eqn-505"><a href="#Eqn-505"><span class="linenos">505</span></a>
+</span><span id="Eqn-506"><a href="#Eqn-506"><span class="linenos">506</span></a>    <span class="c1">#####</span>
+</span><span id="Eqn-507"><a href="#Eqn-507"><span class="linenos">507</span></a>    <span class="c1"># Overrides of binary math operations</span>
+</span><span id="Eqn-508"><a href="#Eqn-508"><span class="linenos">508</span></a>    <span class="c1">#####</span>
+</span><span id="Eqn-509"><a href="#Eqn-509"><span class="linenos">509</span></a>
+</span><span id="Eqn-510"><a href="#Eqn-510"><span class="linenos">510</span></a>    <span class="nd">@classmethod</span>
+</span><span id="Eqn-511"><a href="#Eqn-511"><span class="linenos">511</span></a>    <span class="k">def</span> <span class="nf">_binary_op</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">opfunc_ab</span><span class="p">):</span>
+</span><span id="Eqn-512"><a href="#Eqn-512"><span class="linenos">512</span></a>        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="Eqn-513"><a href="#Eqn-513"><span class="linenos">513</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
+</span><span id="Eqn-514"><a href="#Eqn-514"><span class="linenos">514</span></a>        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="Eqn-515"><a href="#Eqn-515"><span class="linenos">515</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
+</span><span id="Eqn-516"><a href="#Eqn-516"><span class="linenos">516</span></a>        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Equation</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="Eqn-517"><a href="#Eqn-517"><span class="linenos">517</span></a>            <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span> <span class="n">opfunc_ab</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
+</span><span id="Eqn-518"><a href="#Eqn-518"><span class="linenos">518</span></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="Eqn-519"><a href="#Eqn-519"><span class="linenos">519</span></a>            <span class="k">return</span> <span class="bp">NotImplemented</span>
+</span><span id="Eqn-520"><a href="#Eqn-520"><span class="linenos">520</span></a>
+</span><span id="Eqn-521"><a href="#Eqn-521"><span class="linenos">521</span></a>    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-522"><a href="#Eqn-522"><span class="linenos">522</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-523"><a href="#Eqn-523"><span class="linenos">523</span></a>
+</span><span id="Eqn-524"><a href="#Eqn-524"><span class="linenos">524</span></a>    <span class="k">def</span> <span class="fm">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-525"><a href="#Eqn-525"><span class="linenos">525</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-526"><a href="#Eqn-526"><span class="linenos">526</span></a>
+</span><span id="Eqn-527"><a href="#Eqn-527"><span class="linenos">527</span></a>    <span class="k">def</span> <span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-528"><a href="#Eqn-528"><span class="linenos">528</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-529"><a href="#Eqn-529"><span class="linenos">529</span></a>
+</span><span id="Eqn-530"><a href="#Eqn-530"><span class="linenos">530</span></a>    <span class="k">def</span> <span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-531"><a href="#Eqn-531"><span class="linenos">531</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-532"><a href="#Eqn-532"><span class="linenos">532</span></a>
+</span><span id="Eqn-533"><a href="#Eqn-533"><span class="linenos">533</span></a>    <span class="k">def</span> <span class="fm">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-534"><a href="#Eqn-534"><span class="linenos">534</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-535"><a href="#Eqn-535"><span class="linenos">535</span></a>
+</span><span id="Eqn-536"><a href="#Eqn-536"><span class="linenos">536</span></a>    <span class="k">def</span> <span class="fm">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-537"><a href="#Eqn-537"><span class="linenos">537</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-538"><a href="#Eqn-538"><span class="linenos">538</span></a>
+</span><span id="Eqn-539"><a href="#Eqn-539"><span class="linenos">539</span></a>    <span class="k">def</span> <span class="fm">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-540"><a href="#Eqn-540"><span class="linenos">540</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-541"><a href="#Eqn-541"><span class="linenos">541</span></a>
+</span><span id="Eqn-542"><a href="#Eqn-542"><span class="linenos">542</span></a>    <span class="k">def</span> <span class="fm">__rtruediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-543"><a href="#Eqn-543"><span class="linenos">543</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-544"><a href="#Eqn-544"><span class="linenos">544</span></a>
+</span><span id="Eqn-545"><a href="#Eqn-545"><span class="linenos">545</span></a>    <span class="k">def</span> <span class="fm">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-546"><a href="#Eqn-546"><span class="linenos">546</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-547"><a href="#Eqn-547"><span class="linenos">547</span></a>
+</span><span id="Eqn-548"><a href="#Eqn-548"><span class="linenos">548</span></a>    <span class="k">def</span> <span class="fm">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-549"><a href="#Eqn-549"><span class="linenos">549</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-550"><a href="#Eqn-550"><span class="linenos">550</span></a>
+</span><span id="Eqn-551"><a href="#Eqn-551"><span class="linenos">551</span></a>    <span class="k">def</span> <span class="fm">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-552"><a href="#Eqn-552"><span class="linenos">552</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-553"><a href="#Eqn-553"><span class="linenos">553</span></a>
+</span><span id="Eqn-554"><a href="#Eqn-554"><span class="linenos">554</span></a>    <span class="k">def</span> <span class="fm">__rpow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-555"><a href="#Eqn-555"><span class="linenos">555</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_binary_op</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">**</span> <span class="n">b</span><span class="p">)</span>
+</span><span id="Eqn-556"><a href="#Eqn-556"><span class="linenos">556</span></a>
+</span><span id="Eqn-557"><a href="#Eqn-557"><span class="linenos">557</span></a>    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+</span><span id="Eqn-558"><a href="#Eqn-558"><span class="linenos">558</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__pow__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
+</span><span id="Eqn-559"><a href="#Eqn-559"><span class="linenos">559</span></a>
+</span><span id="Eqn-560"><a href="#Eqn-560"><span class="linenos">560</span></a>    <span class="c1">#####</span>
+</span><span id="Eqn-561"><a href="#Eqn-561"><span class="linenos">561</span></a>    <span class="c1"># Operation helper functions</span>
+</span><span id="Eqn-562"><a href="#Eqn-562"><span class="linenos">562</span></a>    <span class="c1">#####</span>
+</span><span id="Eqn-563"><a href="#Eqn-563"><span class="linenos">563</span></a>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-564"><a href="#Eqn-564"><span class="linenos">564</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
+</span><span id="Eqn-565"><a href="#Eqn-565"><span class="linenos">565</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Eqn-566"><a href="#Eqn-566"><span class="linenos">566</span></a>
+</span><span id="Eqn-567"><a href="#Eqn-567"><span class="linenos">567</span></a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-568"><a href="#Eqn-568"><span class="linenos">568</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-569"><a href="#Eqn-569"><span class="linenos">569</span></a>
+</span><span id="Eqn-570"><a href="#Eqn-570"><span class="linenos">570</span></a>    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-571"><a href="#Eqn-571"><span class="linenos">571</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span>
+</span><span id="Eqn-572"><a href="#Eqn-572"><span class="linenos">572</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Eqn-573"><a href="#Eqn-573"><span class="linenos">573</span></a>
+</span><span id="Eqn-574"><a href="#Eqn-574"><span class="linenos">574</span></a>    <span class="k">def</span> <span class="nf">_eval_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-575"><a href="#Eqn-575"><span class="linenos">575</span></a>        <span class="c1"># TODO: cancel out factors common to both sides.</span>
+</span><span id="Eqn-576"><a href="#Eqn-576"><span class="linenos">576</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">factor</span><span class="p">(</span>
+</span><span id="Eqn-577"><a href="#Eqn-577"><span class="linenos">577</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Eqn-578"><a href="#Eqn-578"><span class="linenos">578</span></a>
+</span><span id="Eqn-579"><a href="#Eqn-579"><span class="linenos">579</span></a>    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-580"><a href="#Eqn-580"><span class="linenos">580</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-581"><a href="#Eqn-581"><span class="linenos">581</span></a>
+</span><span id="Eqn-582"><a href="#Eqn-582"><span class="linenos">582</span></a>    <span class="k">def</span> <span class="nf">_eval_collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-583"><a href="#Eqn-583"><span class="linenos">583</span></a>        <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
+</span><span id="Eqn-584"><a href="#Eqn-584"><span class="linenos">584</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="Eqn-585"><a href="#Eqn-585"><span class="linenos">585</span></a>                        <span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Eqn-586"><a href="#Eqn-586"><span class="linenos">586</span></a>
+</span><span id="Eqn-587"><a href="#Eqn-587"><span class="linenos">587</span></a>    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-588"><a href="#Eqn-588"><span class="linenos">588</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-589"><a href="#Eqn-589"><span class="linenos">589</span></a>
+</span><span id="Eqn-590"><a href="#Eqn-590"><span class="linenos">590</span></a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-591"><a href="#Eqn-591"><span class="linenos">591</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="Eqn-592"><a href="#Eqn-592"><span class="linenos">592</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Eqn-593"><a href="#Eqn-593"><span class="linenos">593</span></a>
+</span><span id="Eqn-594"><a href="#Eqn-594"><span class="linenos">594</span></a>    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
+</span><span id="Eqn-595"><a href="#Eqn-595"><span class="linenos">595</span></a>
+</span><span id="Eqn-596"><a href="#Eqn-596"><span class="linenos">596</span></a>    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-597"><a href="#Eqn-597"><span class="linenos">597</span></a>        <span class="c1"># TODO Find why diff and Derivative do not appear to pass through</span>
+</span><span id="Eqn-598"><a href="#Eqn-598"><span class="linenos">598</span></a>        <span class="c1">#  kwargs to this. Since we cannot set evaluation of lhs manually</span>
+</span><span id="Eqn-599"><a href="#Eqn-599"><span class="linenos">599</span></a>        <span class="c1">#  try to be intelligent about when to do it.</span>
+</span><span id="Eqn-600"><a href="#Eqn-600"><span class="linenos">600</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Derivative</span>
+</span><span id="Eqn-601"><a href="#Eqn-601"><span class="linenos">601</span></a>        <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">False</span>
+</span><span id="Eqn-602"><a href="#Eqn-602"><span class="linenos">602</span></a>        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)):</span>
+</span><span id="Eqn-603"><a href="#Eqn-603"><span class="linenos">603</span></a>            <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
+</span><span id="Eqn-604"><a href="#Eqn-604"><span class="linenos">604</span></a>                <span class="k">if</span> <span class="n">sym</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span>
+</span><span id="Eqn-605"><a href="#Eqn-605"><span class="linenos">605</span></a>                        <span class="n">_sympify</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">):</span>
+</span><span id="Eqn-606"><a href="#Eqn-606"><span class="linenos">606</span></a>                    <span class="n">eval_lhs</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="Eqn-607"><a href="#Eqn-607"><span class="linenos">607</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="n">eval_lhs</span><span class="p">),</span>
+</span><span id="Eqn-608"><a href="#Eqn-608"><span class="linenos">608</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Eqn-609"><a href="#Eqn-609"><span class="linenos">609</span></a>
+</span><span id="Eqn-610"><a href="#Eqn-610"><span class="linenos">610</span></a>    <span class="k">def</span> <span class="nf">_eval_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-611"><a href="#Eqn-611"><span class="linenos">611</span></a>        <span class="n">side</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s1">&#39;side&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>  <span class="c1"># Could not seem to pass values for</span>
+</span><span id="Eqn-612"><a href="#Eqn-612"><span class="linenos">612</span></a>        <span class="c1"># `evaluate` through to here.</span>
+</span><span id="Eqn-613"><a href="#Eqn-613"><span class="linenos">613</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="Eqn-614"><a href="#Eqn-614"><span class="linenos">614</span></a>            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;You must specify `side=&quot;lhs&quot;` or `side=&quot;rhs&quot;` &#39;</span>
+</span><span id="Eqn-615"><a href="#Eqn-615"><span class="linenos">615</span></a>                             <span class="s1">&#39;when integrating an Equation&#39;</span><span class="p">)</span>
+</span><span id="Eqn-616"><a href="#Eqn-616"><span class="linenos">616</span></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="Eqn-617"><a href="#Eqn-617"><span class="linenos">617</span></a>            <span class="k">try</span><span class="p">:</span>
+</span><span id="Eqn-618"><a href="#Eqn-618"><span class="linenos">618</span></a>                <span class="k">return</span> <span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">side</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span><span id="Eqn-619"><a href="#Eqn-619"><span class="linenos">619</span></a>            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
+</span><span id="Eqn-620"><a href="#Eqn-620"><span class="linenos">620</span></a>                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s1">&#39;`side` must equal &quot;lhs&quot; or &quot;rhs&quot;.&#39;</span><span class="p">)</span>
+</span><span id="Eqn-621"><a href="#Eqn-621"><span class="linenos">621</span></a>
+</span><span id="Eqn-622"><a href="#Eqn-622"><span class="linenos">622</span></a>    <span class="c1">#####</span>
+</span><span id="Eqn-623"><a href="#Eqn-623"><span class="linenos">623</span></a>    <span class="c1"># Output helper functions</span>
+</span><span id="Eqn-624"><a href="#Eqn-624"><span class="linenos">624</span></a>    <span class="c1">#####</span>
+</span><span id="Eqn-625"><a href="#Eqn-625"><span class="linenos">625</span></a>    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-626"><a href="#Eqn-626"><span class="linenos">626</span></a>        <span class="n">repstr</span> <span class="o">=</span> <span class="s1">&#39;Equation(</span><span class="si">%s</span><span class="s1">, </span><span class="si">%s</span><span class="s1">)&#39;</span> <span class="o">%</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">())</span>
+</span><span id="Eqn-627"><a href="#Eqn-627"><span class="linenos">627</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
+</span><span id="Eqn-628"><a href="#Eqn-628"><span class="linenos">628</span></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__str__</span><span class="p">()</span>
+</span><span id="Eqn-629"><a href="#Eqn-629"><span class="linenos">629</span></a>        <span class="k">return</span> <span class="n">repstr</span>
+</span><span id="Eqn-630"><a href="#Eqn-630"><span class="linenos">630</span></a>
+</span><span id="Eqn-631"><a href="#Eqn-631"><span class="linenos">631</span></a>    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn-632"><a href="#Eqn-632"><span class="linenos">632</span></a>        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
+</span><span id="Eqn-633"><a href="#Eqn-633"><span class="linenos">633</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span> <span class="ow">and</span> <span class="ow">not</span> \
+</span><span id="Eqn-634"><a href="#Eqn-634"><span class="linenos">634</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="p">:</span>
+</span><span id="Eqn-635"><a href="#Eqn-635"><span class="linenos">635</span></a>            <span class="n">tempstr</span> <span class="o">+=</span><span class="s1">&#39;</span><span class="se">\\</span><span class="s1">text{code version: &#39;</span><span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span><span class="o">+</span><span class="s1">&#39;} </span><span class="se">\\</span><span class="s1">newline &#39;</span>
+</span><span id="Eqn-636"><a href="#Eqn-636"><span class="linenos">636</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-637"><a href="#Eqn-637"><span class="linenos">637</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;=&#39;</span>
+</span><span id="Eqn-638"><a href="#Eqn-638"><span class="linenos">638</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn-639"><a href="#Eqn-639"><span class="linenos">639</span></a>        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
+</span><span id="Eqn-640"><a href="#Eqn-640"><span class="linenos">640</span></a>        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span><span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
+</span><span id="Eqn-641"><a href="#Eqn-641"><span class="linenos">641</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,</span><span class="se">\\</span><span class="s1">,&#39;</span>
+</span><span id="Eqn-642"><a href="#Eqn-642"><span class="linenos">642</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;(</span><span class="se">\\</span><span class="s1">text{&#39;</span><span class="o">+</span><span class="n">namestr</span><span class="o">+</span><span class="s1">&#39;})&#39;</span>
+</span><span id="Eqn-643"><a href="#Eqn-643"><span class="linenos">643</span></a>        <span class="k">return</span> <span class="n">tempstr</span>
+</span><span id="Eqn-644"><a href="#Eqn-644"><span class="linenos">644</span></a>
+</span><span id="Eqn-645"><a href="#Eqn-645"><span class="linenos">645</span></a>    <span class="k">def</span> <span class="fm">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn-646"><a href="#Eqn-646"><span class="linenos">646</span></a>        <span class="n">tempstr</span> <span class="o">=</span> <span class="s1">&#39;&#39;</span>
+</span><span id="Eqn-647"><a href="#Eqn-647"><span class="linenos">647</span></a>        <span class="k">if</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">show_code</span><span class="p">:</span>
+</span><span id="Eqn-648"><a href="#Eqn-648"><span class="linenos">648</span></a>            <span class="n">human_text</span> <span class="o">=</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span>
+</span><span id="Eqn-649"><a href="#Eqn-649"><span class="linenos">649</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="kc">False</span>
+</span><span id="Eqn-650"><a href="#Eqn-650"><span class="linenos">650</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;code version: &#39;</span><span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span> <span class="o">+</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
+</span><span id="Eqn-651"><a href="#Eqn-651"><span class="linenos">651</span></a>            <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">human_text</span><span class="o">=</span><span class="n">human_text</span>
+</span><span id="Eqn-652"><a href="#Eqn-652"><span class="linenos">652</span></a>        <span class="n">tempstr</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39; = &#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="Eqn-653"><a href="#Eqn-653"><span class="linenos">653</span></a>        <span class="n">namestr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_eqn_name</span><span class="p">()</span>
+</span><span id="Eqn-654"><a href="#Eqn-654"><span class="linenos">654</span></a>        <span class="k">if</span> <span class="n">namestr</span> <span class="o">!=</span> <span class="s1">&#39;&#39;</span> <span class="ow">and</span> <span class="n">algwsym_config</span><span class="o">.</span><span class="n">output</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
+</span><span id="Eqn-655"><a href="#Eqn-655"><span class="linenos">655</span></a>            <span class="n">tempstr</span> <span class="o">+=</span> <span class="s1">&#39;          (&#39;</span> <span class="o">+</span> <span class="n">namestr</span> <span class="o">+</span> <span class="s1">&#39;)&#39;</span>
+</span><span id="Eqn-656"><a href="#Eqn-656"><span class="linenos">656</span></a>        <span class="k">return</span> <span class="n">tempstr</span>
+</span></pre></div>
+
 
             <div class="docstring"><p>This class defines an equation with a left-hand-side (lhs) and a right-
 hand-side (rhs) connected by the "=" operator (e.g. <code>p*V = n*R*T</code>).</p>
@@ -5607,22 +5667,24 @@ <h1 id="examples">Examples</h1>
 
 
                             <div id="Eqn.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Eqn</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Eqn</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Eqn.__init__"></a>
     
     
 
                             </div>
                             <div id="Eqn.lhs" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Eqn.lhs">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">lhs</span>
 
-        <span class="name">lhs</span>
+        
     </div>
-
+    <a class="headerlink" href="#Eqn.lhs"></a>
     
             <div class="docstring"><p>Returns the lhs of the equation.</p>
 </div>
@@ -5630,11 +5692,12 @@ <h1 id="examples">Examples</h1>
 
                             </div>
                             <div id="Eqn.rhs" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Eqn.rhs">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">rhs</span>
 
-        <span class="name">rhs</span>
+        
     </div>
-
+    <a class="headerlink" href="#Eqn.rhs"></a>
     
             <div class="docstring"><p>Returns the rhs of the equation.</p>
 </div>
@@ -5642,24 +5705,24 @@ <h1 id="examples">Examples</h1>
 
                             </div>
                             <div id="Eqn.as_Boolean" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.as_Boolean">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.as_Boolean-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">as_Boolean</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">as_Boolean</span><span class="signature">(self)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.as_Boolean-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Converts the equation to an Equality.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.as_Boolean"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.as_Boolean-379"><a href="#Eqn.as_Boolean-379"><span class="linenos">379</span></a>    <span class="k">def</span> <span class="nf">as_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eqn.as_Boolean-380"><a href="#Eqn.as_Boolean-380"><span class="linenos">380</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn.as_Boolean-381"><a href="#Eqn.as_Boolean-381"><span class="linenos">381</span></a><span class="sd">        Converts the equation to an Equality.</span>
+</span><span id="Eqn.as_Boolean-382"><a href="#Eqn.as_Boolean-382"><span class="linenos">382</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn.as_Boolean-383"><a href="#Eqn.as_Boolean-383"><span class="linenos">383</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="Eqn.as_Boolean-384"><a href="#Eqn.as_Boolean-384"><span class="linenos">384</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>Converts the equation to an Equality.</p>
 </div>
@@ -5667,31 +5730,31 @@ <h1 id="examples">Examples</h1>
 
                             </div>
                             <div id="Eqn.check" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.check">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.check-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">check</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
+
+                <label class="view-source-button" for="Eqn.check-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Eqn.check"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.check-386"><a href="#Eqn.check-386"><span class="linenos">386</span></a>    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.check-387"><a href="#Eqn.check-387"><span class="linenos">387</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn.check-388"><a href="#Eqn.check-388"><span class="linenos">388</span></a><span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
+</span><span id="Eqn.check-389"><a href="#Eqn.check-389"><span class="linenos">389</span></a><span class="sd">        Parameters</span>
+</span><span id="Eqn.check-390"><a href="#Eqn.check-390"><span class="linenos">390</span></a><span class="sd">        ----------</span>
+</span><span id="Eqn.check-391"><a href="#Eqn.check-391"><span class="linenos">391</span></a><span class="sd">        kwargs any appropriate for `Equality`.</span>
+</span><span id="Eqn.check-392"><a href="#Eqn.check-392"><span class="linenos">392</span></a>
+</span><span id="Eqn.check-393"><a href="#Eqn.check-393"><span class="linenos">393</span></a><span class="sd">        Returns</span>
+</span><span id="Eqn.check-394"><a href="#Eqn.check-394"><span class="linenos">394</span></a><span class="sd">        -------</span>
+</span><span id="Eqn.check-395"><a href="#Eqn.check-395"><span class="linenos">395</span></a><span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
+</span><span id="Eqn.check-396"><a href="#Eqn.check-396"><span class="linenos">396</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn.check-397"><a href="#Eqn.check-397"><span class="linenos">397</span></a>        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
+</span><span id="Eqn.check-398"><a href="#Eqn.check-398"><span class="linenos">398</span></a>        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
+</span></pre></div>
 
-        
-            <span class="def">def</span>
-            <span class="name">check</span><span class="signature">(self, **kwargs)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Forces simplification and casts as `Equality` to check validity.</span>
-<span class="sd">        Parameters</span>
-<span class="sd">        ----------</span>
-<span class="sd">        kwargs any appropriate for `Equality`.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        -------</span>
-<span class="sd">        True, False or an unevaluated `Equality` if truth cannot be determined.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-        <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-</pre></div>
-
-        </details>
 
             <div class="docstring"><p>Forces simplification and casts as <code>Equality</code> to check validity.</p>
 
@@ -5707,11 +5770,12 @@ <h2 id="returns">Returns</h2>
 
                             </div>
                             <div id="Eqn.reversed" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Eqn.reversed">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">reversed</span>
 
-        <span class="name">reversed</span>
+        
     </div>
-
+    <a class="headerlink" href="#Eqn.reversed"></a>
     
             <div class="docstring"><p>Swaps the lhs and the rhs.</p>
 </div>
@@ -5719,11 +5783,12 @@ <h2 id="returns">Returns</h2>
 
                             </div>
                             <div id="Eqn.swap" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Eqn.swap">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">swap</span>
 
-        <span class="name">swap</span>
+        
     </div>
-
+    <a class="headerlink" href="#Eqn.swap"></a>
     
             <div class="docstring"><p>Synonym for <code>.reversed</code></p>
 </div>
@@ -5731,44 +5796,44 @@ <h2 id="returns">Returns</h2>
 
                             </div>
                             <div id="Eqn.apply" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.apply">#&nbsp;&nbsp</a>
-
-        
-            <span class="def">def</span>
-            <span class="name">apply</span><span class="signature">(self, func, *args, side=&#39;both&#39;, **kwargs)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply an operation/function/method to the equation returning the</span>
-<span class="sd">        resulting equation.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        func: object</span>
-<span class="sd">            object to apply usually a function</span>
+                                        <input id="Eqn.apply-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">apply</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="n">func</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
+
+                <label class="view-source-button" for="Eqn.apply-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Eqn.apply"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.apply-426"><a href="#Eqn.apply-426"><span class="linenos">426</span></a>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.apply-427"><a href="#Eqn.apply-427"><span class="linenos">427</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn.apply-428"><a href="#Eqn.apply-428"><span class="linenos">428</span></a><span class="sd">        Apply an operation/function/method to the equation returning the</span>
+</span><span id="Eqn.apply-429"><a href="#Eqn.apply-429"><span class="linenos">429</span></a><span class="sd">        resulting equation.</span>
+</span><span id="Eqn.apply-430"><a href="#Eqn.apply-430"><span class="linenos">430</span></a>
+</span><span id="Eqn.apply-431"><a href="#Eqn.apply-431"><span class="linenos">431</span></a><span class="sd">        Parameters</span>
+</span><span id="Eqn.apply-432"><a href="#Eqn.apply-432"><span class="linenos">432</span></a><span class="sd">        ==========</span>
+</span><span id="Eqn.apply-433"><a href="#Eqn.apply-433"><span class="linenos">433</span></a>
+</span><span id="Eqn.apply-434"><a href="#Eqn.apply-434"><span class="linenos">434</span></a><span class="sd">        func: object</span>
+</span><span id="Eqn.apply-435"><a href="#Eqn.apply-435"><span class="linenos">435</span></a><span class="sd">            object to apply usually a function</span>
+</span><span id="Eqn.apply-436"><a href="#Eqn.apply-436"><span class="linenos">436</span></a>
+</span><span id="Eqn.apply-437"><a href="#Eqn.apply-437"><span class="linenos">437</span></a><span class="sd">        args: as necessary for the function</span>
+</span><span id="Eqn.apply-438"><a href="#Eqn.apply-438"><span class="linenos">438</span></a>
+</span><span id="Eqn.apply-439"><a href="#Eqn.apply-439"><span class="linenos">439</span></a><span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
+</span><span id="Eqn.apply-440"><a href="#Eqn.apply-440"><span class="linenos">440</span></a><span class="sd">            Specifies which side of the equation the operation will be applied</span>
+</span><span id="Eqn.apply-441"><a href="#Eqn.apply-441"><span class="linenos">441</span></a><span class="sd">            to. Default is &#39;both&#39;.</span>
+</span><span id="Eqn.apply-442"><a href="#Eqn.apply-442"><span class="linenos">442</span></a>
+</span><span id="Eqn.apply-443"><a href="#Eqn.apply-443"><span class="linenos">443</span></a><span class="sd">        kwargs: as necessary for the function</span>
+</span><span id="Eqn.apply-444"><a href="#Eqn.apply-444"><span class="linenos">444</span></a><span class="sd">         &quot;&quot;&quot;</span>
+</span><span id="Eqn.apply-445"><a href="#Eqn.apply-445"><span class="linenos">445</span></a>        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
+</span><span id="Eqn.apply-446"><a href="#Eqn.apply-446"><span class="linenos">446</span></a>        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
+</span><span id="Eqn.apply-447"><a href="#Eqn.apply-447"><span class="linenos">447</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
+</span><span id="Eqn.apply-448"><a href="#Eqn.apply-448"><span class="linenos">448</span></a>            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn.apply-449"><a href="#Eqn.apply-449"><span class="linenos">449</span></a>        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
+</span><span id="Eqn.apply-450"><a href="#Eqn.apply-450"><span class="linenos">450</span></a>            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="Eqn.apply-451"><a href="#Eqn.apply-451"><span class="linenos">451</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
+</span></pre></div>
 
-<span class="sd">        args: as necessary for the function</span>
-
-<span class="sd">        side: &#39;both&#39;, &#39;lhs&#39;, &#39;rhs&#39;, optional</span>
-<span class="sd">            Specifies which side of the equation the operation will be applied</span>
-<span class="sd">            to. Default is &#39;both&#39;.</span>
-
-<span class="sd">        kwargs: as necessary for the function</span>
-<span class="sd">         &quot;&quot;&quot;</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;lhs&#39;</span><span class="p">):</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">side</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;both&#39;</span><span class="p">,</span> <span class="s1">&#39;rhs&#39;</span><span class="p">):</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_applytoexpr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-</pre></div>
-
-        </details>
 
             <div class="docstring"><p>Apply an operation/function/method to the equation returning the
 resulting equation.</p>
@@ -5790,25 +5855,25 @@ <h1 id="parameters">Parameters</h1>
 
                             </div>
                             <div id="Eqn.applylhs" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.applylhs">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.applylhs-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">applylhs</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="n">func</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">applylhs</span><span class="signature">(self, func, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.applylhs-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the lhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.applylhs"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.applylhs-453"><a href="#Eqn.applylhs-453"><span class="linenos">453</span></a>    <span class="k">def</span> <span class="nf">applylhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.applylhs-454"><a href="#Eqn.applylhs-454"><span class="linenos">454</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn.applylhs-455"><a href="#Eqn.applylhs-455"><span class="linenos">455</span></a><span class="sd">        If lhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="Eqn.applylhs-456"><a href="#Eqn.applylhs-456"><span class="linenos">456</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="Eqn.applylhs-457"><a href="#Eqn.applylhs-457"><span class="linenos">457</span></a><span class="sd">        The operation is applied to only the lhs.</span>
+</span><span id="Eqn.applylhs-458"><a href="#Eqn.applylhs-458"><span class="linenos">458</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn.applylhs-459"><a href="#Eqn.applylhs-459"><span class="linenos">459</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;lhs&#39;</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>If lhs side of the equation has a defined subfunction (attribute) of
 name <code><a href="#Eqn.func">func</a></code>, that will be applied instead of the global function.
@@ -5818,25 +5883,25 @@ <h1 id="parameters">Parameters</h1>
 
                             </div>
                             <div id="Eqn.applyrhs" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.applyrhs">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.applyrhs-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">applyrhs</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="n">func</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">applyrhs</span><span class="signature">(self, func, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.applyrhs-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
-<span class="sd">        name ``func``, that will be applied instead of the global function.</span>
-<span class="sd">        The operation is applied to only the rhs.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.applyrhs"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.applyrhs-461"><a href="#Eqn.applyrhs-461"><span class="linenos">461</span></a>    <span class="k">def</span> <span class="nf">applyrhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.applyrhs-462"><a href="#Eqn.applyrhs-462"><span class="linenos">462</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eqn.applyrhs-463"><a href="#Eqn.applyrhs-463"><span class="linenos">463</span></a><span class="sd">        If rhs side of the equation has a defined subfunction (attribute) of</span>
+</span><span id="Eqn.applyrhs-464"><a href="#Eqn.applyrhs-464"><span class="linenos">464</span></a><span class="sd">        name ``func``, that will be applied instead of the global function.</span>
+</span><span id="Eqn.applyrhs-465"><a href="#Eqn.applyrhs-465"><span class="linenos">465</span></a><span class="sd">        The operation is applied to only the rhs.</span>
+</span><span id="Eqn.applyrhs-466"><a href="#Eqn.applyrhs-466"><span class="linenos">466</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eqn.applyrhs-467"><a href="#Eqn.applyrhs-467"><span class="linenos">467</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span> <span class="n">side</span><span class="o">=</span><span class="s1">&#39;rhs&#39;</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>If rhs side of the equation has a defined subfunction (attribute) of
 name <code><a href="#Eqn.func">func</a></code>, that will be applied instead of the global function.
@@ -5846,70 +5911,73 @@ <h1 id="parameters">Parameters</h1>
 
                             </div>
                             <div id="Eqn.do" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Eqn.do">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">do</span>
 
-        <span class="name">do</span>
+        
     </div>
-
+    <a class="headerlink" href="#Eqn.do"></a>
     
     
 
                             </div>
                             <div id="Eqn.dolhs" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Eqn.dolhs">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">dolhs</span>
 
-        <span class="name">dolhs</span>
+        
     </div>
-
+    <a class="headerlink" href="#Eqn.dolhs"></a>
     
     
 
                             </div>
                             <div id="Eqn.dorhs" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Eqn.dorhs">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">dorhs</span>
 
-        <span class="name">dorhs</span>
+        
     </div>
-
+    <a class="headerlink" href="#Eqn.dorhs"></a>
     
     
 
                             </div>
                             <div id="Eqn.expand" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.expand">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.expand-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">expand</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">expand</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.expand-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.expand"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.expand-563"><a href="#Eqn.expand-563"><span class="linenos">563</span></a>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.expand-564"><a href="#Eqn.expand-564"><span class="linenos">564</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
+</span><span id="Eqn.expand-565"><a href="#Eqn.expand-565"><span class="linenos">565</span></a>            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span></pre></div>
 
-        </details>
 
     
 
                             </div>
                             <div id="Eqn.simplify" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.simplify">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.simplify-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">simplify</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">simplify</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.simplify-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.simplify"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.simplify-567"><a href="#Eqn.simplify-567"><span class="linenos">567</span></a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.simplify-568"><a href="#Eqn.simplify-568"><span class="linenos">568</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>See the simplify function in sympy.simplify</p>
 </div>
@@ -5917,59 +5985,59 @@ <h1 id="parameters">Parameters</h1>
 
                             </div>
                             <div id="Eqn.factor" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.factor">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.factor-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">factor</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">factor</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.factor-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.factor"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.factor-579"><a href="#Eqn.factor-579"><span class="linenos">579</span></a>    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.factor-580"><a href="#Eqn.factor-580"><span class="linenos">580</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_factor</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
     
 
                             </div>
                             <div id="Eqn.collect" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.collect">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.collect-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">collect</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">collect</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.collect-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.collect"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.collect-587"><a href="#Eqn.collect-587"><span class="linenos">587</span></a>    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.collect-588"><a href="#Eqn.collect-588"><span class="linenos">588</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_collect</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
     
 
                             </div>
                             <div id="Eqn.evalf" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.evalf">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.evalf-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">evalf</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">evalf</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.evalf-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.evalf"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.evalf-590"><a href="#Eqn.evalf-590"><span class="linenos">590</span></a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.evalf-591"><a href="#Eqn.evalf-591"><span class="linenos">591</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="Eqn.evalf-592"><a href="#Eqn.evalf-592"><span class="linenos">592</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>
 
@@ -6039,21 +6107,21 @@ <h1 id="notes">Notes</h1>
 
                             </div>
                             <div id="Eqn.n" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Eqn.n">#&nbsp;&nbsp</a>
+                                        <input id="Eqn.n-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">n</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span>, </span><span class="param"><span class="o">*</span><span class="n">args</span>, </span><span class="param"><span class="o">**</span><span class="n">kwargs</span></span>)</span>
 
-        
-            <span class="def">def</span>
-            <span class="name">n</span><span class="signature">(self, *args, **kwargs)</span>:
-    </div>
+                <label class="view-source-button" for="Eqn.n-view-source"><span>View Source</span></label>
 
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-</pre></div>
+    </div>
+    <a class="headerlink" href="#Eqn.n"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eqn.n-590"><a href="#Eqn.n-590"><span class="linenos">590</span></a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="Eqn.n-591"><a href="#Eqn.n-591"><span class="linenos">591</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
+</span><span id="Eqn.n-592"><a href="#Eqn.n-592"><span class="linenos">592</span></a>                        <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>
 
@@ -6123,11 +6191,12 @@ <h1 id="notes">Notes</h1>
 
                             </div>
                             <div id="Eqn.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Eqn.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Eqn.default_assumptions"></a>
     
     
 
@@ -6199,78 +6268,94 @@ <h5>Inherited Members</h5>
                 <dd id="Eqn.refine" class="function">refine</dd>
                 <dd id="Eqn.rewrite" class="function">rewrite</dd>
                 <dd id="Eqn.could_extract_minus_sign" class="function">could_extract_minus_sign</dd>
-                <dd id="Eqn.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Eqn.is_integer" class="variable">is_integer</dd>
-                <dd id="Eqn.is_complex" class="variable">is_complex</dd>
-                <dd id="Eqn.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Eqn.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Eqn.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Eqn.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Eqn.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Eqn.is_positive" class="variable">is_positive</dd>
+                <dd id="Eqn.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Eqn.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Eqn.is_negative" class="variable">is_negative</dd>
+                <dd id="Eqn.is_composite" class="variable">is_composite</dd>
+                <dd id="Eqn.is_finite" class="variable">is_finite</dd>
                 <dd id="Eqn.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Eqn.is_zero" class="variable">is_zero</dd>
                 <dd id="Eqn.is_even" class="variable">is_even</dd>
-                <dd id="Eqn.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Eqn.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Eqn.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Eqn.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Eqn.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Eqn.is_real" class="variable">is_real</dd>
-                <dd id="Eqn.is_positive" class="variable">is_positive</dd>
-                <dd id="Eqn.is_finite" class="variable">is_finite</dd>
-                <dd id="Eqn.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Eqn.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Eqn.is_rational" class="variable">is_rational</dd>
-                <dd id="Eqn.is_composite" class="variable">is_composite</dd>
-                <dd id="Eqn.is_negative" class="variable">is_negative</dd>
-                <dd id="Eqn.is_polar" class="variable">is_polar</dd>
+                <dd id="Eqn.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Eqn.is_odd" class="variable">is_odd</dd>
-                <dd id="Eqn.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Eqn.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Eqn.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Eqn.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Eqn.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Eqn.is_integer" class="variable">is_integer</dd>
+                <dd id="Eqn.is_rational" class="variable">is_rational</dd>
+                <dd id="Eqn.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Eqn.is_complex" class="variable">is_complex</dd>
                 <dd id="Eqn.is_prime" class="variable">is_prime</dd>
+                <dd id="Eqn.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Eqn.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Eqn.is_real" class="variable">is_real</dd>
+                <dd id="Eqn.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Eqn.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Eqn.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Eqn.is_polar" class="variable">is_polar</dd>
+                <dd id="Eqn.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Eqn.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Eqn.is_zero" class="variable">is_zero</dd>
 
             </div>
                                 </dl>
                             </div>
                 </section>
                 <section id="solve">
-                            <div class="attr function"><a class="headerlink" href="#solve">#&nbsp;&nbsp</a>
-
-        
-            <span class="def">def</span>
-            <span class="name">solve</span><span class="signature">(f, *symbols, **flags)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="k">def</span> <span class="nf">solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Override of sympy `solve()`. If it is passed an equation it will</span>
-<span class="sd">    output the solutions as typeset equations and return</span>
-<span class="sd">    the answer as a list of equations that can be accessed for additional</span>
-<span class="sd">    manipulations.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.solvers.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-    <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-        <span class="n">flags</span><span class="p">[</span><span class="s1">&#39;dict&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">f</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-        <span class="n">solns</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c1"># return result</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">k</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-                <span class="n">val</span> <span class="o">=</span> <span class="n">k</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-                <span class="n">tempeqn</span> <span class="o">=</span> <span class="n">Eqn</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
-                <span class="n">display</span><span class="p">(</span><span class="n">tempeqn</span><span class="p">)</span>
-                <span class="n">solns</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tempeqn</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">solns</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-</pre></div>
-
-        </details>
+                            <input id="solve-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">solve</span><span class="signature pdoc-code condensed">(<span class="param"><span class="n">f</span>, </span><span class="param"><span class="o">*</span><span class="n">symbols</span>, </span><span class="param"><span class="o">**</span><span class="n">flags</span></span>)</span>
+
+                <label class="view-source-button" for="solve-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#solve"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="solve-661"><a href="#solve-661"><span class="linenos">661</span></a><span class="k">def</span> <span class="nf">solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
+</span><span id="solve-662"><a href="#solve-662"><span class="linenos">662</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="solve-663"><a href="#solve-663"><span class="linenos">663</span></a><span class="sd">    Override of sympy `solve()`. If it is passed an equation it will</span>
+</span><span id="solve-664"><a href="#solve-664"><span class="linenos">664</span></a><span class="sd">    output the solutions as typeset equations and return</span>
+</span><span id="solve-665"><a href="#solve-665"><span class="linenos">665</span></a><span class="sd">    the answer as a list of equations that can be accessed for additional</span>
+</span><span id="solve-666"><a href="#solve-666"><span class="linenos">666</span></a><span class="sd">    manipulations.</span>
+</span><span id="solve-667"><a href="#solve-667"><span class="linenos">667</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="solve-668"><a href="#solve-668"><span class="linenos">668</span></a>    <span class="kn">from</span> <span class="nn">sympy.solvers.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
+</span><span id="solve-669"><a href="#solve-669"><span class="linenos">669</span></a>    <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
+</span><span id="solve-670"><a href="#solve-670"><span class="linenos">670</span></a>    <span class="n">newf</span> <span class="o">=</span><span class="p">[]</span>
+</span><span id="solve-671"><a href="#solve-671"><span class="linenos">671</span></a>    <span class="n">solns</span> <span class="o">=</span> <span class="p">[]</span>
+</span><span id="solve-672"><a href="#solve-672"><span class="linenos">672</span></a>    <span class="n">contains_eqn</span> <span class="o">=</span> <span class="kc">False</span>
+</span><span id="solve-673"><a href="#solve-673"><span class="linenos">673</span></a>    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="s1">&#39;__iter__&#39;</span><span class="p">):</span>
+</span><span id="solve-674"><a href="#solve-674"><span class="linenos">674</span></a>        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
+</span><span id="solve-675"><a href="#solve-675"><span class="linenos">675</span></a>            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="solve-676"><a href="#solve-676"><span class="linenos">676</span></a>                <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="o">.</span><span class="n">lhs</span><span class="o">-</span><span class="n">k</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="solve-677"><a href="#solve-677"><span class="linenos">677</span></a>                <span class="n">contains_eqn</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="solve-678"><a href="#solve-678"><span class="linenos">678</span></a>            <span class="k">else</span><span class="p">:</span>
+</span><span id="solve-679"><a href="#solve-679"><span class="linenos">679</span></a>                <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
+</span><span id="solve-680"><a href="#solve-680"><span class="linenos">680</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="solve-681"><a href="#solve-681"><span class="linenos">681</span></a>        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="solve-682"><a href="#solve-682"><span class="linenos">682</span></a>            <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">f</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="solve-683"><a href="#solve-683"><span class="linenos">683</span></a>            <span class="n">contains_eqn</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="solve-684"><a href="#solve-684"><span class="linenos">684</span></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="solve-685"><a href="#solve-685"><span class="linenos">685</span></a>            <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
+</span><span id="solve-686"><a href="#solve-686"><span class="linenos">686</span></a>    <span class="n">flags</span><span class="p">[</span><span class="s1">&#39;dict&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
+</span><span id="solve-687"><a href="#solve-687"><span class="linenos">687</span></a>    <span class="n">result</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">newf</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
+</span><span id="solve-688"><a href="#solve-688"><span class="linenos">688</span></a>    <span class="k">if</span> <span class="n">contains_eqn</span><span class="p">:</span>
+</span><span id="solve-689"><a href="#solve-689"><span class="linenos">689</span></a>        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
+</span><span id="solve-690"><a href="#solve-690"><span class="linenos">690</span></a>            <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">k</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
+</span><span id="solve-691"><a href="#solve-691"><span class="linenos">691</span></a>                <span class="n">val</span> <span class="o">=</span> <span class="n">k</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
+</span><span id="solve-692"><a href="#solve-692"><span class="linenos">692</span></a>                <span class="n">tempeqn</span> <span class="o">=</span> <span class="n">Eqn</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
+</span><span id="solve-693"><a href="#solve-693"><span class="linenos">693</span></a>                <span class="n">display</span><span class="p">(</span><span class="n">tempeqn</span><span class="p">)</span>
+</span><span id="solve-694"><a href="#solve-694"><span class="linenos">694</span></a>                <span class="n">solns</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tempeqn</span><span class="p">)</span>
+</span><span id="solve-695"><a href="#solve-695"><span class="linenos">695</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="solve-696"><a href="#solve-696"><span class="linenos">696</span></a>        <span class="n">solns</span> <span class="o">=</span> <span class="n">result</span>
+</span><span id="solve-697"><a href="#solve-697"><span class="linenos">697</span></a>    <span class="k">return</span> <span class="n">solns</span>
+</span><span id="solve-698"><a href="#solve-698"><span class="linenos">698</span></a>    <span class="c1"># else:</span>
+</span><span id="solve-699"><a href="#solve-699"><span class="linenos">699</span></a>    <span class="c1">#     return solve(f, *symbols, **flags)</span>
+</span></pre></div>
+
 
             <div class="docstring"><p>Override of sympy <code><a href="#solve">solve()</a></code>. If it is passed an equation it will
 output the solutions as typeset equations and return
@@ -6281,30 +6366,30 @@ <h5>Inherited Members</h5>
 
                 </section>
                 <section id="sqrt">
-                            <div class="attr function"><a class="headerlink" href="#sqrt">#&nbsp;&nbsp</a>
-
-        
-            <span class="def">def</span>
-            <span class="name">sqrt</span><span class="signature">(arg, evaluate=None)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="k">def</span> <span class="nf">sqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">evaluate</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Override of sympy convenience function `sqrt`. Simply divides equations</span>
-<span class="sd">    into two sides if `arg` is an instance of `Equation`. This avoids an</span>
-<span class="sd">    issue with the way sympy is delaying specialized applications of _Pow_ on</span>
-<span class="sd">    objects that are not basic sympy expressions.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span> <span class="k">as</span> <span class="n">symsqrt</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">),</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span><span class="n">evaluate</span><span class="p">)</span>
-</pre></div>
+                            <input id="sqrt-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">sqrt</span><span class="signature pdoc-code condensed">(<span class="param"><span class="n">arg</span>, </span><span class="param"><span class="n">evaluate</span><span class="o">=</span><span class="kc">None</span></span>)</span>
+
+                <label class="view-source-button" for="sqrt-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#sqrt"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="sqrt-701"><a href="#sqrt-701"><span class="linenos">701</span></a><span class="k">def</span> <span class="nf">sqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">evaluate</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span>
+</span><span id="sqrt-702"><a href="#sqrt-702"><span class="linenos">702</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="sqrt-703"><a href="#sqrt-703"><span class="linenos">703</span></a><span class="sd">    Override of sympy convenience function `sqrt`. Simply divides equations</span>
+</span><span id="sqrt-704"><a href="#sqrt-704"><span class="linenos">704</span></a><span class="sd">    into two sides if `arg` is an instance of `Equation`. This avoids an</span>
+</span><span id="sqrt-705"><a href="#sqrt-705"><span class="linenos">705</span></a><span class="sd">    issue with the way sympy is delaying specialized applications of _Pow_ on</span>
+</span><span id="sqrt-706"><a href="#sqrt-706"><span class="linenos">706</span></a><span class="sd">    objects that are not basic sympy expressions.</span>
+</span><span id="sqrt-707"><a href="#sqrt-707"><span class="linenos">707</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="sqrt-708"><a href="#sqrt-708"><span class="linenos">708</span></a>    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span> <span class="k">as</span> <span class="n">symsqrt</span>
+</span><span id="sqrt-709"><a href="#sqrt-709"><span class="linenos">709</span></a>    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="sqrt-710"><a href="#sqrt-710"><span class="linenos">710</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">),</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">))</span>
+</span><span id="sqrt-711"><a href="#sqrt-711"><span class="linenos">711</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="sqrt-712"><a href="#sqrt-712"><span class="linenos">712</span></a>        <span class="k">return</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span><span class="n">evaluate</span><span class="p">)</span>
+</span></pre></div>
 
-        </details>
 
             <div class="docstring"><p>Override of sympy convenience function <code><a href="#sqrt">sqrt</a></code>. Simply divides equations
 into two sides if <code><a href="#arg">arg</a></code> is an instance of <code><a href="#Equation">Equation</a></code>. This avoids an
@@ -6397,38 +6482,41 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Square_root
-.. [2] https://en.wikipedia.org/wiki/Principal_value</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                 </section>
                 <section id="root">
-                            <div class="attr function"><a class="headerlink" href="#root">#&nbsp;&nbsp</a>
+                            <input id="root-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">root</span><span class="signature pdoc-code condensed">(<span class="param"><span class="n">arg</span>, </span><span class="param"><span class="n">n</span>, </span><span class="param"><span class="n">k</span><span class="o">=</span><span class="mi">0</span>, </span><span class="param"><span class="n">evaluate</span><span class="o">=</span><span class="kc">None</span></span>)</span>
+
+                <label class="view-source-button" for="root-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#root"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="root-719"><a href="#root-719"><span class="linenos">719</span></a><span class="k">def</span> <span class="nf">root</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">evaluate</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span>
+</span><span id="root-720"><a href="#root-720"><span class="linenos">720</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="root-721"><a href="#root-721"><span class="linenos">721</span></a><span class="sd">    Override of sympy convenience function `root`. Simply divides equations</span>
+</span><span id="root-722"><a href="#root-722"><span class="linenos">722</span></a><span class="sd">    into two sides if `arg` is an instance of `Equation`. This avoids an</span>
+</span><span id="root-723"><a href="#root-723"><span class="linenos">723</span></a><span class="sd">    issue with the way sympy is delaying specialized applications of _Pow_ on</span>
+</span><span id="root-724"><a href="#root-724"><span class="linenos">724</span></a><span class="sd">    objects that are not basic sympy expressions.</span>
+</span><span id="root-725"><a href="#root-725"><span class="linenos">725</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="root-726"><a href="#root-726"><span class="linenos">726</span></a>    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">root</span> <span class="k">as</span> <span class="n">symroot</span>
+</span><span id="root-727"><a href="#root-727"><span class="linenos">727</span></a>    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="root-728"><a href="#root-728"><span class="linenos">728</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">),</span> <span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span>
+</span><span id="root-729"><a href="#root-729"><span class="linenos">729</span></a>                                                            <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">))</span>
+</span><span id="root-730"><a href="#root-730"><span class="linenos">730</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="root-731"><a href="#root-731"><span class="linenos">731</span></a>        <span class="k">return</span> <span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">)</span>
+</span></pre></div>
 
-        
-            <span class="def">def</span>
-            <span class="name">root</span><span class="signature">(arg, n, k=0, evaluate=None)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="k">def</span> <span class="nf">root</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">evaluate</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Override of sympy convenience function `root`. Simply divides equations</span>
-<span class="sd">    into two sides if `arg` is an instance of `Equation`. This avoids an</span>
-<span class="sd">    issue with the way sympy is delaying specialized applications of _Pow_ on</span>
-<span class="sd">    objects that are not basic sympy expressions.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">root</span> <span class="k">as</span> <span class="n">symroot</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Equation</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">),</span> <span class="n">symroot</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span>
-                                                            <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">symsqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">)</span>
-</pre></div>
-
-        </details>
 
             <div class="docstring"><p>Override of sympy convenience function <code><a href="#root">root</a></code>. Simply divides equations
 into two sides if <code><a href="#arg">arg</a></code> is an instance of <code><a href="#Equation">Equation</a></code>. This avoids an
@@ -6529,99 +6617,335 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Square_root
-.. [2] https://en.wikipedia.org/wiki/Real_root
-.. [3] https://en.wikipedia.org/wiki/Root_of_unity
-.. [4] https://en.wikipedia.org/wiki/Principal_value
-.. [5] http://mathworld.wolfram.com/CubeRoot.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                 </section>
                 <section id="collect">
-                            <div class="attr function"><a class="headerlink" href="#collect">#&nbsp;&nbsp</a>
-
-        
-            <span class="def">def</span>
-            <span class="name">collect</span><span class="signature">(
-    expr,
-    syms,
-    func=None,
-    evaluate=None,
-    exact=False,
-    distribute_order_term=True
-)</span>:
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
-            <span class="n">distribute_order_term</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Override of sympy `collect()`.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
-    <span class="n">_eval_collect</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s1">&#39;_eval_collect&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">_eval_collect</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_eval_collect</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span>
-                             <span class="n">exact</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span> <span class="n">exact</span><span class="p">,</span>
-                       <span class="n">distribute_order_term</span><span class="p">)</span>
-</pre></div>
-
-        </details>
+                            <input id="collect-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">collect</span><span class="signature pdoc-code multiline">(<span class="param">	<span class="n">expr</span>,</span><span class="param">	<span class="n">syms</span>,</span><span class="param">	<span class="n">func</span><span class="o">=</span><span class="kc">None</span>,</span><span class="param">	<span class="n">evaluate</span><span class="o">=</span><span class="kc">None</span>,</span><span class="param">	<span class="n">exact</span><span class="o">=</span><span class="kc">False</span>,</span><span class="param">	<span class="n">distribute_order_term</span><span class="o">=</span><span class="kc">True</span></span>)</span>
+
+                <label class="view-source-button" for="collect-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#collect"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="collect-738"><a href="#collect-738"><span class="linenos">738</span></a><span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+</span><span id="collect-739"><a href="#collect-739"><span class="linenos">739</span></a>            <span class="n">distribute_order_term</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+</span><span id="collect-740"><a href="#collect-740"><span class="linenos">740</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="collect-741"><a href="#collect-741"><span class="linenos">741</span></a><span class="sd">    Override of sympy `collect()`.</span>
+</span><span id="collect-742"><a href="#collect-742"><span class="linenos">742</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="collect-743"><a href="#collect-743"><span class="linenos">743</span></a>    <span class="kn">from</span> <span class="nn">sympy.simplify.radsimp</span> <span class="kn">import</span> <span class="n">collect</span>
+</span><span id="collect-744"><a href="#collect-744"><span class="linenos">744</span></a>    <span class="n">_eval_collect</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s1">&#39;_eval_collect&#39;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+</span><span id="collect-745"><a href="#collect-745"><span class="linenos">745</span></a>    <span class="k">if</span> <span class="n">_eval_collect</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+</span><span id="collect-746"><a href="#collect-746"><span class="linenos">746</span></a>        <span class="k">return</span> <span class="n">_eval_collect</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span>
+</span><span id="collect-747"><a href="#collect-747"><span class="linenos">747</span></a>                             <span class="n">exact</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="p">)</span>
+</span><span id="collect-748"><a href="#collect-748"><span class="linenos">748</span></a>    <span class="k">else</span><span class="p">:</span>
+</span><span id="collect-749"><a href="#collect-749"><span class="linenos">749</span></a>        <span class="k">return</span> <span class="n">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span> <span class="n">exact</span><span class="p">,</span>
+</span><span id="collect-750"><a href="#collect-750"><span class="linenos">750</span></a>                       <span class="n">distribute_order_term</span><span class="p">)</span>
+</span></pre></div>
+
 
             <div class="docstring"><p>Override of sympy <code><a href="#collect">collect()</a></code>.</p>
 </div>
 
 
+                </section>
+                <section id="Eq">
+                            <input id="Eq-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Eq</span><wbr>(<span class="base">sympy.core.relational.Equality</span>):
+
+                <label class="view-source-button" for="Eq-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Eq"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eq-752"><a href="#Eq-752"><span class="linenos">752</span></a><span class="k">class</span> <span class="nc">Eq</span><span class="p">(</span><span class="n">Eq</span><span class="p">):</span>
+</span><span id="Eq-753"><a href="#Eq-753"><span class="linenos">753</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eq-754"><a href="#Eq-754"><span class="linenos">754</span></a><span class="sd">    Extension of Equality class to include the ability to convert it to an</span>
+</span><span id="Eq-755"><a href="#Eq-755"><span class="linenos">755</span></a><span class="sd">    Equation.</span>
+</span><span id="Eq-756"><a href="#Eq-756"><span class="linenos">756</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="Eq-757"><a href="#Eq-757"><span class="linenos">757</span></a>    <span class="k">def</span> <span class="nf">to_Equation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eq-758"><a href="#Eq-758"><span class="linenos">758</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eq-759"><a href="#Eq-759"><span class="linenos">759</span></a><span class="sd">        Return: recasts the Equality as an Equation.</span>
+</span><span id="Eq-760"><a href="#Eq-760"><span class="linenos">760</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eq-761"><a href="#Eq-761"><span class="linenos">761</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="Eq-762"><a href="#Eq-762"><span class="linenos">762</span></a>
+</span><span id="Eq-763"><a href="#Eq-763"><span class="linenos">763</span></a>    <span class="k">def</span> <span class="nf">to_Eqn</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eq-764"><a href="#Eq-764"><span class="linenos">764</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eq-765"><a href="#Eq-765"><span class="linenos">765</span></a><span class="sd">        Synonym for to_Equation.</span>
+</span><span id="Eq-766"><a href="#Eq-766"><span class="linenos">766</span></a><span class="sd">        Return: recasts the Equality as an Equation.</span>
+</span><span id="Eq-767"><a href="#Eq-767"><span class="linenos">767</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eq-768"><a href="#Eq-768"><span class="linenos">768</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_Equation</span><span class="p">()</span>
+</span></pre></div>
+
+
+            <div class="docstring"><p>Extension of Equality class to include the ability to convert it to an
+Equation.</p>
+</div>
+
+
+                            <div id="Eq.__init__" class="classattr">
+                                <div class="attr function">
+            
+        <span class="name">Eq</span><span class="signature pdoc-code condensed">()</span>
+
+        
+    </div>
+    <a class="headerlink" href="#Eq.__init__"></a>
+    
+    
+
+                            </div>
+                            <div id="Eq.to_Equation" class="classattr">
+                                        <input id="Eq.to_Equation-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">to_Equation</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span></span>)</span>
+
+                <label class="view-source-button" for="Eq.to_Equation-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Eq.to_Equation"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eq.to_Equation-757"><a href="#Eq.to_Equation-757"><span class="linenos">757</span></a>    <span class="k">def</span> <span class="nf">to_Equation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eq.to_Equation-758"><a href="#Eq.to_Equation-758"><span class="linenos">758</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eq.to_Equation-759"><a href="#Eq.to_Equation-759"><span class="linenos">759</span></a><span class="sd">        Return: recasts the Equality as an Equation.</span>
+</span><span id="Eq.to_Equation-760"><a href="#Eq.to_Equation-760"><span class="linenos">760</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eq.to_Equation-761"><a href="#Eq.to_Equation-761"><span class="linenos">761</span></a>        <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
+</span></pre></div>
+
+
+            <div class="docstring"><p>Return: recasts the Equality as an Equation.</p>
+</div>
+
+
+                            </div>
+                            <div id="Eq.to_Eqn" class="classattr">
+                                        <input id="Eq.to_Eqn-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr function">
+            
+        <span class="def">def</span>
+        <span class="name">to_Eqn</span><span class="signature pdoc-code condensed">(<span class="param"><span class="bp">self</span></span>)</span>
+
+                <label class="view-source-button" for="Eq.to_Eqn-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#Eq.to_Eqn"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="Eq.to_Eqn-763"><a href="#Eq.to_Eqn-763"><span class="linenos">763</span></a>    <span class="k">def</span> <span class="nf">to_Eqn</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+</span><span id="Eq.to_Eqn-764"><a href="#Eq.to_Eqn-764"><span class="linenos">764</span></a>        <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="Eq.to_Eqn-765"><a href="#Eq.to_Eqn-765"><span class="linenos">765</span></a><span class="sd">        Synonym for to_Equation.</span>
+</span><span id="Eq.to_Eqn-766"><a href="#Eq.to_Eqn-766"><span class="linenos">766</span></a><span class="sd">        Return: recasts the Equality as an Equation.</span>
+</span><span id="Eq.to_Eqn-767"><a href="#Eq.to_Eqn-767"><span class="linenos">767</span></a><span class="sd">        &quot;&quot;&quot;</span>
+</span><span id="Eq.to_Eqn-768"><a href="#Eq.to_Eqn-768"><span class="linenos">768</span></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_Equation</span><span class="p">()</span>
+</span></pre></div>
+
+
+            <div class="docstring"><p>Synonym for to_Equation.
+Return: recasts the Equality as an Equation.</p>
+</div>
+
+
+                            </div>
+                            <div id="Eq.default_assumptions" class="classattr">
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+
+        
+    </div>
+    <a class="headerlink" href="#Eq.default_assumptions"></a>
+    
+    
+
+                            </div>
+                            <div class="inherited">
+                                <h5>Inherited Members</h5>
+                                <dl>
+                                    <div><dt>sympy.core.relational.Equality</dt>
+                                <dd id="Eq.rel_op" class="variable">rel_op</dd>
+                <dd id="Eq.is_Equality" class="variable">is_Equality</dd>
+                <dd id="Eq.binary_symbols" class="variable">binary_symbols</dd>
+                <dd id="Eq.integrate" class="function">integrate</dd>
+                <dd id="Eq.as_poly" class="function">as_poly</dd>
+
+            </div>
+            <div><dt>sympy.core.relational.Relational</dt>
+                                <dd id="Eq.ValidRelationOperator" class="variable">ValidRelationOperator</dd>
+                <dd id="Eq.is_Relational" class="variable">is_Relational</dd>
+                <dd id="Eq.lhs" class="variable">lhs</dd>
+                <dd id="Eq.rhs" class="variable">rhs</dd>
+                <dd id="Eq.reversed" class="variable">reversed</dd>
+                <dd id="Eq.reversedsign" class="variable">reversedsign</dd>
+                <dd id="Eq.negated" class="variable">negated</dd>
+                <dd id="Eq.weak" class="variable">weak</dd>
+                <dd id="Eq.strict" class="variable">strict</dd>
+                <dd id="Eq.canonical" class="variable">canonical</dd>
+                <dd id="Eq.equals" class="function">equals</dd>
+                <dd id="Eq.expand" class="function">expand</dd>
+
+            </div>
+            <div><dt>sympy.logic.boolalg.Boolean</dt>
+                                <dd id="Eq.kind" class="variable">kind</dd>
+                <dd id="Eq.to_nnf" class="function">to_nnf</dd>
+                <dd id="Eq.as_set" class="function">as_set</dd>
+
+            </div>
+            <div><dt>sympy.core.basic.Basic</dt>
+                                <dd id="Eq.is_number" class="variable">is_number</dd>
+                <dd id="Eq.is_Atom" class="variable">is_Atom</dd>
+                <dd id="Eq.is_Symbol" class="variable">is_Symbol</dd>
+                <dd id="Eq.is_symbol" class="variable">is_symbol</dd>
+                <dd id="Eq.is_Indexed" class="variable">is_Indexed</dd>
+                <dd id="Eq.is_Dummy" class="variable">is_Dummy</dd>
+                <dd id="Eq.is_Wild" class="variable">is_Wild</dd>
+                <dd id="Eq.is_Function" class="variable">is_Function</dd>
+                <dd id="Eq.is_Add" class="variable">is_Add</dd>
+                <dd id="Eq.is_Mul" class="variable">is_Mul</dd>
+                <dd id="Eq.is_Pow" class="variable">is_Pow</dd>
+                <dd id="Eq.is_Number" class="variable">is_Number</dd>
+                <dd id="Eq.is_Float" class="variable">is_Float</dd>
+                <dd id="Eq.is_Rational" class="variable">is_Rational</dd>
+                <dd id="Eq.is_Integer" class="variable">is_Integer</dd>
+                <dd id="Eq.is_NumberSymbol" class="variable">is_NumberSymbol</dd>
+                <dd id="Eq.is_Order" class="variable">is_Order</dd>
+                <dd id="Eq.is_Derivative" class="variable">is_Derivative</dd>
+                <dd id="Eq.is_Piecewise" class="variable">is_Piecewise</dd>
+                <dd id="Eq.is_Poly" class="variable">is_Poly</dd>
+                <dd id="Eq.is_AlgebraicNumber" class="variable">is_AlgebraicNumber</dd>
+                <dd id="Eq.is_Boolean" class="variable">is_Boolean</dd>
+                <dd id="Eq.is_Not" class="variable">is_Not</dd>
+                <dd id="Eq.is_Matrix" class="variable">is_Matrix</dd>
+                <dd id="Eq.is_Vector" class="variable">is_Vector</dd>
+                <dd id="Eq.is_Point" class="variable">is_Point</dd>
+                <dd id="Eq.is_MatAdd" class="variable">is_MatAdd</dd>
+                <dd id="Eq.is_MatMul" class="variable">is_MatMul</dd>
+                <dd id="Eq.copy" class="function">copy</dd>
+                <dd id="Eq.assumptions0" class="variable">assumptions0</dd>
+                <dd id="Eq.compare" class="function">compare</dd>
+                <dd id="Eq.fromiter" class="function">fromiter</dd>
+                <dd id="Eq.class_key" class="function">class_key</dd>
+                <dd id="Eq.sort_key" class="function">sort_key</dd>
+                <dd id="Eq.dummy_eq" class="function">dummy_eq</dd>
+                <dd id="Eq.atoms" class="function">atoms</dd>
+                <dd id="Eq.free_symbols" class="variable">free_symbols</dd>
+                <dd id="Eq.expr_free_symbols" class="variable">expr_free_symbols</dd>
+                <dd id="Eq.as_dummy" class="function">as_dummy</dd>
+                <dd id="Eq.canonical_variables" class="variable">canonical_variables</dd>
+                <dd id="Eq.rcall" class="function">rcall</dd>
+                <dd id="Eq.is_hypergeometric" class="function">is_hypergeometric</dd>
+                <dd id="Eq.is_comparable" class="variable">is_comparable</dd>
+                <dd id="Eq.func" class="variable">func</dd>
+                <dd id="Eq.args" class="variable">args</dd>
+                <dd id="Eq.as_content_primitive" class="function">as_content_primitive</dd>
+                <dd id="Eq.subs" class="function">subs</dd>
+                <dd id="Eq.xreplace" class="function">xreplace</dd>
+                <dd id="Eq.has" class="function">has</dd>
+                <dd id="Eq.has_free" class="function">has_free</dd>
+                <dd id="Eq.replace" class="function">replace</dd>
+                <dd id="Eq.find" class="function">find</dd>
+                <dd id="Eq.count" class="function">count</dd>
+                <dd id="Eq.matches" class="function">matches</dd>
+                <dd id="Eq.match" class="function">match</dd>
+                <dd id="Eq.count_ops" class="function">count_ops</dd>
+                <dd id="Eq.doit" class="function">doit</dd>
+                <dd id="Eq.simplify" class="function">simplify</dd>
+                <dd id="Eq.refine" class="function">refine</dd>
+                <dd id="Eq.rewrite" class="function">rewrite</dd>
+                <dd id="Eq.could_extract_minus_sign" class="function">could_extract_minus_sign</dd>
+                <dd id="Eq.is_extended_positive" class="variable">is_extended_positive</dd>
+                <dd id="Eq.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Eq.is_positive" class="variable">is_positive</dd>
+                <dd id="Eq.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Eq.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Eq.is_negative" class="variable">is_negative</dd>
+                <dd id="Eq.is_composite" class="variable">is_composite</dd>
+                <dd id="Eq.is_finite" class="variable">is_finite</dd>
+                <dd id="Eq.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
+                <dd id="Eq.is_even" class="variable">is_even</dd>
+                <dd id="Eq.is_nonnegative" class="variable">is_nonnegative</dd>
+                <dd id="Eq.is_infinite" class="variable">is_infinite</dd>
+                <dd id="Eq.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Eq.is_odd" class="variable">is_odd</dd>
+                <dd id="Eq.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Eq.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Eq.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Eq.is_integer" class="variable">is_integer</dd>
+                <dd id="Eq.is_rational" class="variable">is_rational</dd>
+                <dd id="Eq.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Eq.is_complex" class="variable">is_complex</dd>
+                <dd id="Eq.is_prime" class="variable">is_prime</dd>
+                <dd id="Eq.is_noninteger" class="variable">is_noninteger</dd>
+                <dd id="Eq.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Eq.is_real" class="variable">is_real</dd>
+                <dd id="Eq.is_commutative" class="variable">is_commutative</dd>
+                <dd id="Eq.is_hermitian" class="variable">is_hermitian</dd>
+                <dd id="Eq.is_polar" class="variable">is_polar</dd>
+                <dd id="Eq.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Eq.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Eq.is_zero" class="variable">is_zero</dd>
+
+            </div>
+            <div><dt>sympy.core.evalf.EvalfMixin</dt>
+                                <dd id="Eq.evalf" class="function">evalf</dd>
+                <dd id="Eq.n" class="function">n</dd>
+
+            </div>
+                                </dl>
+                            </div>
                 </section>
                 <section id="EqnFunction">
-                                <div class="attr class">
-        <a class="headerlink" href="#EqnFunction">#&nbsp;&nbsp</a>
-
-        
-        <span class="def">class</span>
-        <span class="name">EqnFunction</span><wbr>(<span class="base">sympy.core.function.Function</span>):
-    </div>
-
-            <details>
-            <summary>View Source</summary>
-            <div class="pdoc-code codehilite"><pre><span></span><span class="k">class</span> <span class="nc">EqnFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Extension of the sympy Function class to understand equations. Each</span>
-<span class="sd">    sympy function impacted by this extension is listed in the documentation</span>
-<span class="sd">    that follows.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">eqnloc</span> <span class="o">=</span> <span class="kc">None</span>
-        <span class="n">neqns</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
-                    <span class="n">neqns</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">eqnloc</span> <span class="o">=</span> <span class="n">i</span>
-            <span class="k">if</span> <span class="n">neqns</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Function calls with more than one &#39;</span>
-                                          <span class="s1">&#39;Equation as a parameter are not &#39;</span>
-                                          <span class="s1">&#39;supported. You may be able to get &#39;</span>
-                                          <span class="s1">&#39;your desired outcome using .applyrhs&#39;</span>
-                                          <span class="s1">&#39; and .applylhs.&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">neqns</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span><span class="o">.</span><span class="n">lhs</span>
-                <span class="n">lhs</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-                <span class="n">newargs</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span><span class="o">.</span><span class="n">rhs</span>
-                <span class="n">rhs</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</pre></div>
-
-        </details>
+                            <input id="EqnFunction-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
+<div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">EqnFunction</span><wbr>(<span class="base">sympy.core.function.Function</span>):
+
+                <label class="view-source-button" for="EqnFunction-view-source"><span>View Source</span></label>
+
+    </div>
+    <a class="headerlink" href="#EqnFunction"></a>
+            <div class="pdoc-code codehilite"><pre><span></span><span id="EqnFunction-774"><a href="#EqnFunction-774"><span class="linenos">774</span></a><span class="k">class</span> <span class="nc">EqnFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
+</span><span id="EqnFunction-775"><a href="#EqnFunction-775"><span class="linenos">775</span></a>    <span class="sd">&quot;&quot;&quot;</span>
+</span><span id="EqnFunction-776"><a href="#EqnFunction-776"><span class="linenos">776</span></a><span class="sd">    Extension of the sympy Function class to understand equations. Each</span>
+</span><span id="EqnFunction-777"><a href="#EqnFunction-777"><span class="linenos">777</span></a><span class="sd">    sympy function impacted by this extension is listed in the documentation</span>
+</span><span id="EqnFunction-778"><a href="#EqnFunction-778"><span class="linenos">778</span></a><span class="sd">    that follows.</span>
+</span><span id="EqnFunction-779"><a href="#EqnFunction-779"><span class="linenos">779</span></a><span class="sd">    &quot;&quot;&quot;</span>
+</span><span id="EqnFunction-780"><a href="#EqnFunction-780"><span class="linenos">780</span></a>    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+</span><span id="EqnFunction-781"><a href="#EqnFunction-781"><span class="linenos">781</span></a>        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
+</span><span id="EqnFunction-782"><a href="#EqnFunction-782"><span class="linenos">782</span></a>        <span class="n">eqnloc</span> <span class="o">=</span> <span class="kc">None</span>
+</span><span id="EqnFunction-783"><a href="#EqnFunction-783"><span class="linenos">783</span></a>        <span class="n">neqns</span> <span class="o">=</span> <span class="mi">0</span>
+</span><span id="EqnFunction-784"><a href="#EqnFunction-784"><span class="linenos">784</span></a>        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
+</span><span id="EqnFunction-785"><a href="#EqnFunction-785"><span class="linenos">785</span></a>        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
+</span><span id="EqnFunction-786"><a href="#EqnFunction-786"><span class="linenos">786</span></a>            <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
+</span><span id="EqnFunction-787"><a href="#EqnFunction-787"><span class="linenos">787</span></a>        <span class="k">if</span> <span class="p">(</span><span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
+</span><span id="EqnFunction-788"><a href="#EqnFunction-788"><span class="linenos">788</span></a>            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
+</span><span id="EqnFunction-789"><a href="#EqnFunction-789"><span class="linenos">789</span></a>                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Equation</span><span class="p">):</span>
+</span><span id="EqnFunction-790"><a href="#EqnFunction-790"><span class="linenos">790</span></a>                    <span class="n">neqns</span> <span class="o">+=</span> <span class="mi">1</span>
+</span><span id="EqnFunction-791"><a href="#EqnFunction-791"><span class="linenos">791</span></a>                    <span class="n">eqnloc</span> <span class="o">=</span> <span class="n">i</span>
+</span><span id="EqnFunction-792"><a href="#EqnFunction-792"><span class="linenos">792</span></a>            <span class="k">if</span> <span class="n">neqns</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
+</span><span id="EqnFunction-793"><a href="#EqnFunction-793"><span class="linenos">793</span></a>                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Function calls with more than one &#39;</span>
+</span><span id="EqnFunction-794"><a href="#EqnFunction-794"><span class="linenos">794</span></a>                                          <span class="s1">&#39;Equation as a parameter are not &#39;</span>
+</span><span id="EqnFunction-795"><a href="#EqnFunction-795"><span class="linenos">795</span></a>                                          <span class="s1">&#39;supported. You may be able to get &#39;</span>
+</span><span id="EqnFunction-796"><a href="#EqnFunction-796"><span class="linenos">796</span></a>                                          <span class="s1">&#39;your desired outcome using .applyrhs&#39;</span>
+</span><span id="EqnFunction-797"><a href="#EqnFunction-797"><span class="linenos">797</span></a>                                          <span class="s1">&#39; and .applylhs.&#39;</span><span class="p">)</span>
+</span><span id="EqnFunction-798"><a href="#EqnFunction-798"><span class="linenos">798</span></a>            <span class="k">if</span> <span class="n">neqns</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+</span><span id="EqnFunction-799"><a href="#EqnFunction-799"><span class="linenos">799</span></a>                <span class="n">newargs</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span><span class="o">.</span><span class="n">lhs</span>
+</span><span id="EqnFunction-800"><a href="#EqnFunction-800"><span class="linenos">800</span></a>                <span class="n">lhs</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="EqnFunction-801"><a href="#EqnFunction-801"><span class="linenos">801</span></a>                <span class="n">newargs</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">eqnloc</span><span class="p">]</span><span class="o">.</span><span class="n">rhs</span>
+</span><span id="EqnFunction-802"><a href="#EqnFunction-802"><span class="linenos">802</span></a>                <span class="n">rhs</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span><span id="EqnFunction-803"><a href="#EqnFunction-803"><span class="linenos">803</span></a>                <span class="k">return</span> <span class="n">Equation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span><span class="n">rhs</span><span class="p">)</span>
+</span><span id="EqnFunction-804"><a href="#EqnFunction-804"><span class="linenos">804</span></a>        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+</span></pre></div>
+
 
             <div class="docstring"><p>Extension of the sympy Function class to understand equations. Each
 sympy function impacted by this extension is listed in the documentation
@@ -6630,22 +6954,24 @@ <h1 id="references">References</h1>
 
 
                             <div id="EqnFunction.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#EqnFunction.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">EqnFunction</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">EqnFunction</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#EqnFunction.__init__"></a>
     
     
 
                             </div>
                             <div id="EqnFunction.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#EqnFunction.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#EqnFunction.default_assumptions"></a>
     
     
 
@@ -6797,37 +7123,37 @@ <h5>Inherited Members</h5>
                 <dd id="EqnFunction.simplify" class="function">simplify</dd>
                 <dd id="EqnFunction.refine" class="function">refine</dd>
                 <dd id="EqnFunction.rewrite" class="function">rewrite</dd>
-                <dd id="EqnFunction.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="EqnFunction.is_integer" class="variable">is_integer</dd>
-                <dd id="EqnFunction.is_complex" class="variable">is_complex</dd>
-                <dd id="EqnFunction.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="EqnFunction.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="EqnFunction.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="EqnFunction.is_irrational" class="variable">is_irrational</dd>
+                <dd id="EqnFunction.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="EqnFunction.is_positive" class="variable">is_positive</dd>
+                <dd id="EqnFunction.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="EqnFunction.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="EqnFunction.is_negative" class="variable">is_negative</dd>
+                <dd id="EqnFunction.is_composite" class="variable">is_composite</dd>
+                <dd id="EqnFunction.is_finite" class="variable">is_finite</dd>
                 <dd id="EqnFunction.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="EqnFunction.is_zero" class="variable">is_zero</dd>
                 <dd id="EqnFunction.is_even" class="variable">is_even</dd>
-                <dd id="EqnFunction.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="EqnFunction.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="EqnFunction.is_commutative" class="variable">is_commutative</dd>
-                <dd id="EqnFunction.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="EqnFunction.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="EqnFunction.is_real" class="variable">is_real</dd>
-                <dd id="EqnFunction.is_positive" class="variable">is_positive</dd>
-                <dd id="EqnFunction.is_finite" class="variable">is_finite</dd>
-                <dd id="EqnFunction.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="EqnFunction.is_infinite" class="variable">is_infinite</dd>
-                <dd id="EqnFunction.is_rational" class="variable">is_rational</dd>
-                <dd id="EqnFunction.is_composite" class="variable">is_composite</dd>
-                <dd id="EqnFunction.is_negative" class="variable">is_negative</dd>
-                <dd id="EqnFunction.is_polar" class="variable">is_polar</dd>
+                <dd id="EqnFunction.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="EqnFunction.is_odd" class="variable">is_odd</dd>
-                <dd id="EqnFunction.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="EqnFunction.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="EqnFunction.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="EqnFunction.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="EqnFunction.is_irrational" class="variable">is_irrational</dd>
+                <dd id="EqnFunction.is_integer" class="variable">is_integer</dd>
+                <dd id="EqnFunction.is_rational" class="variable">is_rational</dd>
+                <dd id="EqnFunction.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="EqnFunction.is_complex" class="variable">is_complex</dd>
                 <dd id="EqnFunction.is_prime" class="variable">is_prime</dd>
+                <dd id="EqnFunction.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="EqnFunction.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="EqnFunction.is_real" class="variable">is_real</dd>
+                <dd id="EqnFunction.is_commutative" class="variable">is_commutative</dd>
                 <dd id="EqnFunction.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="EqnFunction.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="EqnFunction.is_polar" class="variable">is_polar</dd>
+                <dd id="EqnFunction.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="EqnFunction.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="EqnFunction.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -6839,14 +7165,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="factorial">
-                                <div class="attr class">
-        <a class="headerlink" href="#factorial">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">factorial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.factorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">factorial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.factorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#factorial"></a>
     
             <div class="docstring"><p>Implementation of factorial function over nonnegative integers.
 By convention (consistent with the gamma function and the binomial
@@ -6903,22 +7229,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="factorial.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#factorial.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">factorial</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">factorial</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#factorial.__init__"></a>
     
     
 
                             </div>
                             <div id="factorial.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#factorial.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#factorial.default_assumptions"></a>
     
     
 
@@ -7073,37 +7401,37 @@ <h5>Inherited Members</h5>
                 <dd id="factorial.simplify" class="function">simplify</dd>
                 <dd id="factorial.refine" class="function">refine</dd>
                 <dd id="factorial.rewrite" class="function">rewrite</dd>
-                <dd id="factorial.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="factorial.is_integer" class="variable">is_integer</dd>
-                <dd id="factorial.is_complex" class="variable">is_complex</dd>
-                <dd id="factorial.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="factorial.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="factorial.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="factorial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="factorial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="factorial.is_positive" class="variable">is_positive</dd>
+                <dd id="factorial.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="factorial.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="factorial.is_negative" class="variable">is_negative</dd>
+                <dd id="factorial.is_composite" class="variable">is_composite</dd>
+                <dd id="factorial.is_finite" class="variable">is_finite</dd>
                 <dd id="factorial.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="factorial.is_zero" class="variable">is_zero</dd>
                 <dd id="factorial.is_even" class="variable">is_even</dd>
-                <dd id="factorial.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="factorial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="factorial.is_commutative" class="variable">is_commutative</dd>
-                <dd id="factorial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="factorial.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="factorial.is_real" class="variable">is_real</dd>
-                <dd id="factorial.is_positive" class="variable">is_positive</dd>
-                <dd id="factorial.is_finite" class="variable">is_finite</dd>
-                <dd id="factorial.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="factorial.is_infinite" class="variable">is_infinite</dd>
-                <dd id="factorial.is_rational" class="variable">is_rational</dd>
-                <dd id="factorial.is_composite" class="variable">is_composite</dd>
-                <dd id="factorial.is_negative" class="variable">is_negative</dd>
-                <dd id="factorial.is_polar" class="variable">is_polar</dd>
+                <dd id="factorial.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="factorial.is_odd" class="variable">is_odd</dd>
-                <dd id="factorial.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="factorial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="factorial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="factorial.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="factorial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="factorial.is_integer" class="variable">is_integer</dd>
+                <dd id="factorial.is_rational" class="variable">is_rational</dd>
+                <dd id="factorial.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="factorial.is_complex" class="variable">is_complex</dd>
                 <dd id="factorial.is_prime" class="variable">is_prime</dd>
+                <dd id="factorial.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="factorial.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="factorial.is_real" class="variable">is_real</dd>
+                <dd id="factorial.is_commutative" class="variable">is_commutative</dd>
                 <dd id="factorial.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="factorial.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="factorial.is_polar" class="variable">is_polar</dd>
+                <dd id="factorial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="factorial.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="factorial.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -7115,29 +7443,26 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="factorial2">
-                                <div class="attr class">
-        <a class="headerlink" href="#factorial2">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">factorial2</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.factorial2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">factorial2</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.factorial2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#factorial2"></a>
     
             <div class="docstring"><p>The double factorial <code>n!!</code>, not to be confused with <code>(n!)!</code></p>
 
 <p>The double factorial is defined for nonnegative integers and for odd
 negative integers as:</p>
 
-<p>.. math:: n!! = \begin{cases} 1 &amp; n = 0 \
-                n(n-2)(n-4) \cdots 1 &amp; n\ \text{positive odd} \
-                n(n-2)(n-4) \cdots 2 &amp; n\ \text{positive even} \
-                (n+2)!!/(n+2) &amp; n\ \text{negative odd} \end{cases}</p>
+<p>$$n!! = \begin{cases} 1 &amp; n = 0 \n(n-2)(n-4) \cdots 1 &amp; n\ \text{positive odd} \
+n(n-2)(n-4) \cdots 2 &amp; n\ \text{positive even} \
+(n+2)!!/(n+2) &amp; n\ \text{negative odd} \end{cases}$$</p>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Double_factorial</p>
-
 <h1 id="examples">Examples</h1>
 
 <div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial2</span><span class="p">,</span> <span class="n">var</span>
@@ -7157,26 +7482,34 @@ <h1 id="examples">Examples</h1>
 <h1 id="see-also">See Also</h1>
 
 <p>factorial, RisingFactorial, FallingFactorial</p>
+
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="factorial2.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#factorial2.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">factorial2</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">factorial2</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#factorial2.__init__"></a>
     
     
 
                             </div>
                             <div id="factorial2.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#factorial2.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#factorial2.default_assumptions"></a>
     
     
 
@@ -7331,37 +7664,37 @@ <h5>Inherited Members</h5>
                 <dd id="factorial2.simplify" class="function">simplify</dd>
                 <dd id="factorial2.refine" class="function">refine</dd>
                 <dd id="factorial2.rewrite" class="function">rewrite</dd>
-                <dd id="factorial2.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="factorial2.is_integer" class="variable">is_integer</dd>
-                <dd id="factorial2.is_complex" class="variable">is_complex</dd>
-                <dd id="factorial2.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="factorial2.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="factorial2.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="factorial2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="factorial2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="factorial2.is_positive" class="variable">is_positive</dd>
+                <dd id="factorial2.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="factorial2.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="factorial2.is_negative" class="variable">is_negative</dd>
+                <dd id="factorial2.is_composite" class="variable">is_composite</dd>
+                <dd id="factorial2.is_finite" class="variable">is_finite</dd>
                 <dd id="factorial2.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="factorial2.is_zero" class="variable">is_zero</dd>
                 <dd id="factorial2.is_even" class="variable">is_even</dd>
-                <dd id="factorial2.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="factorial2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="factorial2.is_commutative" class="variable">is_commutative</dd>
-                <dd id="factorial2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="factorial2.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="factorial2.is_real" class="variable">is_real</dd>
-                <dd id="factorial2.is_positive" class="variable">is_positive</dd>
-                <dd id="factorial2.is_finite" class="variable">is_finite</dd>
-                <dd id="factorial2.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="factorial2.is_infinite" class="variable">is_infinite</dd>
-                <dd id="factorial2.is_rational" class="variable">is_rational</dd>
-                <dd id="factorial2.is_composite" class="variable">is_composite</dd>
-                <dd id="factorial2.is_negative" class="variable">is_negative</dd>
-                <dd id="factorial2.is_polar" class="variable">is_polar</dd>
+                <dd id="factorial2.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="factorial2.is_odd" class="variable">is_odd</dd>
-                <dd id="factorial2.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="factorial2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="factorial2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="factorial2.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="factorial2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="factorial2.is_integer" class="variable">is_integer</dd>
+                <dd id="factorial2.is_rational" class="variable">is_rational</dd>
+                <dd id="factorial2.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="factorial2.is_complex" class="variable">is_complex</dd>
                 <dd id="factorial2.is_prime" class="variable">is_prime</dd>
+                <dd id="factorial2.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="factorial2.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="factorial2.is_real" class="variable">is_real</dd>
+                <dd id="factorial2.is_commutative" class="variable">is_commutative</dd>
                 <dd id="factorial2.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="factorial2.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="factorial2.is_polar" class="variable">is_polar</dd>
+                <dd id="factorial2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="factorial2.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="factorial2.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -7373,20 +7706,20 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="rf">
-                                <div class="attr class">
-        <a class="headerlink" href="#rf">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">rf</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.RisingFactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">rf</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.RisingFactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#rf"></a>
     
-            <div class="docstring"><p>Rising factorial (also called Pochhammer symbol [1]_) is a double valued
+            <div class="docstring"><p>Rising factorial (also called Pochhammer symbol <sup class="footnote-ref" id="fnref-1"><a href="#fn-1">1</a></sup>) is a double valued
 function arising in concrete mathematics, hypergeometric functions
 and series expansions. It is defined by:</p>
 
-<p>.. math:: rf(x,k) = x \cdot (x+1) \cdots (x+k-1)</p>
+<p>$$rf(x,k) = x \cdot (x+1) \cdots (x+k-1)$$</p>
 
 <p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For
 more information check "Concrete mathematics" by Graham, pp. 66
@@ -7394,7 +7727,7 @@ <h5>Inherited Members</h5>
 
 <p>When <code>x</code> is a Poly instance of degree $\ge 1$ with a single variable,
 <code>rf(x,k) = x(y) \cdot x(y+1) \cdots x(y+k-1)</code>, where <code>y</code> is the
-variable of <code>x</code>. This is as described in [2]_.</p>
+variable of <code>x</code>. This is as described in <sup class="footnote-ref" id="fnref-2"><a href="#fn-2">2</a></sup>.</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -7434,30 +7767,42 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol
-.. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic
-       Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268,
-       1995.</p>
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-1">
+<p>https://en.wikipedia.org/wiki/Pochhammer_symbol&#160;<a href="#fnref-1" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+
+<li id="fn-2">
+<p>Peter Paule, "Greatest Factorial Factorization and Symbolic
+Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268,
+1995.&#160;<a href="#fnref-2" class="footnoteBackLink" title="Jump back to footnote 2 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="rf.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#rf.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">rf</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">rf</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#rf.__init__"></a>
     
     
 
                             </div>
                             <div id="rf.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#rf.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#rf.default_assumptions"></a>
     
     
 
@@ -7612,37 +7957,37 @@ <h5>Inherited Members</h5>
                 <dd id="rf.simplify" class="function">simplify</dd>
                 <dd id="rf.refine" class="function">refine</dd>
                 <dd id="rf.rewrite" class="function">rewrite</dd>
-                <dd id="rf.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="rf.is_integer" class="variable">is_integer</dd>
-                <dd id="rf.is_complex" class="variable">is_complex</dd>
-                <dd id="rf.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="rf.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="rf.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="rf.is_irrational" class="variable">is_irrational</dd>
+                <dd id="rf.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="rf.is_positive" class="variable">is_positive</dd>
+                <dd id="rf.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="rf.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="rf.is_negative" class="variable">is_negative</dd>
+                <dd id="rf.is_composite" class="variable">is_composite</dd>
+                <dd id="rf.is_finite" class="variable">is_finite</dd>
                 <dd id="rf.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="rf.is_zero" class="variable">is_zero</dd>
                 <dd id="rf.is_even" class="variable">is_even</dd>
-                <dd id="rf.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="rf.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="rf.is_commutative" class="variable">is_commutative</dd>
-                <dd id="rf.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="rf.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="rf.is_real" class="variable">is_real</dd>
-                <dd id="rf.is_positive" class="variable">is_positive</dd>
-                <dd id="rf.is_finite" class="variable">is_finite</dd>
-                <dd id="rf.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="rf.is_infinite" class="variable">is_infinite</dd>
-                <dd id="rf.is_rational" class="variable">is_rational</dd>
-                <dd id="rf.is_composite" class="variable">is_composite</dd>
-                <dd id="rf.is_negative" class="variable">is_negative</dd>
-                <dd id="rf.is_polar" class="variable">is_polar</dd>
+                <dd id="rf.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="rf.is_odd" class="variable">is_odd</dd>
-                <dd id="rf.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="rf.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="rf.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="rf.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="rf.is_irrational" class="variable">is_irrational</dd>
+                <dd id="rf.is_integer" class="variable">is_integer</dd>
+                <dd id="rf.is_rational" class="variable">is_rational</dd>
+                <dd id="rf.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="rf.is_complex" class="variable">is_complex</dd>
                 <dd id="rf.is_prime" class="variable">is_prime</dd>
+                <dd id="rf.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="rf.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="rf.is_real" class="variable">is_real</dd>
+                <dd id="rf.is_commutative" class="variable">is_commutative</dd>
                 <dd id="rf.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="rf.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="rf.is_polar" class="variable">is_polar</dd>
+                <dd id="rf.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="rf.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="rf.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -7654,20 +7999,20 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="ff">
-                                <div class="attr class">
-        <a class="headerlink" href="#ff">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">ff</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.FallingFactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">ff</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.FallingFactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#ff"></a>
     
             <div class="docstring"><p>Falling factorial (related to rising factorial) is a double valued
 function arising in concrete mathematics, hypergeometric functions
 and series expansions. It is defined by</p>
 
-<p>.. math:: ff(x,k) = x \cdot (x-1) \cdots (x-k+1)</p>
+<p>$$ff(x,k) = x \cdot (x-1) \cdots (x-k+1)$$</p>
 
 <p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For
 more information check "Concrete mathematics" by Graham, pp. 66
@@ -7720,27 +8065,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://mathworld.wolfram.com/FallingFactorial.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="ff.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#ff.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">ff</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">ff</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#ff.__init__"></a>
     
     
 
                             </div>
                             <div id="ff.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#ff.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#ff.default_assumptions"></a>
     
     
 
@@ -7895,37 +8246,37 @@ <h5>Inherited Members</h5>
                 <dd id="ff.simplify" class="function">simplify</dd>
                 <dd id="ff.refine" class="function">refine</dd>
                 <dd id="ff.rewrite" class="function">rewrite</dd>
-                <dd id="ff.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="ff.is_integer" class="variable">is_integer</dd>
-                <dd id="ff.is_complex" class="variable">is_complex</dd>
-                <dd id="ff.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="ff.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="ff.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="ff.is_irrational" class="variable">is_irrational</dd>
+                <dd id="ff.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="ff.is_positive" class="variable">is_positive</dd>
+                <dd id="ff.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="ff.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="ff.is_negative" class="variable">is_negative</dd>
+                <dd id="ff.is_composite" class="variable">is_composite</dd>
+                <dd id="ff.is_finite" class="variable">is_finite</dd>
                 <dd id="ff.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="ff.is_zero" class="variable">is_zero</dd>
                 <dd id="ff.is_even" class="variable">is_even</dd>
-                <dd id="ff.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="ff.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="ff.is_commutative" class="variable">is_commutative</dd>
-                <dd id="ff.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="ff.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="ff.is_real" class="variable">is_real</dd>
-                <dd id="ff.is_positive" class="variable">is_positive</dd>
-                <dd id="ff.is_finite" class="variable">is_finite</dd>
-                <dd id="ff.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="ff.is_infinite" class="variable">is_infinite</dd>
-                <dd id="ff.is_rational" class="variable">is_rational</dd>
-                <dd id="ff.is_composite" class="variable">is_composite</dd>
-                <dd id="ff.is_negative" class="variable">is_negative</dd>
-                <dd id="ff.is_polar" class="variable">is_polar</dd>
+                <dd id="ff.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="ff.is_odd" class="variable">is_odd</dd>
-                <dd id="ff.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="ff.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="ff.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="ff.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="ff.is_irrational" class="variable">is_irrational</dd>
+                <dd id="ff.is_integer" class="variable">is_integer</dd>
+                <dd id="ff.is_rational" class="variable">is_rational</dd>
+                <dd id="ff.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="ff.is_complex" class="variable">is_complex</dd>
                 <dd id="ff.is_prime" class="variable">is_prime</dd>
+                <dd id="ff.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="ff.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="ff.is_real" class="variable">is_real</dd>
+                <dd id="ff.is_commutative" class="variable">is_commutative</dd>
                 <dd id="ff.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="ff.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="ff.is_polar" class="variable">is_polar</dd>
+                <dd id="ff.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="ff.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="ff.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -7937,20 +8288,19 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="binomial">
-                                <div class="attr class">
-        <a class="headerlink" href="#binomial">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">binomial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.binomial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">binomial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.binomial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#binomial"></a>
     
             <div class="docstring"><p>Implementation of the binomial coefficient. It can be defined
 in two ways depending on its desired interpretation:</p>
 
-<p>.. math:: \binom{n}{k} = \frac{n!}{k!(n-k)!}\ \text{or}\
-            \binom{n}{k} = \frac{ff(n, k)}{k!}</p>
+<p>$$\binom{n}{k} = \frac{n!}{k!(n-k)!}\ \text{or}\binom{n}{k} = \frac{ff(n, k)}{k!}$$</p>
 
 <p>First, in a strict combinatorial sense it defines the
 number of ways we can choose <code>k</code> elements from a set of
@@ -8026,27 +8376,33 @@ <h1 id="examples">Examples</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://www.johndcook.com/blog/binomial_coefficients/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="binomial.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#binomial.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">binomial</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">binomial</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#binomial.__init__"></a>
     
     
 
                             </div>
                             <div id="binomial.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#binomial.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#binomial.default_assumptions"></a>
     
     
 
@@ -8201,37 +8557,37 @@ <h5>Inherited Members</h5>
                 <dd id="binomial.simplify" class="function">simplify</dd>
                 <dd id="binomial.refine" class="function">refine</dd>
                 <dd id="binomial.rewrite" class="function">rewrite</dd>
-                <dd id="binomial.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="binomial.is_integer" class="variable">is_integer</dd>
-                <dd id="binomial.is_complex" class="variable">is_complex</dd>
-                <dd id="binomial.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="binomial.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="binomial.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="binomial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="binomial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="binomial.is_positive" class="variable">is_positive</dd>
+                <dd id="binomial.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="binomial.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="binomial.is_negative" class="variable">is_negative</dd>
+                <dd id="binomial.is_composite" class="variable">is_composite</dd>
+                <dd id="binomial.is_finite" class="variable">is_finite</dd>
                 <dd id="binomial.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="binomial.is_zero" class="variable">is_zero</dd>
                 <dd id="binomial.is_even" class="variable">is_even</dd>
-                <dd id="binomial.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="binomial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="binomial.is_commutative" class="variable">is_commutative</dd>
-                <dd id="binomial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="binomial.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="binomial.is_real" class="variable">is_real</dd>
-                <dd id="binomial.is_positive" class="variable">is_positive</dd>
-                <dd id="binomial.is_finite" class="variable">is_finite</dd>
-                <dd id="binomial.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="binomial.is_infinite" class="variable">is_infinite</dd>
-                <dd id="binomial.is_rational" class="variable">is_rational</dd>
-                <dd id="binomial.is_composite" class="variable">is_composite</dd>
-                <dd id="binomial.is_negative" class="variable">is_negative</dd>
-                <dd id="binomial.is_polar" class="variable">is_polar</dd>
+                <dd id="binomial.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="binomial.is_odd" class="variable">is_odd</dd>
-                <dd id="binomial.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="binomial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="binomial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="binomial.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="binomial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="binomial.is_integer" class="variable">is_integer</dd>
+                <dd id="binomial.is_rational" class="variable">is_rational</dd>
+                <dd id="binomial.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="binomial.is_complex" class="variable">is_complex</dd>
                 <dd id="binomial.is_prime" class="variable">is_prime</dd>
+                <dd id="binomial.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="binomial.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="binomial.is_real" class="variable">is_real</dd>
+                <dd id="binomial.is_commutative" class="variable">is_commutative</dd>
                 <dd id="binomial.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="binomial.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="binomial.is_polar" class="variable">is_polar</dd>
+                <dd id="binomial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="binomial.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="binomial.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -8243,20 +8599,20 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="RisingFactorial">
-                                <div class="attr class">
-        <a class="headerlink" href="#RisingFactorial">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">RisingFactorial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.RisingFactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">RisingFactorial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.RisingFactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#RisingFactorial"></a>
     
-            <div class="docstring"><p>Rising factorial (also called Pochhammer symbol [1]_) is a double valued
+            <div class="docstring"><p>Rising factorial (also called Pochhammer symbol <sup class="footnote-ref" id="fnref-1"><a href="#fn-1">1</a></sup>) is a double valued
 function arising in concrete mathematics, hypergeometric functions
 and series expansions. It is defined by:</p>
 
-<p>.. math:: rf(x,k) = x \cdot (x+1) \cdots (x+k-1)</p>
+<p>$$rf(x,k) = x \cdot (x+1) \cdots (x+k-1)$$</p>
 
 <p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For
 more information check "Concrete mathematics" by Graham, pp. 66
@@ -8264,7 +8620,7 @@ <h5>Inherited Members</h5>
 
 <p>When <code>x</code> is a Poly instance of degree $\ge 1$ with a single variable,
 <code>rf(x,k) = x(y) \cdot x(y+1) \cdots x(y+k-1)</code>, where <code>y</code> is the
-variable of <code>x</code>. This is as described in [2]_.</p>
+variable of <code>x</code>. This is as described in <sup class="footnote-ref" id="fnref-2"><a href="#fn-2">2</a></sup>.</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -8304,30 +8660,42 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol
-.. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic
-       Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268,
-       1995.</p>
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-1">
+<p>https://en.wikipedia.org/wiki/Pochhammer_symbol&#160;<a href="#fnref-1" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+
+<li id="fn-2">
+<p>Peter Paule, "Greatest Factorial Factorization and Symbolic
+Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268,
+1995.&#160;<a href="#fnref-2" class="footnoteBackLink" title="Jump back to footnote 2 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="RisingFactorial.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#RisingFactorial.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">RisingFactorial</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">RisingFactorial</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#RisingFactorial.__init__"></a>
     
     
 
                             </div>
                             <div id="RisingFactorial.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#RisingFactorial.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#RisingFactorial.default_assumptions"></a>
     
     
 
@@ -8482,37 +8850,37 @@ <h5>Inherited Members</h5>
                 <dd id="RisingFactorial.simplify" class="function">simplify</dd>
                 <dd id="RisingFactorial.refine" class="function">refine</dd>
                 <dd id="RisingFactorial.rewrite" class="function">rewrite</dd>
-                <dd id="RisingFactorial.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="RisingFactorial.is_integer" class="variable">is_integer</dd>
-                <dd id="RisingFactorial.is_complex" class="variable">is_complex</dd>
-                <dd id="RisingFactorial.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="RisingFactorial.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="RisingFactorial.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="RisingFactorial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="RisingFactorial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="RisingFactorial.is_positive" class="variable">is_positive</dd>
+                <dd id="RisingFactorial.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="RisingFactorial.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="RisingFactorial.is_negative" class="variable">is_negative</dd>
+                <dd id="RisingFactorial.is_composite" class="variable">is_composite</dd>
+                <dd id="RisingFactorial.is_finite" class="variable">is_finite</dd>
                 <dd id="RisingFactorial.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="RisingFactorial.is_zero" class="variable">is_zero</dd>
                 <dd id="RisingFactorial.is_even" class="variable">is_even</dd>
-                <dd id="RisingFactorial.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="RisingFactorial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="RisingFactorial.is_commutative" class="variable">is_commutative</dd>
-                <dd id="RisingFactorial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="RisingFactorial.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="RisingFactorial.is_real" class="variable">is_real</dd>
-                <dd id="RisingFactorial.is_positive" class="variable">is_positive</dd>
-                <dd id="RisingFactorial.is_finite" class="variable">is_finite</dd>
-                <dd id="RisingFactorial.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="RisingFactorial.is_infinite" class="variable">is_infinite</dd>
-                <dd id="RisingFactorial.is_rational" class="variable">is_rational</dd>
-                <dd id="RisingFactorial.is_composite" class="variable">is_composite</dd>
-                <dd id="RisingFactorial.is_negative" class="variable">is_negative</dd>
-                <dd id="RisingFactorial.is_polar" class="variable">is_polar</dd>
+                <dd id="RisingFactorial.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="RisingFactorial.is_odd" class="variable">is_odd</dd>
-                <dd id="RisingFactorial.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="RisingFactorial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="RisingFactorial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="RisingFactorial.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="RisingFactorial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="RisingFactorial.is_integer" class="variable">is_integer</dd>
+                <dd id="RisingFactorial.is_rational" class="variable">is_rational</dd>
+                <dd id="RisingFactorial.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="RisingFactorial.is_complex" class="variable">is_complex</dd>
                 <dd id="RisingFactorial.is_prime" class="variable">is_prime</dd>
+                <dd id="RisingFactorial.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="RisingFactorial.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="RisingFactorial.is_real" class="variable">is_real</dd>
+                <dd id="RisingFactorial.is_commutative" class="variable">is_commutative</dd>
                 <dd id="RisingFactorial.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="RisingFactorial.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="RisingFactorial.is_polar" class="variable">is_polar</dd>
+                <dd id="RisingFactorial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="RisingFactorial.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="RisingFactorial.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -8524,20 +8892,20 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="FallingFactorial">
-                                <div class="attr class">
-        <a class="headerlink" href="#FallingFactorial">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">FallingFactorial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.FallingFactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">FallingFactorial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.FallingFactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#FallingFactorial"></a>
     
             <div class="docstring"><p>Falling factorial (related to rising factorial) is a double valued
 function arising in concrete mathematics, hypergeometric functions
 and series expansions. It is defined by</p>
 
-<p>.. math:: ff(x,k) = x \cdot (x-1) \cdots (x-k+1)</p>
+<p>$$ff(x,k) = x \cdot (x-1) \cdots (x-k+1)$$</p>
 
 <p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For
 more information check "Concrete mathematics" by Graham, pp. 66
@@ -8590,27 +8958,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://mathworld.wolfram.com/FallingFactorial.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="FallingFactorial.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#FallingFactorial.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">FallingFactorial</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">FallingFactorial</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#FallingFactorial.__init__"></a>
     
     
 
                             </div>
                             <div id="FallingFactorial.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#FallingFactorial.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#FallingFactorial.default_assumptions"></a>
     
     
 
@@ -8765,37 +9139,37 @@ <h5>Inherited Members</h5>
                 <dd id="FallingFactorial.simplify" class="function">simplify</dd>
                 <dd id="FallingFactorial.refine" class="function">refine</dd>
                 <dd id="FallingFactorial.rewrite" class="function">rewrite</dd>
-                <dd id="FallingFactorial.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="FallingFactorial.is_integer" class="variable">is_integer</dd>
-                <dd id="FallingFactorial.is_complex" class="variable">is_complex</dd>
-                <dd id="FallingFactorial.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="FallingFactorial.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="FallingFactorial.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="FallingFactorial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="FallingFactorial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="FallingFactorial.is_positive" class="variable">is_positive</dd>
+                <dd id="FallingFactorial.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="FallingFactorial.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="FallingFactorial.is_negative" class="variable">is_negative</dd>
+                <dd id="FallingFactorial.is_composite" class="variable">is_composite</dd>
+                <dd id="FallingFactorial.is_finite" class="variable">is_finite</dd>
                 <dd id="FallingFactorial.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="FallingFactorial.is_zero" class="variable">is_zero</dd>
                 <dd id="FallingFactorial.is_even" class="variable">is_even</dd>
-                <dd id="FallingFactorial.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="FallingFactorial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="FallingFactorial.is_commutative" class="variable">is_commutative</dd>
-                <dd id="FallingFactorial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="FallingFactorial.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="FallingFactorial.is_real" class="variable">is_real</dd>
-                <dd id="FallingFactorial.is_positive" class="variable">is_positive</dd>
-                <dd id="FallingFactorial.is_finite" class="variable">is_finite</dd>
-                <dd id="FallingFactorial.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="FallingFactorial.is_infinite" class="variable">is_infinite</dd>
-                <dd id="FallingFactorial.is_rational" class="variable">is_rational</dd>
-                <dd id="FallingFactorial.is_composite" class="variable">is_composite</dd>
-                <dd id="FallingFactorial.is_negative" class="variable">is_negative</dd>
-                <dd id="FallingFactorial.is_polar" class="variable">is_polar</dd>
+                <dd id="FallingFactorial.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="FallingFactorial.is_odd" class="variable">is_odd</dd>
-                <dd id="FallingFactorial.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="FallingFactorial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="FallingFactorial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="FallingFactorial.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="FallingFactorial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="FallingFactorial.is_integer" class="variable">is_integer</dd>
+                <dd id="FallingFactorial.is_rational" class="variable">is_rational</dd>
+                <dd id="FallingFactorial.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="FallingFactorial.is_complex" class="variable">is_complex</dd>
                 <dd id="FallingFactorial.is_prime" class="variable">is_prime</dd>
+                <dd id="FallingFactorial.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="FallingFactorial.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="FallingFactorial.is_real" class="variable">is_real</dd>
+                <dd id="FallingFactorial.is_commutative" class="variable">is_commutative</dd>
                 <dd id="FallingFactorial.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="FallingFactorial.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="FallingFactorial.is_polar" class="variable">is_polar</dd>
+                <dd id="FallingFactorial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="FallingFactorial.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="FallingFactorial.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -8807,38 +9181,34 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="subfactorial">
-                                <div class="attr class">
-        <a class="headerlink" href="#subfactorial">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">subfactorial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.subfactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">subfactorial</span><wbr>(<span class="base">sympy.functions.combinatorial.factorials.subfactorial</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#subfactorial"></a>
     
             <div class="docstring"><p>The subfactorial counts the derangements of n items and is
 defined for non-negative integers as:</p>
 
-<p>.. math:: !n = \begin{cases} 1 &amp; n = 0 \ 0 &amp; n = 1 \
-                (n-1)(!(n-1) + !(n-2)) &amp; n &gt; 1 \end{cases}</p>
+<p>$$!n = \begin{cases} 1 &amp; n = 0 \ 0 &amp; n = 1 \(n-1)(!(n-1) + !(n-2)) &amp; n &gt; 1 \end{cases}$$</p>
 
 <p>It can also be written as <code>int(round(n!/exp(1)))</code> but the
 recursive definition with caching is implemented for this function.</p>
 
-<p>An interesting analytic expression is the following [2]_</p>
+<p>An interesting analytic expression is the following <sup class="footnote-ref" id="fnref-2"><a href="#fn-2">1</a></sup></p>
 
-<p>.. math:: !x = \Gamma(x + 1, -1)/e</p>
+<p>$$!x = \Gamma(x + 1, -1)/e$$</p>
 
 <p>which is valid for non-negative integers <code>x</code>. The above formula
-is not very useful incase of non-integers. :math:<code>\Gamma(x + 1, -1)</code> is
+is not very useful incase of non-integers. \( \Gamma(x + 1, -1) \) is
 single-valued only for integral arguments <code>x</code>, elsewhere on the positive
 real axis it has an infinite number of branches none of which are real.</p>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Subfactorial
-.. [2] http://mathworld.wolfram.com/Subfactorial.html</p>
-
 <h1 id="examples">Examples</h1>
 
 <div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">subfactorial</span>
@@ -8854,26 +9224,37 @@ <h1 id="see-also">See Also</h1>
 <p>sympy.functions.combinatorial.factorials.factorial,
 sympy.utilities.iterables.generate_derangements,
 sympy.functions.special.gamma_functions.uppergamma</p>
+
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-2">
+<p>http://mathworld.wolfram.com/Subfactorial.html&#160;<a href="#fnref-2" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="subfactorial.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#subfactorial.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">subfactorial</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">subfactorial</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#subfactorial.__init__"></a>
     
     
 
                             </div>
                             <div id="subfactorial.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#subfactorial.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#subfactorial.default_assumptions"></a>
     
     
 
@@ -9028,37 +9409,37 @@ <h5>Inherited Members</h5>
                 <dd id="subfactorial.simplify" class="function">simplify</dd>
                 <dd id="subfactorial.refine" class="function">refine</dd>
                 <dd id="subfactorial.rewrite" class="function">rewrite</dd>
-                <dd id="subfactorial.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="subfactorial.is_integer" class="variable">is_integer</dd>
-                <dd id="subfactorial.is_complex" class="variable">is_complex</dd>
-                <dd id="subfactorial.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="subfactorial.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="subfactorial.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="subfactorial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="subfactorial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="subfactorial.is_positive" class="variable">is_positive</dd>
+                <dd id="subfactorial.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="subfactorial.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="subfactorial.is_negative" class="variable">is_negative</dd>
+                <dd id="subfactorial.is_composite" class="variable">is_composite</dd>
+                <dd id="subfactorial.is_finite" class="variable">is_finite</dd>
                 <dd id="subfactorial.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="subfactorial.is_zero" class="variable">is_zero</dd>
                 <dd id="subfactorial.is_even" class="variable">is_even</dd>
-                <dd id="subfactorial.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="subfactorial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="subfactorial.is_commutative" class="variable">is_commutative</dd>
-                <dd id="subfactorial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="subfactorial.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="subfactorial.is_real" class="variable">is_real</dd>
-                <dd id="subfactorial.is_positive" class="variable">is_positive</dd>
-                <dd id="subfactorial.is_finite" class="variable">is_finite</dd>
-                <dd id="subfactorial.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="subfactorial.is_infinite" class="variable">is_infinite</dd>
-                <dd id="subfactorial.is_rational" class="variable">is_rational</dd>
-                <dd id="subfactorial.is_composite" class="variable">is_composite</dd>
-                <dd id="subfactorial.is_negative" class="variable">is_negative</dd>
-                <dd id="subfactorial.is_polar" class="variable">is_polar</dd>
+                <dd id="subfactorial.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="subfactorial.is_odd" class="variable">is_odd</dd>
-                <dd id="subfactorial.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="subfactorial.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="subfactorial.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="subfactorial.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="subfactorial.is_irrational" class="variable">is_irrational</dd>
+                <dd id="subfactorial.is_integer" class="variable">is_integer</dd>
+                <dd id="subfactorial.is_rational" class="variable">is_rational</dd>
+                <dd id="subfactorial.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="subfactorial.is_complex" class="variable">is_complex</dd>
                 <dd id="subfactorial.is_prime" class="variable">is_prime</dd>
+                <dd id="subfactorial.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="subfactorial.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="subfactorial.is_real" class="variable">is_real</dd>
+                <dd id="subfactorial.is_commutative" class="variable">is_commutative</dd>
                 <dd id="subfactorial.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="subfactorial.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="subfactorial.is_polar" class="variable">is_polar</dd>
+                <dd id="subfactorial.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="subfactorial.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="subfactorial.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -9070,14 +9451,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="carmichael">
-                                <div class="attr class">
-        <a class="headerlink" href="#carmichael">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">carmichael</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.carmichael</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">carmichael</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.carmichael</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#carmichael"></a>
     
             <div class="docstring"><p>Carmichael Numbers:</p>
 
@@ -9108,45 +9489,49 @@ <h5>Inherited Members</h5>
 <h1 id="examples">Examples</h1>
 
 <div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">carmichael</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">carmichael</span><span class="o">.</span><span class="n">find_first_n_carmichaels</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#carmichael.find_first_n_carmichaels">carmichael</span><span class="o">.</span><span class="n">find_first_n_carmichaels</a></span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <span class="go">[561, 1105, 1729, 2465, 2821]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">carmichael</span><span class="o">.</span><span class="n">is_prime</span><span class="p">(</span><span class="mi">2465</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#carmichael.is_prime">carmichael</span><span class="o">.</span><span class="n">is_prime</a></span><span class="p">(</span><span class="mi">2465</span><span class="p">)</span>
 <span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">carmichael</span><span class="o">.</span><span class="n">is_prime</span><span class="p">(</span><span class="mi">1729</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#carmichael.is_prime">carmichael</span><span class="o">.</span><span class="n">is_prime</a></span><span class="p">(</span><span class="mi">1729</span><span class="p">)</span>
 <span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">carmichael</span><span class="o">.</span><span class="n">find_carmichael_numbers_in_range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">562</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#carmichael.find_carmichael_numbers_in_range">carmichael</span><span class="o">.</span><span class="n">find_carmichael_numbers_in_range</a></span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">562</span><span class="p">)</span>
 <span class="go">[561]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">carmichael</span><span class="o">.</span><span class="n">find_carmichael_numbers_in_range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1000</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#carmichael.find_carmichael_numbers_in_range">carmichael</span><span class="o">.</span><span class="n">find_carmichael_numbers_in_range</a></span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1000</span><span class="p">)</span>
 <span class="go">[561]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">carmichael</span><span class="o">.</span><span class="n">find_carmichael_numbers_in_range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2000</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#carmichael.find_carmichael_numbers_in_range">carmichael</span><span class="o">.</span><span class="n">find_carmichael_numbers_in_range</a></span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2000</span><span class="p">)</span>
 <span class="go">[561, 1105, 1729]</span>
 </code></pre></div>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Carmichael_number
-.. [2] https://en.wikipedia.org/wiki/Fermat_primality_test
-.. [3] https://www.jstor.org/stable/23248683?seq=1#metadata_info_tab_contents</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="carmichael.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#carmichael.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">carmichael</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">carmichael</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#carmichael.__init__"></a>
     
     
 
                             </div>
                             <div id="carmichael.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#carmichael.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#carmichael.default_assumptions"></a>
     
     
 
@@ -9307,36 +9692,36 @@ <h5>Inherited Members</h5>
                 <dd id="carmichael.simplify" class="function">simplify</dd>
                 <dd id="carmichael.refine" class="function">refine</dd>
                 <dd id="carmichael.rewrite" class="function">rewrite</dd>
-                <dd id="carmichael.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="carmichael.is_integer" class="variable">is_integer</dd>
-                <dd id="carmichael.is_complex" class="variable">is_complex</dd>
-                <dd id="carmichael.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="carmichael.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="carmichael.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="carmichael.is_irrational" class="variable">is_irrational</dd>
+                <dd id="carmichael.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="carmichael.is_positive" class="variable">is_positive</dd>
+                <dd id="carmichael.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="carmichael.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="carmichael.is_negative" class="variable">is_negative</dd>
+                <dd id="carmichael.is_composite" class="variable">is_composite</dd>
+                <dd id="carmichael.is_finite" class="variable">is_finite</dd>
                 <dd id="carmichael.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="carmichael.is_zero" class="variable">is_zero</dd>
                 <dd id="carmichael.is_even" class="variable">is_even</dd>
-                <dd id="carmichael.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="carmichael.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="carmichael.is_commutative" class="variable">is_commutative</dd>
-                <dd id="carmichael.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="carmichael.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="carmichael.is_real" class="variable">is_real</dd>
-                <dd id="carmichael.is_positive" class="variable">is_positive</dd>
-                <dd id="carmichael.is_finite" class="variable">is_finite</dd>
-                <dd id="carmichael.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="carmichael.is_infinite" class="variable">is_infinite</dd>
-                <dd id="carmichael.is_rational" class="variable">is_rational</dd>
-                <dd id="carmichael.is_composite" class="variable">is_composite</dd>
-                <dd id="carmichael.is_negative" class="variable">is_negative</dd>
-                <dd id="carmichael.is_polar" class="variable">is_polar</dd>
+                <dd id="carmichael.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="carmichael.is_odd" class="variable">is_odd</dd>
-                <dd id="carmichael.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="carmichael.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="carmichael.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="carmichael.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="carmichael.is_irrational" class="variable">is_irrational</dd>
+                <dd id="carmichael.is_integer" class="variable">is_integer</dd>
+                <dd id="carmichael.is_rational" class="variable">is_rational</dd>
+                <dd id="carmichael.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="carmichael.is_complex" class="variable">is_complex</dd>
+                <dd id="carmichael.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="carmichael.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="carmichael.is_real" class="variable">is_real</dd>
+                <dd id="carmichael.is_commutative" class="variable">is_commutative</dd>
                 <dd id="carmichael.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="carmichael.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="carmichael.is_polar" class="variable">is_polar</dd>
+                <dd id="carmichael.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="carmichael.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="carmichael.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -9348,14 +9733,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="fibonacci">
-                                <div class="attr class">
-        <a class="headerlink" href="#fibonacci">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">fibonacci</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.fibonacci</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">fibonacci</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.fibonacci</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#fibonacci"></a>
     
             <div class="docstring"><p>Fibonacci numbers / Fibonacci polynomials</p>
 
@@ -9393,28 +9778,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Fibonacci_number
-.. [2] http://mathworld.wolfram.com/FibonacciNumber.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="fibonacci.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#fibonacci.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">fibonacci</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">fibonacci</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#fibonacci.__init__"></a>
     
     
 
                             </div>
                             <div id="fibonacci.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#fibonacci.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#fibonacci.default_assumptions"></a>
     
     
 
@@ -9569,37 +9959,37 @@ <h5>Inherited Members</h5>
                 <dd id="fibonacci.simplify" class="function">simplify</dd>
                 <dd id="fibonacci.refine" class="function">refine</dd>
                 <dd id="fibonacci.rewrite" class="function">rewrite</dd>
-                <dd id="fibonacci.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="fibonacci.is_integer" class="variable">is_integer</dd>
-                <dd id="fibonacci.is_complex" class="variable">is_complex</dd>
-                <dd id="fibonacci.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="fibonacci.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="fibonacci.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="fibonacci.is_irrational" class="variable">is_irrational</dd>
+                <dd id="fibonacci.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="fibonacci.is_positive" class="variable">is_positive</dd>
+                <dd id="fibonacci.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="fibonacci.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="fibonacci.is_negative" class="variable">is_negative</dd>
+                <dd id="fibonacci.is_composite" class="variable">is_composite</dd>
+                <dd id="fibonacci.is_finite" class="variable">is_finite</dd>
                 <dd id="fibonacci.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="fibonacci.is_zero" class="variable">is_zero</dd>
                 <dd id="fibonacci.is_even" class="variable">is_even</dd>
-                <dd id="fibonacci.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="fibonacci.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="fibonacci.is_commutative" class="variable">is_commutative</dd>
-                <dd id="fibonacci.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="fibonacci.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="fibonacci.is_real" class="variable">is_real</dd>
-                <dd id="fibonacci.is_positive" class="variable">is_positive</dd>
-                <dd id="fibonacci.is_finite" class="variable">is_finite</dd>
-                <dd id="fibonacci.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="fibonacci.is_infinite" class="variable">is_infinite</dd>
-                <dd id="fibonacci.is_rational" class="variable">is_rational</dd>
-                <dd id="fibonacci.is_composite" class="variable">is_composite</dd>
-                <dd id="fibonacci.is_negative" class="variable">is_negative</dd>
-                <dd id="fibonacci.is_polar" class="variable">is_polar</dd>
+                <dd id="fibonacci.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="fibonacci.is_odd" class="variable">is_odd</dd>
-                <dd id="fibonacci.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="fibonacci.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="fibonacci.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="fibonacci.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="fibonacci.is_irrational" class="variable">is_irrational</dd>
+                <dd id="fibonacci.is_integer" class="variable">is_integer</dd>
+                <dd id="fibonacci.is_rational" class="variable">is_rational</dd>
+                <dd id="fibonacci.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="fibonacci.is_complex" class="variable">is_complex</dd>
                 <dd id="fibonacci.is_prime" class="variable">is_prime</dd>
+                <dd id="fibonacci.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="fibonacci.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="fibonacci.is_real" class="variable">is_real</dd>
+                <dd id="fibonacci.is_commutative" class="variable">is_commutative</dd>
                 <dd id="fibonacci.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="fibonacci.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="fibonacci.is_polar" class="variable">is_polar</dd>
+                <dd id="fibonacci.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="fibonacci.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="fibonacci.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -9611,14 +10001,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="lucas">
-                                <div class="attr class">
-        <a class="headerlink" href="#lucas">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">lucas</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.lucas</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">lucas</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.lucas</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#lucas"></a>
     
             <div class="docstring"><p>Lucas numbers</p>
 
@@ -9646,28 +10036,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Lucas_number
-.. [2] http://mathworld.wolfram.com/LucasNumber.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="lucas.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#lucas.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">lucas</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">lucas</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#lucas.__init__"></a>
     
     
 
                             </div>
                             <div id="lucas.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#lucas.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#lucas.default_assumptions"></a>
     
     
 
@@ -9822,37 +10217,37 @@ <h5>Inherited Members</h5>
                 <dd id="lucas.simplify" class="function">simplify</dd>
                 <dd id="lucas.refine" class="function">refine</dd>
                 <dd id="lucas.rewrite" class="function">rewrite</dd>
-                <dd id="lucas.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="lucas.is_integer" class="variable">is_integer</dd>
-                <dd id="lucas.is_complex" class="variable">is_complex</dd>
-                <dd id="lucas.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="lucas.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="lucas.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="lucas.is_irrational" class="variable">is_irrational</dd>
+                <dd id="lucas.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="lucas.is_positive" class="variable">is_positive</dd>
+                <dd id="lucas.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="lucas.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="lucas.is_negative" class="variable">is_negative</dd>
+                <dd id="lucas.is_composite" class="variable">is_composite</dd>
+                <dd id="lucas.is_finite" class="variable">is_finite</dd>
                 <dd id="lucas.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="lucas.is_zero" class="variable">is_zero</dd>
                 <dd id="lucas.is_even" class="variable">is_even</dd>
-                <dd id="lucas.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="lucas.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="lucas.is_commutative" class="variable">is_commutative</dd>
-                <dd id="lucas.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="lucas.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="lucas.is_real" class="variable">is_real</dd>
-                <dd id="lucas.is_positive" class="variable">is_positive</dd>
-                <dd id="lucas.is_finite" class="variable">is_finite</dd>
-                <dd id="lucas.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="lucas.is_infinite" class="variable">is_infinite</dd>
-                <dd id="lucas.is_rational" class="variable">is_rational</dd>
-                <dd id="lucas.is_composite" class="variable">is_composite</dd>
-                <dd id="lucas.is_negative" class="variable">is_negative</dd>
-                <dd id="lucas.is_polar" class="variable">is_polar</dd>
+                <dd id="lucas.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="lucas.is_odd" class="variable">is_odd</dd>
-                <dd id="lucas.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="lucas.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="lucas.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="lucas.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="lucas.is_irrational" class="variable">is_irrational</dd>
+                <dd id="lucas.is_integer" class="variable">is_integer</dd>
+                <dd id="lucas.is_rational" class="variable">is_rational</dd>
+                <dd id="lucas.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="lucas.is_complex" class="variable">is_complex</dd>
                 <dd id="lucas.is_prime" class="variable">is_prime</dd>
+                <dd id="lucas.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="lucas.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="lucas.is_real" class="variable">is_real</dd>
+                <dd id="lucas.is_commutative" class="variable">is_commutative</dd>
                 <dd id="lucas.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="lucas.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="lucas.is_polar" class="variable">is_polar</dd>
+                <dd id="lucas.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="lucas.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="lucas.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -9864,14 +10259,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="motzkin">
-                                <div class="attr class">
-        <a class="headerlink" href="#motzkin">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">motzkin</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.motzkin</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">motzkin</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.motzkin</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#motzkin"></a>
     
             <div class="docstring"><p>The nth Motzkin number is the number
 of ways of drawing non-intersecting chords
@@ -9889,40 +10284,45 @@ <h1 id="examples">Examples</h1>
 <div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">motzkin</span>
 </code></pre></div>
 
-<div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="n">motzkin</span><span class="o">.</span><span class="n">is_motzkin</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
+<div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#motzkin.is_motzkin">motzkin</span><span class="o">.</span><span class="n">is_motzkin</a></span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">motzkin</span><span class="o">.</span><span class="n">find_motzkin_numbers_in_range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">300</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#motzkin.find_motzkin_numbers_in_range">motzkin</span><span class="o">.</span><span class="n">find_motzkin_numbers_in_range</a></span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">300</span><span class="p">)</span>
 <span class="go">[2, 4, 9, 21, 51, 127]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">motzkin</span><span class="o">.</span><span class="n">find_motzkin_numbers_in_range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">900</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#motzkin.find_motzkin_numbers_in_range">motzkin</span><span class="o">.</span><span class="n">find_motzkin_numbers_in_range</a></span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">900</span><span class="p">)</span>
 <span class="go">[2, 4, 9, 21, 51, 127, 323, 835]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">motzkin</span><span class="o">.</span><span class="n">find_first_n_motzkins</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n"><a href="#motzkin.find_first_n_motzkins">motzkin</span><span class="o">.</span><span class="n">find_first_n_motzkins</a></span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
 <span class="go">[1, 1, 2, 4, 9, 21, 51, 127, 323, 835]</span>
 </code></pre></div>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Motzkin_number
-.. [2] https://mathworld.wolfram.com/MotzkinNumber.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="motzkin.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#motzkin.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">motzkin</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">motzkin</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#motzkin.__init__"></a>
     
     
 
                             </div>
                             <div id="motzkin.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#motzkin.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#motzkin.default_assumptions"></a>
     
     
 
@@ -10080,37 +10480,37 @@ <h5>Inherited Members</h5>
                 <dd id="motzkin.simplify" class="function">simplify</dd>
                 <dd id="motzkin.refine" class="function">refine</dd>
                 <dd id="motzkin.rewrite" class="function">rewrite</dd>
-                <dd id="motzkin.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="motzkin.is_integer" class="variable">is_integer</dd>
-                <dd id="motzkin.is_complex" class="variable">is_complex</dd>
-                <dd id="motzkin.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="motzkin.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="motzkin.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="motzkin.is_irrational" class="variable">is_irrational</dd>
+                <dd id="motzkin.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="motzkin.is_positive" class="variable">is_positive</dd>
+                <dd id="motzkin.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="motzkin.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="motzkin.is_negative" class="variable">is_negative</dd>
+                <dd id="motzkin.is_composite" class="variable">is_composite</dd>
+                <dd id="motzkin.is_finite" class="variable">is_finite</dd>
                 <dd id="motzkin.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="motzkin.is_zero" class="variable">is_zero</dd>
                 <dd id="motzkin.is_even" class="variable">is_even</dd>
-                <dd id="motzkin.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="motzkin.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="motzkin.is_commutative" class="variable">is_commutative</dd>
-                <dd id="motzkin.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="motzkin.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="motzkin.is_real" class="variable">is_real</dd>
-                <dd id="motzkin.is_positive" class="variable">is_positive</dd>
-                <dd id="motzkin.is_finite" class="variable">is_finite</dd>
-                <dd id="motzkin.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="motzkin.is_infinite" class="variable">is_infinite</dd>
-                <dd id="motzkin.is_rational" class="variable">is_rational</dd>
-                <dd id="motzkin.is_composite" class="variable">is_composite</dd>
-                <dd id="motzkin.is_negative" class="variable">is_negative</dd>
-                <dd id="motzkin.is_polar" class="variable">is_polar</dd>
+                <dd id="motzkin.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="motzkin.is_odd" class="variable">is_odd</dd>
-                <dd id="motzkin.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="motzkin.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="motzkin.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="motzkin.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="motzkin.is_irrational" class="variable">is_irrational</dd>
+                <dd id="motzkin.is_integer" class="variable">is_integer</dd>
+                <dd id="motzkin.is_rational" class="variable">is_rational</dd>
+                <dd id="motzkin.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="motzkin.is_complex" class="variable">is_complex</dd>
                 <dd id="motzkin.is_prime" class="variable">is_prime</dd>
+                <dd id="motzkin.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="motzkin.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="motzkin.is_real" class="variable">is_real</dd>
+                <dd id="motzkin.is_commutative" class="variable">is_commutative</dd>
                 <dd id="motzkin.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="motzkin.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="motzkin.is_polar" class="variable">is_polar</dd>
+                <dd id="motzkin.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="motzkin.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="motzkin.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -10122,14 +10522,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="tribonacci">
-                                <div class="attr class">
-        <a class="headerlink" href="#tribonacci">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">tribonacci</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.tribonacci</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">tribonacci</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.tribonacci</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#tribonacci"></a>
     
             <div class="docstring"><p>Tribonacci numbers / Tribonacci polynomials</p>
 
@@ -10163,29 +10563,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers
-.. [2] http://mathworld.wolfram.com/TribonacciNumber.html
-.. [3] https://oeis.org/A000073</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="tribonacci.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#tribonacci.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">tribonacci</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">tribonacci</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#tribonacci.__init__"></a>
     
     
 
                             </div>
                             <div id="tribonacci.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#tribonacci.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#tribonacci.default_assumptions"></a>
     
     
 
@@ -10340,37 +10744,37 @@ <h5>Inherited Members</h5>
                 <dd id="tribonacci.simplify" class="function">simplify</dd>
                 <dd id="tribonacci.refine" class="function">refine</dd>
                 <dd id="tribonacci.rewrite" class="function">rewrite</dd>
-                <dd id="tribonacci.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="tribonacci.is_integer" class="variable">is_integer</dd>
-                <dd id="tribonacci.is_complex" class="variable">is_complex</dd>
-                <dd id="tribonacci.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="tribonacci.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="tribonacci.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="tribonacci.is_irrational" class="variable">is_irrational</dd>
+                <dd id="tribonacci.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="tribonacci.is_positive" class="variable">is_positive</dd>
+                <dd id="tribonacci.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="tribonacci.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="tribonacci.is_negative" class="variable">is_negative</dd>
+                <dd id="tribonacci.is_composite" class="variable">is_composite</dd>
+                <dd id="tribonacci.is_finite" class="variable">is_finite</dd>
                 <dd id="tribonacci.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="tribonacci.is_zero" class="variable">is_zero</dd>
                 <dd id="tribonacci.is_even" class="variable">is_even</dd>
-                <dd id="tribonacci.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="tribonacci.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="tribonacci.is_commutative" class="variable">is_commutative</dd>
-                <dd id="tribonacci.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="tribonacci.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="tribonacci.is_real" class="variable">is_real</dd>
-                <dd id="tribonacci.is_positive" class="variable">is_positive</dd>
-                <dd id="tribonacci.is_finite" class="variable">is_finite</dd>
-                <dd id="tribonacci.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="tribonacci.is_infinite" class="variable">is_infinite</dd>
-                <dd id="tribonacci.is_rational" class="variable">is_rational</dd>
-                <dd id="tribonacci.is_composite" class="variable">is_composite</dd>
-                <dd id="tribonacci.is_negative" class="variable">is_negative</dd>
-                <dd id="tribonacci.is_polar" class="variable">is_polar</dd>
+                <dd id="tribonacci.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="tribonacci.is_odd" class="variable">is_odd</dd>
-                <dd id="tribonacci.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="tribonacci.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="tribonacci.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="tribonacci.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="tribonacci.is_irrational" class="variable">is_irrational</dd>
+                <dd id="tribonacci.is_integer" class="variable">is_integer</dd>
+                <dd id="tribonacci.is_rational" class="variable">is_rational</dd>
+                <dd id="tribonacci.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="tribonacci.is_complex" class="variable">is_complex</dd>
                 <dd id="tribonacci.is_prime" class="variable">is_prime</dd>
+                <dd id="tribonacci.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="tribonacci.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="tribonacci.is_real" class="variable">is_real</dd>
+                <dd id="tribonacci.is_commutative" class="variable">is_commutative</dd>
                 <dd id="tribonacci.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="tribonacci.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="tribonacci.is_polar" class="variable">is_polar</dd>
+                <dd id="tribonacci.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="tribonacci.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="tribonacci.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -10382,14 +10786,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="harmonic">
-                                <div class="attr class">
-        <a class="headerlink" href="#harmonic">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">harmonic</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.harmonic</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">harmonic</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.harmonic</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#harmonic"></a>
     
             <div class="docstring"><p>Harmonic numbers</p>
 
@@ -10398,7 +10802,7 @@ <h5>Inherited Members</h5>
 
 <p>More generally:</p>
 
-<p>.. math:: \operatorname{H}_{n,m} = \sum_{k=1}^{n} \frac{1}{k^m}</p>
+<p>$$\operatorname{H}_{n,m} = \sum_{k=1}^{n} \frac{1}{k^m}$$</p>
 
 <p>As <code>n \rightarrow \infty</code>, <code>\operatorname{H}_{n,m} \rightarrow \zeta(m)</code>,
 the Riemann zeta function.</p>
@@ -10526,29 +10930,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Harmonic_number
-.. [2] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber/
-.. [3] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="harmonic.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#harmonic.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">harmonic</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">harmonic</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#harmonic.__init__"></a>
     
     
 
                             </div>
                             <div id="harmonic.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#harmonic.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#harmonic.default_assumptions"></a>
     
     
 
@@ -10703,37 +11111,37 @@ <h5>Inherited Members</h5>
                 <dd id="harmonic.simplify" class="function">simplify</dd>
                 <dd id="harmonic.refine" class="function">refine</dd>
                 <dd id="harmonic.rewrite" class="function">rewrite</dd>
-                <dd id="harmonic.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="harmonic.is_integer" class="variable">is_integer</dd>
-                <dd id="harmonic.is_complex" class="variable">is_complex</dd>
-                <dd id="harmonic.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="harmonic.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="harmonic.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="harmonic.is_irrational" class="variable">is_irrational</dd>
+                <dd id="harmonic.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="harmonic.is_positive" class="variable">is_positive</dd>
+                <dd id="harmonic.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="harmonic.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="harmonic.is_negative" class="variable">is_negative</dd>
+                <dd id="harmonic.is_composite" class="variable">is_composite</dd>
+                <dd id="harmonic.is_finite" class="variable">is_finite</dd>
                 <dd id="harmonic.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="harmonic.is_zero" class="variable">is_zero</dd>
                 <dd id="harmonic.is_even" class="variable">is_even</dd>
-                <dd id="harmonic.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="harmonic.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="harmonic.is_commutative" class="variable">is_commutative</dd>
-                <dd id="harmonic.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="harmonic.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="harmonic.is_real" class="variable">is_real</dd>
-                <dd id="harmonic.is_positive" class="variable">is_positive</dd>
-                <dd id="harmonic.is_finite" class="variable">is_finite</dd>
-                <dd id="harmonic.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="harmonic.is_infinite" class="variable">is_infinite</dd>
-                <dd id="harmonic.is_rational" class="variable">is_rational</dd>
-                <dd id="harmonic.is_composite" class="variable">is_composite</dd>
-                <dd id="harmonic.is_negative" class="variable">is_negative</dd>
-                <dd id="harmonic.is_polar" class="variable">is_polar</dd>
+                <dd id="harmonic.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="harmonic.is_odd" class="variable">is_odd</dd>
-                <dd id="harmonic.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="harmonic.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="harmonic.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="harmonic.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="harmonic.is_irrational" class="variable">is_irrational</dd>
+                <dd id="harmonic.is_integer" class="variable">is_integer</dd>
+                <dd id="harmonic.is_rational" class="variable">is_rational</dd>
+                <dd id="harmonic.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="harmonic.is_complex" class="variable">is_complex</dd>
                 <dd id="harmonic.is_prime" class="variable">is_prime</dd>
+                <dd id="harmonic.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="harmonic.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="harmonic.is_real" class="variable">is_real</dd>
+                <dd id="harmonic.is_commutative" class="variable">is_commutative</dd>
                 <dd id="harmonic.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="harmonic.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="harmonic.is_polar" class="variable">is_polar</dd>
+                <dd id="harmonic.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="harmonic.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="harmonic.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -10745,14 +11153,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="bernoulli">
-                                <div class="attr class">
-        <a class="headerlink" href="#bernoulli">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">bernoulli</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.bernoulli</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">bernoulli</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.bernoulli</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#bernoulli"></a>
     
             <div class="docstring"><p>Bernoulli numbers / Bernoulli polynomials</p>
 
@@ -10812,30 +11220,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Bernoulli_number
-.. [2] https://en.wikipedia.org/wiki/Bernoulli_polynomial
-.. [3] http://mathworld.wolfram.com/BernoulliNumber.html
-.. [4] http://mathworld.wolfram.com/BernoulliPolynomial.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="bernoulli.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#bernoulli.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">bernoulli</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">bernoulli</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#bernoulli.__init__"></a>
     
     
 
                             </div>
                             <div id="bernoulli.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#bernoulli.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#bernoulli.default_assumptions"></a>
     
     
 
@@ -10990,37 +11401,37 @@ <h5>Inherited Members</h5>
                 <dd id="bernoulli.simplify" class="function">simplify</dd>
                 <dd id="bernoulli.refine" class="function">refine</dd>
                 <dd id="bernoulli.rewrite" class="function">rewrite</dd>
-                <dd id="bernoulli.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="bernoulli.is_integer" class="variable">is_integer</dd>
-                <dd id="bernoulli.is_complex" class="variable">is_complex</dd>
-                <dd id="bernoulli.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="bernoulli.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="bernoulli.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="bernoulli.is_irrational" class="variable">is_irrational</dd>
+                <dd id="bernoulli.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="bernoulli.is_positive" class="variable">is_positive</dd>
+                <dd id="bernoulli.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="bernoulli.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="bernoulli.is_negative" class="variable">is_negative</dd>
+                <dd id="bernoulli.is_composite" class="variable">is_composite</dd>
+                <dd id="bernoulli.is_finite" class="variable">is_finite</dd>
                 <dd id="bernoulli.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="bernoulli.is_zero" class="variable">is_zero</dd>
                 <dd id="bernoulli.is_even" class="variable">is_even</dd>
-                <dd id="bernoulli.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="bernoulli.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="bernoulli.is_commutative" class="variable">is_commutative</dd>
-                <dd id="bernoulli.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="bernoulli.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="bernoulli.is_real" class="variable">is_real</dd>
-                <dd id="bernoulli.is_positive" class="variable">is_positive</dd>
-                <dd id="bernoulli.is_finite" class="variable">is_finite</dd>
-                <dd id="bernoulli.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="bernoulli.is_infinite" class="variable">is_infinite</dd>
-                <dd id="bernoulli.is_rational" class="variable">is_rational</dd>
-                <dd id="bernoulli.is_composite" class="variable">is_composite</dd>
-                <dd id="bernoulli.is_negative" class="variable">is_negative</dd>
-                <dd id="bernoulli.is_polar" class="variable">is_polar</dd>
+                <dd id="bernoulli.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="bernoulli.is_odd" class="variable">is_odd</dd>
-                <dd id="bernoulli.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="bernoulli.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="bernoulli.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="bernoulli.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="bernoulli.is_irrational" class="variable">is_irrational</dd>
+                <dd id="bernoulli.is_integer" class="variable">is_integer</dd>
+                <dd id="bernoulli.is_rational" class="variable">is_rational</dd>
+                <dd id="bernoulli.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="bernoulli.is_complex" class="variable">is_complex</dd>
                 <dd id="bernoulli.is_prime" class="variable">is_prime</dd>
+                <dd id="bernoulli.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="bernoulli.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="bernoulli.is_real" class="variable">is_real</dd>
+                <dd id="bernoulli.is_commutative" class="variable">is_commutative</dd>
                 <dd id="bernoulli.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="bernoulli.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="bernoulli.is_polar" class="variable">is_polar</dd>
+                <dd id="bernoulli.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="bernoulli.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="bernoulli.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -11032,38 +11443,37 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="bell">
-                                <div class="attr class">
-        <a class="headerlink" href="#bell">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">bell</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.bell</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">bell</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.bell</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#bell"></a>
     
             <div class="docstring"><p>Bell numbers / Bell polynomials</p>
 
 <p>The Bell numbers satisfy <code>B_0 = 1</code> and</p>
 
-<p>.. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.</p>
+<p>$$B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.$$</p>
 
 <p>They are also given by:</p>
 
-<p>.. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.</p>
+<p>$$B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.$$</p>
 
 <p>The Bell polynomials are given by <code>B_0(x) = 1</code> and</p>
 
-<p>.. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).</p>
+<p>$$B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).$$</p>
 
 <p>The second kind of Bell polynomials (are sometimes called "partial" Bell
 polynomials or incomplete Bell polynomials) are defined as</p>
 
-<p>.. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
-        \sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
-            \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
-            \left(\frac{x_1}{1!} \right)^{j_1}
-            \left(\frac{x_2}{2!} \right)^{j_2} \dotsb
-            \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.</p>
+<p>$$B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =\sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
+    \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
+    \left(\frac{x_1}{1!} \right)^{j_1}
+    \left(\frac{x_2}{2!} \right)^{j_2} \dotsb
+    \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.$$</p>
 
 <ul>
 <li><code>bell(n)</code> gives the <code>n^{th}</code> Bell number, <code>B_n</code>.</li>
@@ -11098,29 +11508,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Bell_number
-.. [2] http://mathworld.wolfram.com/BellNumber.html
-.. [3] http://mathworld.wolfram.com/BellPolynomial.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="bell.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#bell.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">bell</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">bell</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#bell.__init__"></a>
     
     
 
                             </div>
                             <div id="bell.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#bell.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#bell.default_assumptions"></a>
     
     
 
@@ -11275,37 +11689,37 @@ <h5>Inherited Members</h5>
                 <dd id="bell.simplify" class="function">simplify</dd>
                 <dd id="bell.refine" class="function">refine</dd>
                 <dd id="bell.rewrite" class="function">rewrite</dd>
-                <dd id="bell.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="bell.is_integer" class="variable">is_integer</dd>
-                <dd id="bell.is_complex" class="variable">is_complex</dd>
-                <dd id="bell.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="bell.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="bell.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="bell.is_irrational" class="variable">is_irrational</dd>
+                <dd id="bell.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="bell.is_positive" class="variable">is_positive</dd>
+                <dd id="bell.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="bell.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="bell.is_negative" class="variable">is_negative</dd>
+                <dd id="bell.is_composite" class="variable">is_composite</dd>
+                <dd id="bell.is_finite" class="variable">is_finite</dd>
                 <dd id="bell.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="bell.is_zero" class="variable">is_zero</dd>
                 <dd id="bell.is_even" class="variable">is_even</dd>
-                <dd id="bell.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="bell.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="bell.is_commutative" class="variable">is_commutative</dd>
-                <dd id="bell.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="bell.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="bell.is_real" class="variable">is_real</dd>
-                <dd id="bell.is_positive" class="variable">is_positive</dd>
-                <dd id="bell.is_finite" class="variable">is_finite</dd>
-                <dd id="bell.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="bell.is_infinite" class="variable">is_infinite</dd>
-                <dd id="bell.is_rational" class="variable">is_rational</dd>
-                <dd id="bell.is_composite" class="variable">is_composite</dd>
-                <dd id="bell.is_negative" class="variable">is_negative</dd>
-                <dd id="bell.is_polar" class="variable">is_polar</dd>
+                <dd id="bell.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="bell.is_odd" class="variable">is_odd</dd>
-                <dd id="bell.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="bell.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="bell.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="bell.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="bell.is_irrational" class="variable">is_irrational</dd>
+                <dd id="bell.is_integer" class="variable">is_integer</dd>
+                <dd id="bell.is_rational" class="variable">is_rational</dd>
+                <dd id="bell.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="bell.is_complex" class="variable">is_complex</dd>
                 <dd id="bell.is_prime" class="variable">is_prime</dd>
+                <dd id="bell.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="bell.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="bell.is_real" class="variable">is_real</dd>
+                <dd id="bell.is_commutative" class="variable">is_commutative</dd>
                 <dd id="bell.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="bell.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="bell.is_polar" class="variable">is_polar</dd>
+                <dd id="bell.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="bell.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="bell.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -11317,32 +11731,30 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="euler">
-                                <div class="attr class">
-        <a class="headerlink" href="#euler">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">euler</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.euler</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">euler</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.euler</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#euler"></a>
     
             <div class="docstring"><p>Euler numbers / Euler polynomials</p>
 
 <p>The Euler numbers are given by:</p>
 
-<p>.. math:: E_{2n} = I \sum_{k=1}^{2n+1} \sum_{j=0}^k \binom{k}{j}
-    \frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k}</p>
+<p>$$E_{2n} = I \sum_{k=1}^{2n+1} \sum_{j=0}^k \binom{k}{j}\frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k}$$</p>
 
-<p>.. math:: E_{2n+1} = 0</p>
+<p>$$E_{2n+1} = 0$$</p>
 
 <p>Euler numbers and Euler polynomials are related by</p>
 
-<p>.. math:: E_n = 2^n E_n\left(\frac{1}{2}\right).</p>
+<p>$$E_n = 2^n E_n\left(\frac{1}{2}\right).$$</p>
 
-<p>We compute symbolic Euler polynomials using [5]_</p>
+<p>We compute symbolic Euler polynomials using <sup class="footnote-ref" id="fnref-5"><a href="#fn-5">1</a></sup></p>
 
-<p>.. math:: E_n(x) = \sum_{k=0}^n \binom{n}{k} \frac{E_k}{2^k}
-                   \left(x - \frac{1}{2}\right)^{n-k}.</p>
+<p>$$E_n(x) = \sum_{k=0}^n \binom{n}{k} \frac{E_k}{2^k}\left(x - \frac{1}{2}\right)^{n-k}.$$</p>
 
 <p>However, numerical evaluation of the Euler polynomial is computed
 more efficiently (and more accurately) using the mpmath library.</p>
@@ -11391,31 +11803,36 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Euler_numbers
-.. [2] http://mathworld.wolfram.com/EulerNumber.html
-.. [3] https://en.wikipedia.org/wiki/Alternating_permutation
-.. [4] http://mathworld.wolfram.com/AlternatingPermutation.html
-.. [5] http://dlmf.nist.gov/24.2#ii</p>
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-5">
+<p>http://dlmf.nist.gov/24.2#ii&#160;<a href="#fnref-5" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="euler.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#euler.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">euler</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">euler</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#euler.__init__"></a>
     
     
 
                             </div>
                             <div id="euler.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#euler.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#euler.default_assumptions"></a>
     
     
 
@@ -11570,37 +11987,37 @@ <h5>Inherited Members</h5>
                 <dd id="euler.simplify" class="function">simplify</dd>
                 <dd id="euler.refine" class="function">refine</dd>
                 <dd id="euler.rewrite" class="function">rewrite</dd>
-                <dd id="euler.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="euler.is_integer" class="variable">is_integer</dd>
-                <dd id="euler.is_complex" class="variable">is_complex</dd>
-                <dd id="euler.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="euler.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="euler.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="euler.is_irrational" class="variable">is_irrational</dd>
+                <dd id="euler.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="euler.is_positive" class="variable">is_positive</dd>
+                <dd id="euler.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="euler.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="euler.is_negative" class="variable">is_negative</dd>
+                <dd id="euler.is_composite" class="variable">is_composite</dd>
+                <dd id="euler.is_finite" class="variable">is_finite</dd>
                 <dd id="euler.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="euler.is_zero" class="variable">is_zero</dd>
                 <dd id="euler.is_even" class="variable">is_even</dd>
-                <dd id="euler.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="euler.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="euler.is_commutative" class="variable">is_commutative</dd>
-                <dd id="euler.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="euler.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="euler.is_real" class="variable">is_real</dd>
-                <dd id="euler.is_positive" class="variable">is_positive</dd>
-                <dd id="euler.is_finite" class="variable">is_finite</dd>
-                <dd id="euler.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="euler.is_infinite" class="variable">is_infinite</dd>
-                <dd id="euler.is_rational" class="variable">is_rational</dd>
-                <dd id="euler.is_composite" class="variable">is_composite</dd>
-                <dd id="euler.is_negative" class="variable">is_negative</dd>
-                <dd id="euler.is_polar" class="variable">is_polar</dd>
+                <dd id="euler.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="euler.is_odd" class="variable">is_odd</dd>
-                <dd id="euler.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="euler.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="euler.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="euler.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="euler.is_irrational" class="variable">is_irrational</dd>
+                <dd id="euler.is_integer" class="variable">is_integer</dd>
+                <dd id="euler.is_rational" class="variable">is_rational</dd>
+                <dd id="euler.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="euler.is_complex" class="variable">is_complex</dd>
                 <dd id="euler.is_prime" class="variable">is_prime</dd>
+                <dd id="euler.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="euler.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="euler.is_real" class="variable">is_real</dd>
+                <dd id="euler.is_commutative" class="variable">is_commutative</dd>
                 <dd id="euler.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="euler.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="euler.is_polar" class="variable">is_polar</dd>
+                <dd id="euler.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="euler.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="euler.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -11612,14 +12029,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="catalan">
-                                <div class="attr class">
-        <a class="headerlink" href="#catalan">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">catalan</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.catalan</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">catalan</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.catalan</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#catalan"></a>
     
             <div class="docstring"><p>Catalan numbers</p>
 
@@ -11703,30 +12120,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Catalan_number
-.. [2] http://mathworld.wolfram.com/CatalanNumber.html
-.. [3] http://functions.wolfram.com/GammaBetaErf/CatalanNumber/
-.. [4] http://geometer.org/mathcircles/catalan.pdf</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="catalan.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#catalan.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">catalan</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">catalan</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#catalan.__init__"></a>
     
     
 
                             </div>
                             <div id="catalan.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#catalan.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#catalan.default_assumptions"></a>
     
     
 
@@ -11881,37 +12301,37 @@ <h5>Inherited Members</h5>
                 <dd id="catalan.simplify" class="function">simplify</dd>
                 <dd id="catalan.refine" class="function">refine</dd>
                 <dd id="catalan.rewrite" class="function">rewrite</dd>
-                <dd id="catalan.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="catalan.is_integer" class="variable">is_integer</dd>
-                <dd id="catalan.is_complex" class="variable">is_complex</dd>
-                <dd id="catalan.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="catalan.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="catalan.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="catalan.is_irrational" class="variable">is_irrational</dd>
+                <dd id="catalan.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="catalan.is_positive" class="variable">is_positive</dd>
+                <dd id="catalan.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="catalan.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="catalan.is_negative" class="variable">is_negative</dd>
+                <dd id="catalan.is_composite" class="variable">is_composite</dd>
+                <dd id="catalan.is_finite" class="variable">is_finite</dd>
                 <dd id="catalan.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="catalan.is_zero" class="variable">is_zero</dd>
                 <dd id="catalan.is_even" class="variable">is_even</dd>
-                <dd id="catalan.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="catalan.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="catalan.is_commutative" class="variable">is_commutative</dd>
-                <dd id="catalan.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="catalan.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="catalan.is_real" class="variable">is_real</dd>
-                <dd id="catalan.is_positive" class="variable">is_positive</dd>
-                <dd id="catalan.is_finite" class="variable">is_finite</dd>
-                <dd id="catalan.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="catalan.is_infinite" class="variable">is_infinite</dd>
-                <dd id="catalan.is_rational" class="variable">is_rational</dd>
-                <dd id="catalan.is_composite" class="variable">is_composite</dd>
-                <dd id="catalan.is_negative" class="variable">is_negative</dd>
-                <dd id="catalan.is_polar" class="variable">is_polar</dd>
+                <dd id="catalan.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="catalan.is_odd" class="variable">is_odd</dd>
-                <dd id="catalan.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="catalan.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="catalan.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="catalan.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="catalan.is_irrational" class="variable">is_irrational</dd>
+                <dd id="catalan.is_integer" class="variable">is_integer</dd>
+                <dd id="catalan.is_rational" class="variable">is_rational</dd>
+                <dd id="catalan.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="catalan.is_complex" class="variable">is_complex</dd>
                 <dd id="catalan.is_prime" class="variable">is_prime</dd>
+                <dd id="catalan.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="catalan.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="catalan.is_real" class="variable">is_real</dd>
+                <dd id="catalan.is_commutative" class="variable">is_commutative</dd>
                 <dd id="catalan.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="catalan.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="catalan.is_polar" class="variable">is_polar</dd>
+                <dd id="catalan.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="catalan.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="catalan.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -11923,21 +12343,21 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="genocchi">
-                                <div class="attr class">
-        <a class="headerlink" href="#genocchi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">genocchi</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.genocchi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">genocchi</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.genocchi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#genocchi"></a>
     
             <div class="docstring"><p>Genocchi numbers</p>
 
 <p>The Genocchi numbers are a sequence of integers <code>G_n</code> that satisfy the
 relation:</p>
 
-<p>.. math:: \frac{2t}{e^t + 1} = \sum_{n=1}^\infty \frac{G_n t^n}{n!}</p>
+<p>$$\frac{2t}{e^t + 1} = \sum_{n=1}^\infty \frac{G_n t^n}{n!}$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -11955,28 +12375,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Genocchi_number
-.. [2] http://mathworld.wolfram.com/GenocchiNumber.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="genocchi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#genocchi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">genocchi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">genocchi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#genocchi.__init__"></a>
     
     
 
                             </div>
                             <div id="genocchi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#genocchi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#genocchi.default_assumptions"></a>
     
     
 
@@ -12131,37 +12556,37 @@ <h5>Inherited Members</h5>
                 <dd id="genocchi.simplify" class="function">simplify</dd>
                 <dd id="genocchi.refine" class="function">refine</dd>
                 <dd id="genocchi.rewrite" class="function">rewrite</dd>
-                <dd id="genocchi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="genocchi.is_integer" class="variable">is_integer</dd>
-                <dd id="genocchi.is_complex" class="variable">is_complex</dd>
-                <dd id="genocchi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="genocchi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="genocchi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="genocchi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="genocchi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="genocchi.is_positive" class="variable">is_positive</dd>
+                <dd id="genocchi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="genocchi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="genocchi.is_negative" class="variable">is_negative</dd>
+                <dd id="genocchi.is_composite" class="variable">is_composite</dd>
+                <dd id="genocchi.is_finite" class="variable">is_finite</dd>
                 <dd id="genocchi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="genocchi.is_zero" class="variable">is_zero</dd>
                 <dd id="genocchi.is_even" class="variable">is_even</dd>
-                <dd id="genocchi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="genocchi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="genocchi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="genocchi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="genocchi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="genocchi.is_real" class="variable">is_real</dd>
-                <dd id="genocchi.is_positive" class="variable">is_positive</dd>
-                <dd id="genocchi.is_finite" class="variable">is_finite</dd>
-                <dd id="genocchi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="genocchi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="genocchi.is_rational" class="variable">is_rational</dd>
-                <dd id="genocchi.is_composite" class="variable">is_composite</dd>
-                <dd id="genocchi.is_negative" class="variable">is_negative</dd>
-                <dd id="genocchi.is_polar" class="variable">is_polar</dd>
+                <dd id="genocchi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="genocchi.is_odd" class="variable">is_odd</dd>
-                <dd id="genocchi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="genocchi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="genocchi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="genocchi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="genocchi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="genocchi.is_integer" class="variable">is_integer</dd>
+                <dd id="genocchi.is_rational" class="variable">is_rational</dd>
+                <dd id="genocchi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="genocchi.is_complex" class="variable">is_complex</dd>
                 <dd id="genocchi.is_prime" class="variable">is_prime</dd>
+                <dd id="genocchi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="genocchi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="genocchi.is_real" class="variable">is_real</dd>
+                <dd id="genocchi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="genocchi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="genocchi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="genocchi.is_polar" class="variable">is_polar</dd>
+                <dd id="genocchi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="genocchi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="genocchi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -12173,14 +12598,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="partition">
-                                <div class="attr class">
-        <a class="headerlink" href="#partition">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">partition</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.partition</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">partition</span><wbr>(<span class="base">sympy.functions.combinatorial.numbers.partition</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#partition"></a>
     
             <div class="docstring"><p>Partition numbers</p>
 
@@ -12188,7 +12613,7 @@ <h5>Inherited Members</h5>
 number of distinct ways of representing <code><a href="#partition.n">n</a></code> as a sum of natural numbers
 (with order irrelevant). The generating function for <code>p_n</code> is given by:</p>
 
-<p>.. math:: \sum_{n=0}^\infty p_n x^n = \prod_{k=1}^\infty (1 - x^k)^{-1}</p>
+<p>$$\sum_{n=0}^\infty p_n x^n = \prod_{k=1}^\infty (1 - x^k)^{-1}$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -12206,28 +12631,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Partition_(number_theory%29
-.. [2] https://en.wikipedia.org/wiki/Pentagonal_number_theorem</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="partition.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#partition.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">partition</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">partition</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#partition.__init__"></a>
     
     
 
                             </div>
                             <div id="partition.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#partition.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#partition.default_assumptions"></a>
     
     
 
@@ -12382,37 +12812,37 @@ <h5>Inherited Members</h5>
                 <dd id="partition.simplify" class="function">simplify</dd>
                 <dd id="partition.refine" class="function">refine</dd>
                 <dd id="partition.rewrite" class="function">rewrite</dd>
-                <dd id="partition.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="partition.is_integer" class="variable">is_integer</dd>
-                <dd id="partition.is_complex" class="variable">is_complex</dd>
-                <dd id="partition.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="partition.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="partition.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="partition.is_irrational" class="variable">is_irrational</dd>
+                <dd id="partition.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="partition.is_positive" class="variable">is_positive</dd>
+                <dd id="partition.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="partition.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="partition.is_negative" class="variable">is_negative</dd>
+                <dd id="partition.is_composite" class="variable">is_composite</dd>
+                <dd id="partition.is_finite" class="variable">is_finite</dd>
                 <dd id="partition.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="partition.is_zero" class="variable">is_zero</dd>
                 <dd id="partition.is_even" class="variable">is_even</dd>
-                <dd id="partition.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="partition.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="partition.is_commutative" class="variable">is_commutative</dd>
-                <dd id="partition.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="partition.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="partition.is_real" class="variable">is_real</dd>
-                <dd id="partition.is_positive" class="variable">is_positive</dd>
-                <dd id="partition.is_finite" class="variable">is_finite</dd>
-                <dd id="partition.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="partition.is_infinite" class="variable">is_infinite</dd>
-                <dd id="partition.is_rational" class="variable">is_rational</dd>
-                <dd id="partition.is_composite" class="variable">is_composite</dd>
-                <dd id="partition.is_negative" class="variable">is_negative</dd>
-                <dd id="partition.is_polar" class="variable">is_polar</dd>
+                <dd id="partition.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="partition.is_odd" class="variable">is_odd</dd>
-                <dd id="partition.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="partition.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="partition.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="partition.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="partition.is_irrational" class="variable">is_irrational</dd>
+                <dd id="partition.is_integer" class="variable">is_integer</dd>
+                <dd id="partition.is_rational" class="variable">is_rational</dd>
+                <dd id="partition.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="partition.is_complex" class="variable">is_complex</dd>
                 <dd id="partition.is_prime" class="variable">is_prime</dd>
+                <dd id="partition.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="partition.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="partition.is_real" class="variable">is_real</dd>
+                <dd id="partition.is_commutative" class="variable">is_commutative</dd>
                 <dd id="partition.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="partition.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="partition.is_polar" class="variable">is_polar</dd>
+                <dd id="partition.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="partition.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="partition.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -12424,14 +12854,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Rem">
-                                <div class="attr class">
-        <a class="headerlink" href="#Rem">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Rem</span><wbr>(<span class="base">sympy.functions.elementary.miscellaneous.Rem</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Rem</span><wbr>(<span class="base">sympy.functions.elementary.miscellaneous.Rem</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Rem"></a>
     
             <div class="docstring"><p>Returns the remainder when <code>p</code> is divided by <code>q</code> where <code>p</code> is finite
 and <code>q</code> is not equal to zero. The result, <code>p - int(p/q)*q</code>, has the same sign
@@ -12466,22 +12896,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="Rem.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Rem.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Rem</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Rem</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Rem.__init__"></a>
     
     
 
                             </div>
                             <div id="Rem.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Rem.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Rem.default_assumptions"></a>
     
     
 
@@ -12636,37 +13068,37 @@ <h5>Inherited Members</h5>
                 <dd id="Rem.simplify" class="function">simplify</dd>
                 <dd id="Rem.refine" class="function">refine</dd>
                 <dd id="Rem.rewrite" class="function">rewrite</dd>
-                <dd id="Rem.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Rem.is_integer" class="variable">is_integer</dd>
-                <dd id="Rem.is_complex" class="variable">is_complex</dd>
-                <dd id="Rem.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Rem.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Rem.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Rem.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Rem.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Rem.is_positive" class="variable">is_positive</dd>
+                <dd id="Rem.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Rem.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Rem.is_negative" class="variable">is_negative</dd>
+                <dd id="Rem.is_composite" class="variable">is_composite</dd>
+                <dd id="Rem.is_finite" class="variable">is_finite</dd>
                 <dd id="Rem.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Rem.is_zero" class="variable">is_zero</dd>
                 <dd id="Rem.is_even" class="variable">is_even</dd>
-                <dd id="Rem.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Rem.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Rem.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Rem.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Rem.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Rem.is_real" class="variable">is_real</dd>
-                <dd id="Rem.is_positive" class="variable">is_positive</dd>
-                <dd id="Rem.is_finite" class="variable">is_finite</dd>
-                <dd id="Rem.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Rem.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Rem.is_rational" class="variable">is_rational</dd>
-                <dd id="Rem.is_composite" class="variable">is_composite</dd>
-                <dd id="Rem.is_negative" class="variable">is_negative</dd>
-                <dd id="Rem.is_polar" class="variable">is_polar</dd>
+                <dd id="Rem.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Rem.is_odd" class="variable">is_odd</dd>
-                <dd id="Rem.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Rem.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Rem.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Rem.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Rem.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Rem.is_integer" class="variable">is_integer</dd>
+                <dd id="Rem.is_rational" class="variable">is_rational</dd>
+                <dd id="Rem.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Rem.is_complex" class="variable">is_complex</dd>
                 <dd id="Rem.is_prime" class="variable">is_prime</dd>
+                <dd id="Rem.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Rem.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Rem.is_real" class="variable">is_real</dd>
+                <dd id="Rem.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Rem.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Rem.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Rem.is_polar" class="variable">is_polar</dd>
+                <dd id="Rem.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Rem.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Rem.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -12678,14 +13110,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="re">
-                                <div class="attr class">
-        <a class="headerlink" href="#re">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">re</span><wbr>(<span class="base">sympy.functions.elementary.complexes.re</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">re</span><wbr>(<span class="base">sympy.functions.elementary.complexes.re</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#re"></a>
     
             <div class="docstring"><p>Returns real part of expression. This function performs only
 elementary analysis and so it will fail to decompose properly
@@ -12726,52 +13158,57 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="re.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#re.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">re</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">re</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#re.__init__"></a>
     
     
 
                             </div>
                             <div id="re.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#re.is_extended_real">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#re.is_extended_real"></a>
     
     
 
                             </div>
                             <div id="re.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#re.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;extended_real&#39;: True, &#39;imaginary&#39;: False, &#39;commutative&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;extended_real&#39;: True, &#39;commutative&#39;: True, &#39;imaginary&#39;: False}</span>
+        
     </div>
-
+    <a class="headerlink" href="#re.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="re.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#re.is_commutative">#&nbsp;&nbsp</a>
+                            <div id="re.is_imaginary" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#re.is_imaginary"></a>
     
     
 
                             </div>
-                            <div id="re.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#re.is_imaginary">#&nbsp;&nbsp</a>
+                            <div id="re.is_commutative" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#re.is_commutative"></a>
     
     
 
@@ -12927,34 +13364,34 @@ <h5>Inherited Members</h5>
                 <dd id="re.simplify" class="function">simplify</dd>
                 <dd id="re.refine" class="function">refine</dd>
                 <dd id="re.rewrite" class="function">rewrite</dd>
-                <dd id="re.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="re.is_integer" class="variable">is_integer</dd>
-                <dd id="re.is_complex" class="variable">is_complex</dd>
-                <dd id="re.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="re.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="re.is_irrational" class="variable">is_irrational</dd>
+                <dd id="re.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="re.is_positive" class="variable">is_positive</dd>
+                <dd id="re.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="re.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="re.is_negative" class="variable">is_negative</dd>
+                <dd id="re.is_composite" class="variable">is_composite</dd>
+                <dd id="re.is_finite" class="variable">is_finite</dd>
                 <dd id="re.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="re.is_zero" class="variable">is_zero</dd>
                 <dd id="re.is_even" class="variable">is_even</dd>
-                <dd id="re.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="re.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="re.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="re.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="re.is_real" class="variable">is_real</dd>
-                <dd id="re.is_positive" class="variable">is_positive</dd>
-                <dd id="re.is_finite" class="variable">is_finite</dd>
-                <dd id="re.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="re.is_infinite" class="variable">is_infinite</dd>
-                <dd id="re.is_rational" class="variable">is_rational</dd>
-                <dd id="re.is_composite" class="variable">is_composite</dd>
-                <dd id="re.is_negative" class="variable">is_negative</dd>
-                <dd id="re.is_polar" class="variable">is_polar</dd>
                 <dd id="re.is_odd" class="variable">is_odd</dd>
-                <dd id="re.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="re.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="re.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="re.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="re.is_irrational" class="variable">is_irrational</dd>
+                <dd id="re.is_integer" class="variable">is_integer</dd>
+                <dd id="re.is_rational" class="variable">is_rational</dd>
+                <dd id="re.is_complex" class="variable">is_complex</dd>
                 <dd id="re.is_prime" class="variable">is_prime</dd>
+                <dd id="re.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="re.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="re.is_real" class="variable">is_real</dd>
                 <dd id="re.is_hermitian" class="variable">is_hermitian</dd>
+                <dd id="re.is_polar" class="variable">is_polar</dd>
+                <dd id="re.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="re.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="re.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -12966,14 +13403,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="im">
-                                <div class="attr class">
-        <a class="headerlink" href="#im">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">im</span><wbr>(<span class="base">sympy.functions.elementary.complexes.im</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">im</span><wbr>(<span class="base">sympy.functions.elementary.complexes.im</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#im"></a>
     
             <div class="docstring"><p>Returns imaginary part of expression. This function performs only
 elementary analysis and so it will fail to decompose properly more
@@ -13014,52 +13451,57 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="im.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#im.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">im</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">im</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#im.__init__"></a>
     
     
 
                             </div>
                             <div id="im.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#im.is_extended_real">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#im.is_extended_real"></a>
     
     
 
                             </div>
                             <div id="im.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#im.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;extended_real&#39;: True, &#39;imaginary&#39;: False, &#39;commutative&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;extended_real&#39;: True, &#39;commutative&#39;: True, &#39;imaginary&#39;: False}</span>
+        
     </div>
-
+    <a class="headerlink" href="#im.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="im.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#im.is_commutative">#&nbsp;&nbsp</a>
+                            <div id="im.is_imaginary" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#im.is_imaginary"></a>
     
     
 
                             </div>
-                            <div id="im.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#im.is_imaginary">#&nbsp;&nbsp</a>
+                            <div id="im.is_commutative" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#im.is_commutative"></a>
     
     
 
@@ -13215,34 +13657,34 @@ <h5>Inherited Members</h5>
                 <dd id="im.simplify" class="function">simplify</dd>
                 <dd id="im.refine" class="function">refine</dd>
                 <dd id="im.rewrite" class="function">rewrite</dd>
-                <dd id="im.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="im.is_integer" class="variable">is_integer</dd>
-                <dd id="im.is_complex" class="variable">is_complex</dd>
-                <dd id="im.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="im.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="im.is_irrational" class="variable">is_irrational</dd>
+                <dd id="im.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="im.is_positive" class="variable">is_positive</dd>
+                <dd id="im.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="im.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="im.is_negative" class="variable">is_negative</dd>
+                <dd id="im.is_composite" class="variable">is_composite</dd>
+                <dd id="im.is_finite" class="variable">is_finite</dd>
                 <dd id="im.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="im.is_zero" class="variable">is_zero</dd>
                 <dd id="im.is_even" class="variable">is_even</dd>
-                <dd id="im.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="im.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="im.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="im.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="im.is_real" class="variable">is_real</dd>
-                <dd id="im.is_positive" class="variable">is_positive</dd>
-                <dd id="im.is_finite" class="variable">is_finite</dd>
-                <dd id="im.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="im.is_infinite" class="variable">is_infinite</dd>
-                <dd id="im.is_rational" class="variable">is_rational</dd>
-                <dd id="im.is_composite" class="variable">is_composite</dd>
-                <dd id="im.is_negative" class="variable">is_negative</dd>
-                <dd id="im.is_polar" class="variable">is_polar</dd>
                 <dd id="im.is_odd" class="variable">is_odd</dd>
-                <dd id="im.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="im.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="im.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="im.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="im.is_irrational" class="variable">is_irrational</dd>
+                <dd id="im.is_integer" class="variable">is_integer</dd>
+                <dd id="im.is_rational" class="variable">is_rational</dd>
+                <dd id="im.is_complex" class="variable">is_complex</dd>
                 <dd id="im.is_prime" class="variable">is_prime</dd>
+                <dd id="im.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="im.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="im.is_real" class="variable">is_real</dd>
                 <dd id="im.is_hermitian" class="variable">is_hermitian</dd>
+                <dd id="im.is_polar" class="variable">is_polar</dd>
+                <dd id="im.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="im.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="im.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -13254,14 +13696,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="sign">
-                                <div class="attr class">
-        <a class="headerlink" href="#sign">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">sign</span><wbr>(<span class="base">sympy.functions.elementary.complexes.sign</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">sign</span><wbr>(<span class="base">sympy.functions.elementary.complexes.sign</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#sign"></a>
     
             <div class="docstring"><p>Returns the complex sign of an expression:</p>
 
@@ -13317,62 +13759,68 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="sign.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#sign.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">sign</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">sign</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#sign.__init__"></a>
     
     
 
                             </div>
                             <div id="sign.is_complex" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sign.is_complex">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_complex</span><span class="default_value"> = True</span>
 
-        <span class="name">is_complex</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#sign.is_complex"></a>
     
     
 
                             </div>
                             <div id="sign.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sign.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;complex&#39;: True, &#39;commutative&#39;: True, &#39;infinite&#39;: False, &#39;finite&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;complex&#39;: True, &#39;finite&#39;: True, &#39;commutative&#39;: True, &#39;infinite&#39;: False}</span>
+        
     </div>
-
+    <a class="headerlink" href="#sign.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="sign.is_finite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sign.is_finite">#&nbsp;&nbsp</a>
+                            <div id="sign.is_commutative" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_finite</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#sign.is_commutative"></a>
     
     
 
                             </div>
-                            <div id="sign.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sign.is_commutative">#&nbsp;&nbsp</a>
+                            <div id="sign.is_infinite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_infinite</span><span class="default_value"> = False</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#sign.is_infinite"></a>
     
     
 
                             </div>
-                            <div id="sign.is_infinite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sign.is_infinite">#&nbsp;&nbsp</a>
+                            <div id="sign.is_finite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_finite</span><span class="default_value"> = True</span>
 
-        <span class="name">is_infinite</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#sign.is_finite"></a>
     
     
 
@@ -13527,33 +13975,33 @@ <h5>Inherited Members</h5>
                 <dd id="sign.simplify" class="function">simplify</dd>
                 <dd id="sign.refine" class="function">refine</dd>
                 <dd id="sign.rewrite" class="function">rewrite</dd>
-                <dd id="sign.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="sign.is_integer" class="variable">is_integer</dd>
-                <dd id="sign.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="sign.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="sign.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="sign.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sign.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sign.is_positive" class="variable">is_positive</dd>
+                <dd id="sign.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="sign.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="sign.is_negative" class="variable">is_negative</dd>
+                <dd id="sign.is_composite" class="variable">is_composite</dd>
                 <dd id="sign.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="sign.is_zero" class="variable">is_zero</dd>
                 <dd id="sign.is_even" class="variable">is_even</dd>
-                <dd id="sign.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="sign.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="sign.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="sign.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="sign.is_real" class="variable">is_real</dd>
-                <dd id="sign.is_positive" class="variable">is_positive</dd>
+                <dd id="sign.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="sign.is_odd" class="variable">is_odd</dd>
+                <dd id="sign.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="sign.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="sign.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sign.is_integer" class="variable">is_integer</dd>
                 <dd id="sign.is_rational" class="variable">is_rational</dd>
-                <dd id="sign.is_composite" class="variable">is_composite</dd>
-                <dd id="sign.is_negative" class="variable">is_negative</dd>
-                <dd id="sign.is_polar" class="variable">is_polar</dd>
-                <dd id="sign.is_odd" class="variable">is_odd</dd>
-                <dd id="sign.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="sign.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sign.is_extended_real" class="variable">is_extended_real</dd>
                 <dd id="sign.is_prime" class="variable">is_prime</dd>
+                <dd id="sign.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="sign.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="sign.is_real" class="variable">is_real</dd>
                 <dd id="sign.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="sign.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="sign.is_polar" class="variable">is_polar</dd>
+                <dd id="sign.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="sign.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="sign.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -13565,14 +14013,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Abs">
-                                <div class="attr class">
-        <a class="headerlink" href="#Abs">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Abs</span><wbr>(<span class="base">sympy.functions.elementary.complexes.Abs</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Abs</span><wbr>(<span class="base">sympy.functions.elementary.complexes.Abs</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Abs"></a>
     
             <div class="docstring"><p>Return the absolute value of the argument.</p>
 
@@ -13629,82 +14077,90 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="Abs.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Abs.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Abs</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Abs</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Abs.__init__"></a>
     
     
 
                             </div>
                             <div id="Abs.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Abs.is_extended_real">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Abs.is_extended_real"></a>
     
     
 
                             </div>
                             <div id="Abs.is_extended_negative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Abs.is_extended_negative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_extended_negative</span><span class="default_value"> = False</span>
 
-        <span class="name">is_extended_negative</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#Abs.is_extended_negative"></a>
     
     
 
                             </div>
                             <div id="Abs.is_extended_nonnegative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Abs.is_extended_nonnegative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_extended_nonnegative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_extended_nonnegative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Abs.is_extended_nonnegative"></a>
     
     
 
                             </div>
                             <div id="Abs.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Abs.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;extended_negative&#39;: False, &#39;negative&#39;: False, &#39;extended_real&#39;: True, &#39;imaginary&#39;: False, &#39;commutative&#39;: True, &#39;extended_nonnegative&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;extended_real&#39;: True, &#39;commutative&#39;: True, &#39;imaginary&#39;: False, &#39;extended_nonnegative&#39;: True, &#39;extended_negative&#39;: False, &#39;negative&#39;: False}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Abs.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="Abs.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Abs.is_commutative">#&nbsp;&nbsp</a>
+                            <div id="Abs.is_negative" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_negative</span><span class="default_value"> = False</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Abs.is_negative"></a>
     
     
 
                             </div>
                             <div id="Abs.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Abs.is_imaginary">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#Abs.is_imaginary"></a>
     
     
 
                             </div>
-                            <div id="Abs.is_negative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Abs.is_negative">#&nbsp;&nbsp</a>
+                            <div id="Abs.is_commutative" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_negative</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#Abs.is_commutative"></a>
     
     
 
@@ -13860,31 +14316,31 @@ <h5>Inherited Members</h5>
                 <dd id="Abs.simplify" class="function">simplify</dd>
                 <dd id="Abs.refine" class="function">refine</dd>
                 <dd id="Abs.rewrite" class="function">rewrite</dd>
-                <dd id="Abs.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Abs.is_integer" class="variable">is_integer</dd>
-                <dd id="Abs.is_complex" class="variable">is_complex</dd>
-                <dd id="Abs.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Abs.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Abs.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Abs.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Abs.is_positive" class="variable">is_positive</dd>
+                <dd id="Abs.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Abs.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Abs.is_composite" class="variable">is_composite</dd>
+                <dd id="Abs.is_finite" class="variable">is_finite</dd>
                 <dd id="Abs.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Abs.is_zero" class="variable">is_zero</dd>
                 <dd id="Abs.is_even" class="variable">is_even</dd>
-                <dd id="Abs.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Abs.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Abs.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Abs.is_real" class="variable">is_real</dd>
-                <dd id="Abs.is_positive" class="variable">is_positive</dd>
-                <dd id="Abs.is_finite" class="variable">is_finite</dd>
                 <dd id="Abs.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Abs.is_rational" class="variable">is_rational</dd>
-                <dd id="Abs.is_composite" class="variable">is_composite</dd>
-                <dd id="Abs.is_polar" class="variable">is_polar</dd>
                 <dd id="Abs.is_odd" class="variable">is_odd</dd>
-                <dd id="Abs.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Abs.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Abs.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Abs.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Abs.is_integer" class="variable">is_integer</dd>
+                <dd id="Abs.is_rational" class="variable">is_rational</dd>
+                <dd id="Abs.is_complex" class="variable">is_complex</dd>
                 <dd id="Abs.is_prime" class="variable">is_prime</dd>
+                <dd id="Abs.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Abs.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Abs.is_real" class="variable">is_real</dd>
                 <dd id="Abs.is_hermitian" class="variable">is_hermitian</dd>
+                <dd id="Abs.is_polar" class="variable">is_polar</dd>
+                <dd id="Abs.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Abs.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -13896,21 +14352,21 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="conjugate">
-                                <div class="attr class">
-        <a class="headerlink" href="#conjugate">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">conjugate</span><wbr>(<span class="base">sympy.functions.elementary.complexes.conjugate</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">conjugate</span><wbr>(<span class="base">sympy.functions.elementary.complexes.conjugate</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#conjugate"></a>
     
-            <div class="docstring"><p>Returns the <em>complex conjugate</em> [1]_ of an argument.
+            <div class="docstring"><p>Returns the <em>complex conjugate</em> <sup class="footnote-ref" id="fnref-1"><a href="#fn-1">1</a></sup> of an argument.
 In mathematics, the complex conjugate of a complex number
 is given by changing the sign of the imaginary part.</p>
 
 <p>Thus, the conjugate of the complex number
-:math:<code>a + ib</code> (where $a$ and $b$ are real numbers) is :math:<code>a - ib</code></p>
+\( a + ib \) (where $a$ and $b$ are real numbers) is \( a - ib \)</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -13941,27 +14397,36 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Complex_conjugation</p>
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-1">
+<p>https://en.wikipedia.org/wiki/Complex_conjugation&#160;<a href="#fnref-1" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="conjugate.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#conjugate.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">conjugate</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">conjugate</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#conjugate.__init__"></a>
     
     
 
                             </div>
                             <div id="conjugate.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#conjugate.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#conjugate.default_assumptions"></a>
     
     
 
@@ -14117,37 +14582,37 @@ <h5>Inherited Members</h5>
                 <dd id="conjugate.simplify" class="function">simplify</dd>
                 <dd id="conjugate.refine" class="function">refine</dd>
                 <dd id="conjugate.rewrite" class="function">rewrite</dd>
-                <dd id="conjugate.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="conjugate.is_integer" class="variable">is_integer</dd>
-                <dd id="conjugate.is_complex" class="variable">is_complex</dd>
-                <dd id="conjugate.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="conjugate.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="conjugate.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="conjugate.is_irrational" class="variable">is_irrational</dd>
+                <dd id="conjugate.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="conjugate.is_positive" class="variable">is_positive</dd>
+                <dd id="conjugate.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="conjugate.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="conjugate.is_negative" class="variable">is_negative</dd>
+                <dd id="conjugate.is_composite" class="variable">is_composite</dd>
+                <dd id="conjugate.is_finite" class="variable">is_finite</dd>
                 <dd id="conjugate.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="conjugate.is_zero" class="variable">is_zero</dd>
                 <dd id="conjugate.is_even" class="variable">is_even</dd>
-                <dd id="conjugate.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="conjugate.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="conjugate.is_commutative" class="variable">is_commutative</dd>
-                <dd id="conjugate.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="conjugate.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="conjugate.is_real" class="variable">is_real</dd>
-                <dd id="conjugate.is_positive" class="variable">is_positive</dd>
-                <dd id="conjugate.is_finite" class="variable">is_finite</dd>
-                <dd id="conjugate.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="conjugate.is_infinite" class="variable">is_infinite</dd>
-                <dd id="conjugate.is_rational" class="variable">is_rational</dd>
-                <dd id="conjugate.is_composite" class="variable">is_composite</dd>
-                <dd id="conjugate.is_negative" class="variable">is_negative</dd>
-                <dd id="conjugate.is_polar" class="variable">is_polar</dd>
+                <dd id="conjugate.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="conjugate.is_odd" class="variable">is_odd</dd>
-                <dd id="conjugate.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="conjugate.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="conjugate.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="conjugate.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="conjugate.is_irrational" class="variable">is_irrational</dd>
+                <dd id="conjugate.is_integer" class="variable">is_integer</dd>
+                <dd id="conjugate.is_rational" class="variable">is_rational</dd>
+                <dd id="conjugate.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="conjugate.is_complex" class="variable">is_complex</dd>
                 <dd id="conjugate.is_prime" class="variable">is_prime</dd>
+                <dd id="conjugate.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="conjugate.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="conjugate.is_real" class="variable">is_real</dd>
+                <dd id="conjugate.is_commutative" class="variable">is_commutative</dd>
                 <dd id="conjugate.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="conjugate.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="conjugate.is_polar" class="variable">is_polar</dd>
+                <dd id="conjugate.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="conjugate.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="conjugate.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -14159,14 +14624,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="arg">
-                                <div class="attr class">
-        <a class="headerlink" href="#arg">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">arg</span><wbr>(<span class="base">sympy.functions.elementary.complexes.arg</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">arg</span><wbr>(<span class="base">sympy.functions.elementary.complexes.arg</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#arg"></a>
     
             <div class="docstring"><p>Returns the argument (in radians) of a complex number. The argument is
 evaluated in consistent convention with <code><a href="#atan2">atan2</a></code> where the branch-cut is
@@ -14213,102 +14678,112 @@ <h1 id="returns">Returns</h1>
 
 
                             <div id="arg.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#arg.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">arg</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">arg</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#arg.__init__"></a>
     
     
 
                             </div>
                             <div id="arg.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.is_extended_real">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.is_extended_real"></a>
     
     
 
                             </div>
                             <div id="arg.is_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.is_real">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.is_real"></a>
     
     
 
                             </div>
                             <div id="arg.is_finite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.is_finite">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_finite</span><span class="default_value"> = True</span>
 
-        <span class="name">is_finite</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.is_finite"></a>
     
     
 
                             </div>
                             <div id="arg.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;finite&#39;: True, &#39;infinite&#39;: False, &#39;extended_real&#39;: True, &#39;imaginary&#39;: False, &#39;commutative&#39;: True, &#39;real&#39;: True, &#39;hermitian&#39;: True, &#39;complex&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;extended_real&#39;: True, &#39;commutative&#39;: True, &#39;imaginary&#39;: False, &#39;real&#39;: True, &#39;hermitian&#39;: True, &#39;complex&#39;: True, &#39;infinite&#39;: False, &#39;finite&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="arg.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.is_commutative">#&nbsp;&nbsp</a>
+                            <div id="arg.is_infinite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_infinite</span><span class="default_value"> = False</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.is_infinite"></a>
     
     
 
                             </div>
                             <div id="arg.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.is_imaginary">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.is_imaginary"></a>
     
     
 
                             </div>
-                            <div id="arg.is_hermitian" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.is_hermitian">#&nbsp;&nbsp</a>
+                            <div id="arg.is_commutative" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_hermitian</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.is_commutative"></a>
     
     
 
                             </div>
-                            <div id="arg.is_complex" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.is_complex">#&nbsp;&nbsp</a>
+                            <div id="arg.is_hermitian" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_hermitian</span><span class="default_value"> = True</span>
 
-        <span class="name">is_complex</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.is_hermitian"></a>
     
     
 
                             </div>
-                            <div id="arg.is_infinite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#arg.is_infinite">#&nbsp;&nbsp</a>
+                            <div id="arg.is_complex" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_complex</span><span class="default_value"> = True</span>
 
-        <span class="name">is_infinite</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#arg.is_complex"></a>
     
     
 
@@ -14463,29 +14938,29 @@ <h5>Inherited Members</h5>
                 <dd id="arg.simplify" class="function">simplify</dd>
                 <dd id="arg.refine" class="function">refine</dd>
                 <dd id="arg.rewrite" class="function">rewrite</dd>
-                <dd id="arg.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="arg.is_integer" class="variable">is_integer</dd>
-                <dd id="arg.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="arg.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="arg.is_irrational" class="variable">is_irrational</dd>
+                <dd id="arg.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="arg.is_positive" class="variable">is_positive</dd>
+                <dd id="arg.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="arg.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="arg.is_negative" class="variable">is_negative</dd>
+                <dd id="arg.is_composite" class="variable">is_composite</dd>
                 <dd id="arg.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="arg.is_zero" class="variable">is_zero</dd>
                 <dd id="arg.is_even" class="variable">is_even</dd>
-                <dd id="arg.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="arg.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="arg.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="arg.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="arg.is_positive" class="variable">is_positive</dd>
+                <dd id="arg.is_odd" class="variable">is_odd</dd>
+                <dd id="arg.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="arg.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="arg.is_irrational" class="variable">is_irrational</dd>
+                <dd id="arg.is_integer" class="variable">is_integer</dd>
                 <dd id="arg.is_rational" class="variable">is_rational</dd>
-                <dd id="arg.is_composite" class="variable">is_composite</dd>
-                <dd id="arg.is_negative" class="variable">is_negative</dd>
-                <dd id="arg.is_polar" class="variable">is_polar</dd>
-                <dd id="arg.is_odd" class="variable">is_odd</dd>
-                <dd id="arg.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="arg.is_nonzero" class="variable">is_nonzero</dd>
                 <dd id="arg.is_prime" class="variable">is_prime</dd>
+                <dd id="arg.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="arg.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="arg.is_polar" class="variable">is_polar</dd>
+                <dd id="arg.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="arg.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="arg.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -14497,14 +14972,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="polar_lift">
-                                <div class="attr class">
-        <a class="headerlink" href="#polar_lift">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">polar_lift</span><wbr>(<span class="base">sympy.functions.elementary.complexes.polar_lift</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">polar_lift</span><wbr>(<span class="base">sympy.functions.elementary.complexes.polar_lift</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#polar_lift"></a>
     
             <div class="docstring"><p>Lift argument to the Riemann surface of the logarithm, using the
 standard branch.</p>
@@ -14543,32 +15018,35 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="polar_lift.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#polar_lift.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">polar_lift</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">polar_lift</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#polar_lift.__init__"></a>
     
     
 
                             </div>
                             <div id="polar_lift.is_polar" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#polar_lift.is_polar">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_polar</span><span class="default_value"> = True</span>
 
-        <span class="name">is_polar</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#polar_lift.is_polar"></a>
     
     
 
                             </div>
                             <div id="polar_lift.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#polar_lift.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;polar&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;polar&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#polar_lift.default_assumptions"></a>
     
     
 
@@ -14723,36 +15201,36 @@ <h5>Inherited Members</h5>
                 <dd id="polar_lift.simplify" class="function">simplify</dd>
                 <dd id="polar_lift.refine" class="function">refine</dd>
                 <dd id="polar_lift.rewrite" class="function">rewrite</dd>
-                <dd id="polar_lift.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="polar_lift.is_integer" class="variable">is_integer</dd>
-                <dd id="polar_lift.is_complex" class="variable">is_complex</dd>
-                <dd id="polar_lift.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="polar_lift.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="polar_lift.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="polar_lift.is_irrational" class="variable">is_irrational</dd>
+                <dd id="polar_lift.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="polar_lift.is_positive" class="variable">is_positive</dd>
+                <dd id="polar_lift.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="polar_lift.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="polar_lift.is_negative" class="variable">is_negative</dd>
+                <dd id="polar_lift.is_composite" class="variable">is_composite</dd>
+                <dd id="polar_lift.is_finite" class="variable">is_finite</dd>
                 <dd id="polar_lift.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="polar_lift.is_zero" class="variable">is_zero</dd>
                 <dd id="polar_lift.is_even" class="variable">is_even</dd>
-                <dd id="polar_lift.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="polar_lift.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="polar_lift.is_commutative" class="variable">is_commutative</dd>
-                <dd id="polar_lift.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="polar_lift.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="polar_lift.is_real" class="variable">is_real</dd>
-                <dd id="polar_lift.is_positive" class="variable">is_positive</dd>
-                <dd id="polar_lift.is_finite" class="variable">is_finite</dd>
-                <dd id="polar_lift.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="polar_lift.is_infinite" class="variable">is_infinite</dd>
-                <dd id="polar_lift.is_rational" class="variable">is_rational</dd>
-                <dd id="polar_lift.is_composite" class="variable">is_composite</dd>
-                <dd id="polar_lift.is_negative" class="variable">is_negative</dd>
+                <dd id="polar_lift.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="polar_lift.is_odd" class="variable">is_odd</dd>
-                <dd id="polar_lift.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="polar_lift.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="polar_lift.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="polar_lift.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="polar_lift.is_irrational" class="variable">is_irrational</dd>
+                <dd id="polar_lift.is_integer" class="variable">is_integer</dd>
+                <dd id="polar_lift.is_rational" class="variable">is_rational</dd>
+                <dd id="polar_lift.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="polar_lift.is_complex" class="variable">is_complex</dd>
                 <dd id="polar_lift.is_prime" class="variable">is_prime</dd>
+                <dd id="polar_lift.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="polar_lift.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="polar_lift.is_real" class="variable">is_real</dd>
+                <dd id="polar_lift.is_commutative" class="variable">is_commutative</dd>
                 <dd id="polar_lift.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="polar_lift.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="polar_lift.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="polar_lift.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="polar_lift.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -14764,14 +15242,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="periodic_argument">
-                                <div class="attr class">
-        <a class="headerlink" href="#periodic_argument">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">periodic_argument</span><wbr>(<span class="base">sympy.functions.elementary.complexes.periodic_argument</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">periodic_argument</span><wbr>(<span class="base">sympy.functions.elementary.complexes.periodic_argument</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#periodic_argument"></a>
     
             <div class="docstring"><p>Represent the argument on a quotient of the Riemann surface of the
 logarithm. That is, given a period $P$, always return a value in
@@ -14812,22 +15290,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="periodic_argument.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#periodic_argument.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">periodic_argument</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">periodic_argument</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#periodic_argument.__init__"></a>
     
     
 
                             </div>
                             <div id="periodic_argument.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#periodic_argument.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#periodic_argument.default_assumptions"></a>
     
     
 
@@ -14982,37 +15462,37 @@ <h5>Inherited Members</h5>
                 <dd id="periodic_argument.simplify" class="function">simplify</dd>
                 <dd id="periodic_argument.refine" class="function">refine</dd>
                 <dd id="periodic_argument.rewrite" class="function">rewrite</dd>
-                <dd id="periodic_argument.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="periodic_argument.is_integer" class="variable">is_integer</dd>
-                <dd id="periodic_argument.is_complex" class="variable">is_complex</dd>
-                <dd id="periodic_argument.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="periodic_argument.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="periodic_argument.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="periodic_argument.is_irrational" class="variable">is_irrational</dd>
+                <dd id="periodic_argument.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="periodic_argument.is_positive" class="variable">is_positive</dd>
+                <dd id="periodic_argument.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="periodic_argument.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="periodic_argument.is_negative" class="variable">is_negative</dd>
+                <dd id="periodic_argument.is_composite" class="variable">is_composite</dd>
+                <dd id="periodic_argument.is_finite" class="variable">is_finite</dd>
                 <dd id="periodic_argument.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="periodic_argument.is_zero" class="variable">is_zero</dd>
                 <dd id="periodic_argument.is_even" class="variable">is_even</dd>
-                <dd id="periodic_argument.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="periodic_argument.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="periodic_argument.is_commutative" class="variable">is_commutative</dd>
-                <dd id="periodic_argument.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="periodic_argument.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="periodic_argument.is_real" class="variable">is_real</dd>
-                <dd id="periodic_argument.is_positive" class="variable">is_positive</dd>
-                <dd id="periodic_argument.is_finite" class="variable">is_finite</dd>
-                <dd id="periodic_argument.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="periodic_argument.is_infinite" class="variable">is_infinite</dd>
-                <dd id="periodic_argument.is_rational" class="variable">is_rational</dd>
-                <dd id="periodic_argument.is_composite" class="variable">is_composite</dd>
-                <dd id="periodic_argument.is_negative" class="variable">is_negative</dd>
-                <dd id="periodic_argument.is_polar" class="variable">is_polar</dd>
+                <dd id="periodic_argument.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="periodic_argument.is_odd" class="variable">is_odd</dd>
-                <dd id="periodic_argument.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="periodic_argument.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="periodic_argument.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="periodic_argument.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="periodic_argument.is_irrational" class="variable">is_irrational</dd>
+                <dd id="periodic_argument.is_integer" class="variable">is_integer</dd>
+                <dd id="periodic_argument.is_rational" class="variable">is_rational</dd>
+                <dd id="periodic_argument.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="periodic_argument.is_complex" class="variable">is_complex</dd>
                 <dd id="periodic_argument.is_prime" class="variable">is_prime</dd>
+                <dd id="periodic_argument.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="periodic_argument.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="periodic_argument.is_real" class="variable">is_real</dd>
+                <dd id="periodic_argument.is_commutative" class="variable">is_commutative</dd>
                 <dd id="periodic_argument.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="periodic_argument.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="periodic_argument.is_polar" class="variable">is_polar</dd>
+                <dd id="periodic_argument.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="periodic_argument.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="periodic_argument.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -15024,14 +15504,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="principal_branch">
-                                <div class="attr class">
-        <a class="headerlink" href="#principal_branch">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">principal_branch</span><wbr>(<span class="base">sympy.functions.elementary.complexes.principal_branch</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">principal_branch</span><wbr>(<span class="base">sympy.functions.elementary.complexes.principal_branch</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#principal_branch"></a>
     
             <div class="docstring"><p>Represent a polar number reduced to its principal branch on a quotient
 of the Riemann surface of the logarithm.</p>
@@ -15071,32 +15551,35 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="principal_branch.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#principal_branch.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">principal_branch</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">principal_branch</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#principal_branch.__init__"></a>
     
     
 
                             </div>
                             <div id="principal_branch.is_polar" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#principal_branch.is_polar">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_polar</span><span class="default_value"> = True</span>
 
-        <span class="name">is_polar</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#principal_branch.is_polar"></a>
     
     
 
                             </div>
                             <div id="principal_branch.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#principal_branch.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;polar&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;polar&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#principal_branch.default_assumptions"></a>
     
     
 
@@ -15251,36 +15734,36 @@ <h5>Inherited Members</h5>
                 <dd id="principal_branch.simplify" class="function">simplify</dd>
                 <dd id="principal_branch.refine" class="function">refine</dd>
                 <dd id="principal_branch.rewrite" class="function">rewrite</dd>
-                <dd id="principal_branch.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="principal_branch.is_integer" class="variable">is_integer</dd>
-                <dd id="principal_branch.is_complex" class="variable">is_complex</dd>
-                <dd id="principal_branch.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="principal_branch.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="principal_branch.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="principal_branch.is_irrational" class="variable">is_irrational</dd>
+                <dd id="principal_branch.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="principal_branch.is_positive" class="variable">is_positive</dd>
+                <dd id="principal_branch.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="principal_branch.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="principal_branch.is_negative" class="variable">is_negative</dd>
+                <dd id="principal_branch.is_composite" class="variable">is_composite</dd>
+                <dd id="principal_branch.is_finite" class="variable">is_finite</dd>
                 <dd id="principal_branch.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="principal_branch.is_zero" class="variable">is_zero</dd>
                 <dd id="principal_branch.is_even" class="variable">is_even</dd>
-                <dd id="principal_branch.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="principal_branch.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="principal_branch.is_commutative" class="variable">is_commutative</dd>
-                <dd id="principal_branch.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="principal_branch.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="principal_branch.is_real" class="variable">is_real</dd>
-                <dd id="principal_branch.is_positive" class="variable">is_positive</dd>
-                <dd id="principal_branch.is_finite" class="variable">is_finite</dd>
-                <dd id="principal_branch.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="principal_branch.is_infinite" class="variable">is_infinite</dd>
-                <dd id="principal_branch.is_rational" class="variable">is_rational</dd>
-                <dd id="principal_branch.is_composite" class="variable">is_composite</dd>
-                <dd id="principal_branch.is_negative" class="variable">is_negative</dd>
+                <dd id="principal_branch.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="principal_branch.is_odd" class="variable">is_odd</dd>
-                <dd id="principal_branch.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="principal_branch.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="principal_branch.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="principal_branch.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="principal_branch.is_irrational" class="variable">is_irrational</dd>
+                <dd id="principal_branch.is_integer" class="variable">is_integer</dd>
+                <dd id="principal_branch.is_rational" class="variable">is_rational</dd>
+                <dd id="principal_branch.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="principal_branch.is_complex" class="variable">is_complex</dd>
                 <dd id="principal_branch.is_prime" class="variable">is_prime</dd>
+                <dd id="principal_branch.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="principal_branch.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="principal_branch.is_real" class="variable">is_real</dd>
+                <dd id="principal_branch.is_commutative" class="variable">is_commutative</dd>
                 <dd id="principal_branch.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="principal_branch.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="principal_branch.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="principal_branch.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="principal_branch.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -15292,14 +15775,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="transpose">
-                                <div class="attr class">
-        <a class="headerlink" href="#transpose">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">transpose</span><wbr>(<span class="base">sympy.functions.elementary.complexes.transpose</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">transpose</span><wbr>(<span class="base">sympy.functions.elementary.complexes.transpose</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#transpose"></a>
     
             <div class="docstring"><p>Linear map transposition.</p>
 
@@ -15339,22 +15822,24 @@ <h1 id="returns">Returns</h1>
 
 
                             <div id="transpose.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#transpose.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">transpose</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">transpose</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#transpose.__init__"></a>
     
     
 
                             </div>
                             <div id="transpose.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#transpose.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#transpose.default_assumptions"></a>
     
     
 
@@ -15509,37 +15994,37 @@ <h5>Inherited Members</h5>
                 <dd id="transpose.simplify" class="function">simplify</dd>
                 <dd id="transpose.refine" class="function">refine</dd>
                 <dd id="transpose.rewrite" class="function">rewrite</dd>
-                <dd id="transpose.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="transpose.is_integer" class="variable">is_integer</dd>
-                <dd id="transpose.is_complex" class="variable">is_complex</dd>
-                <dd id="transpose.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="transpose.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="transpose.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="transpose.is_irrational" class="variable">is_irrational</dd>
+                <dd id="transpose.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="transpose.is_positive" class="variable">is_positive</dd>
+                <dd id="transpose.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="transpose.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="transpose.is_negative" class="variable">is_negative</dd>
+                <dd id="transpose.is_composite" class="variable">is_composite</dd>
+                <dd id="transpose.is_finite" class="variable">is_finite</dd>
                 <dd id="transpose.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="transpose.is_zero" class="variable">is_zero</dd>
                 <dd id="transpose.is_even" class="variable">is_even</dd>
-                <dd id="transpose.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="transpose.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="transpose.is_commutative" class="variable">is_commutative</dd>
-                <dd id="transpose.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="transpose.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="transpose.is_real" class="variable">is_real</dd>
-                <dd id="transpose.is_positive" class="variable">is_positive</dd>
-                <dd id="transpose.is_finite" class="variable">is_finite</dd>
-                <dd id="transpose.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="transpose.is_infinite" class="variable">is_infinite</dd>
-                <dd id="transpose.is_rational" class="variable">is_rational</dd>
-                <dd id="transpose.is_composite" class="variable">is_composite</dd>
-                <dd id="transpose.is_negative" class="variable">is_negative</dd>
-                <dd id="transpose.is_polar" class="variable">is_polar</dd>
+                <dd id="transpose.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="transpose.is_odd" class="variable">is_odd</dd>
-                <dd id="transpose.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="transpose.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="transpose.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="transpose.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="transpose.is_irrational" class="variable">is_irrational</dd>
+                <dd id="transpose.is_integer" class="variable">is_integer</dd>
+                <dd id="transpose.is_rational" class="variable">is_rational</dd>
+                <dd id="transpose.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="transpose.is_complex" class="variable">is_complex</dd>
                 <dd id="transpose.is_prime" class="variable">is_prime</dd>
+                <dd id="transpose.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="transpose.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="transpose.is_real" class="variable">is_real</dd>
+                <dd id="transpose.is_commutative" class="variable">is_commutative</dd>
                 <dd id="transpose.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="transpose.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="transpose.is_polar" class="variable">is_polar</dd>
+                <dd id="transpose.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="transpose.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="transpose.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -15551,14 +16036,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="adjoint">
-                                <div class="attr class">
-        <a class="headerlink" href="#adjoint">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">adjoint</span><wbr>(<span class="base">sympy.functions.elementary.complexes.adjoint</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">adjoint</span><wbr>(<span class="base">sympy.functions.elementary.complexes.adjoint</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#adjoint"></a>
     
             <div class="docstring"><p>Conjugate transpose or Hermite conjugation.</p>
 
@@ -15584,22 +16069,24 @@ <h1 id="returns">Returns</h1>
 
 
                             <div id="adjoint.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#adjoint.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">adjoint</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">adjoint</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#adjoint.__init__"></a>
     
     
 
                             </div>
                             <div id="adjoint.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#adjoint.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#adjoint.default_assumptions"></a>
     
     
 
@@ -15754,37 +16241,37 @@ <h5>Inherited Members</h5>
                 <dd id="adjoint.simplify" class="function">simplify</dd>
                 <dd id="adjoint.refine" class="function">refine</dd>
                 <dd id="adjoint.rewrite" class="function">rewrite</dd>
-                <dd id="adjoint.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="adjoint.is_integer" class="variable">is_integer</dd>
-                <dd id="adjoint.is_complex" class="variable">is_complex</dd>
-                <dd id="adjoint.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="adjoint.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="adjoint.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="adjoint.is_irrational" class="variable">is_irrational</dd>
+                <dd id="adjoint.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="adjoint.is_positive" class="variable">is_positive</dd>
+                <dd id="adjoint.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="adjoint.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="adjoint.is_negative" class="variable">is_negative</dd>
+                <dd id="adjoint.is_composite" class="variable">is_composite</dd>
+                <dd id="adjoint.is_finite" class="variable">is_finite</dd>
                 <dd id="adjoint.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="adjoint.is_zero" class="variable">is_zero</dd>
                 <dd id="adjoint.is_even" class="variable">is_even</dd>
-                <dd id="adjoint.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="adjoint.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="adjoint.is_commutative" class="variable">is_commutative</dd>
-                <dd id="adjoint.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="adjoint.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="adjoint.is_real" class="variable">is_real</dd>
-                <dd id="adjoint.is_positive" class="variable">is_positive</dd>
-                <dd id="adjoint.is_finite" class="variable">is_finite</dd>
-                <dd id="adjoint.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="adjoint.is_infinite" class="variable">is_infinite</dd>
-                <dd id="adjoint.is_rational" class="variable">is_rational</dd>
-                <dd id="adjoint.is_composite" class="variable">is_composite</dd>
-                <dd id="adjoint.is_negative" class="variable">is_negative</dd>
-                <dd id="adjoint.is_polar" class="variable">is_polar</dd>
+                <dd id="adjoint.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="adjoint.is_odd" class="variable">is_odd</dd>
-                <dd id="adjoint.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="adjoint.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="adjoint.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="adjoint.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="adjoint.is_irrational" class="variable">is_irrational</dd>
+                <dd id="adjoint.is_integer" class="variable">is_integer</dd>
+                <dd id="adjoint.is_rational" class="variable">is_rational</dd>
+                <dd id="adjoint.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="adjoint.is_complex" class="variable">is_complex</dd>
                 <dd id="adjoint.is_prime" class="variable">is_prime</dd>
+                <dd id="adjoint.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="adjoint.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="adjoint.is_real" class="variable">is_real</dd>
+                <dd id="adjoint.is_commutative" class="variable">is_commutative</dd>
                 <dd id="adjoint.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="adjoint.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="adjoint.is_polar" class="variable">is_polar</dd>
+                <dd id="adjoint.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="adjoint.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="adjoint.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -15796,14 +16283,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="sin">
-                                <div class="attr class">
-        <a class="headerlink" href="#sin">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">sin</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.sin</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">sin</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.sin</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#sin"></a>
     
             <div class="docstring"><p>The sine function.</p>
 
@@ -15812,7 +16299,7 @@ <h5>Inherited Members</h5>
 <h1 id="explanation">Explanation</h1>
 
 <p>This function will evaluate automatically in the
-case $x/\pi$ is some rational number [4]_.  For example,
+case $x/\pi$ is some rational number <sup class="footnote-ref" id="fnref-4"><a href="#fn-4">1</a></sup>.  For example,
 if $x$ is a multiple of $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, and $\pi/6$.</p>
 
 <h1 id="examples">Examples</h1>
@@ -15840,30 +16327,36 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.14
-.. [3] http://functions.wolfram.com/ElementaryFunctions/Sin
-.. [4] http://mathworld.wolfram.com/TrigonometryAngles.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-4">
+<p>http://mathworld.wolfram.com/TrigonometryAngles.html&#160;<a href="#fnref-4" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="sin.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#sin.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">sin</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">sin</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#sin.__init__"></a>
     
     
 
                             </div>
                             <div id="sin.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sin.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#sin.default_assumptions"></a>
     
     
 
@@ -16023,37 +16516,37 @@ <h5>Inherited Members</h5>
                 <dd id="sin.simplify" class="function">simplify</dd>
                 <dd id="sin.refine" class="function">refine</dd>
                 <dd id="sin.rewrite" class="function">rewrite</dd>
-                <dd id="sin.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="sin.is_integer" class="variable">is_integer</dd>
-                <dd id="sin.is_complex" class="variable">is_complex</dd>
-                <dd id="sin.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="sin.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="sin.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="sin.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sin.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sin.is_positive" class="variable">is_positive</dd>
+                <dd id="sin.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="sin.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="sin.is_negative" class="variable">is_negative</dd>
+                <dd id="sin.is_composite" class="variable">is_composite</dd>
+                <dd id="sin.is_finite" class="variable">is_finite</dd>
                 <dd id="sin.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="sin.is_zero" class="variable">is_zero</dd>
                 <dd id="sin.is_even" class="variable">is_even</dd>
-                <dd id="sin.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="sin.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="sin.is_commutative" class="variable">is_commutative</dd>
-                <dd id="sin.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="sin.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="sin.is_real" class="variable">is_real</dd>
-                <dd id="sin.is_positive" class="variable">is_positive</dd>
-                <dd id="sin.is_finite" class="variable">is_finite</dd>
-                <dd id="sin.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="sin.is_infinite" class="variable">is_infinite</dd>
-                <dd id="sin.is_rational" class="variable">is_rational</dd>
-                <dd id="sin.is_composite" class="variable">is_composite</dd>
-                <dd id="sin.is_negative" class="variable">is_negative</dd>
-                <dd id="sin.is_polar" class="variable">is_polar</dd>
+                <dd id="sin.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="sin.is_odd" class="variable">is_odd</dd>
-                <dd id="sin.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="sin.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sin.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="sin.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="sin.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sin.is_integer" class="variable">is_integer</dd>
+                <dd id="sin.is_rational" class="variable">is_rational</dd>
+                <dd id="sin.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="sin.is_complex" class="variable">is_complex</dd>
                 <dd id="sin.is_prime" class="variable">is_prime</dd>
+                <dd id="sin.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="sin.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="sin.is_real" class="variable">is_real</dd>
+                <dd id="sin.is_commutative" class="variable">is_commutative</dd>
                 <dd id="sin.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="sin.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="sin.is_polar" class="variable">is_polar</dd>
+                <dd id="sin.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="sin.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="sin.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -16065,14 +16558,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="cos">
-                                <div class="attr class">
-        <a class="headerlink" href="#cos">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">cos</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.cos</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">cos</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.cos</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#cos"></a>
     
             <div class="docstring"><p>The cosine function.</p>
 
@@ -16080,7 +16573,7 @@ <h5>Inherited Members</h5>
 
 <h1 id="explanation">Explanation</h1>
 
-<p>See :func:<code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
+<p>See <code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -16107,29 +16600,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.14
-.. [3] http://functions.wolfram.com/ElementaryFunctions/Cos</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="cos.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#cos.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">cos</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">cos</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#cos.__init__"></a>
     
     
 
                             </div>
                             <div id="cos.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#cos.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#cos.default_assumptions"></a>
     
     
 
@@ -16289,37 +16786,37 @@ <h5>Inherited Members</h5>
                 <dd id="cos.simplify" class="function">simplify</dd>
                 <dd id="cos.refine" class="function">refine</dd>
                 <dd id="cos.rewrite" class="function">rewrite</dd>
-                <dd id="cos.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="cos.is_integer" class="variable">is_integer</dd>
-                <dd id="cos.is_complex" class="variable">is_complex</dd>
-                <dd id="cos.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="cos.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="cos.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="cos.is_irrational" class="variable">is_irrational</dd>
+                <dd id="cos.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="cos.is_positive" class="variable">is_positive</dd>
+                <dd id="cos.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="cos.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="cos.is_negative" class="variable">is_negative</dd>
+                <dd id="cos.is_composite" class="variable">is_composite</dd>
+                <dd id="cos.is_finite" class="variable">is_finite</dd>
                 <dd id="cos.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="cos.is_zero" class="variable">is_zero</dd>
                 <dd id="cos.is_even" class="variable">is_even</dd>
-                <dd id="cos.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="cos.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="cos.is_commutative" class="variable">is_commutative</dd>
-                <dd id="cos.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="cos.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="cos.is_real" class="variable">is_real</dd>
-                <dd id="cos.is_positive" class="variable">is_positive</dd>
-                <dd id="cos.is_finite" class="variable">is_finite</dd>
-                <dd id="cos.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="cos.is_infinite" class="variable">is_infinite</dd>
-                <dd id="cos.is_rational" class="variable">is_rational</dd>
-                <dd id="cos.is_composite" class="variable">is_composite</dd>
-                <dd id="cos.is_negative" class="variable">is_negative</dd>
-                <dd id="cos.is_polar" class="variable">is_polar</dd>
+                <dd id="cos.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="cos.is_odd" class="variable">is_odd</dd>
-                <dd id="cos.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="cos.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="cos.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="cos.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="cos.is_irrational" class="variable">is_irrational</dd>
+                <dd id="cos.is_integer" class="variable">is_integer</dd>
+                <dd id="cos.is_rational" class="variable">is_rational</dd>
+                <dd id="cos.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="cos.is_complex" class="variable">is_complex</dd>
                 <dd id="cos.is_prime" class="variable">is_prime</dd>
+                <dd id="cos.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="cos.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="cos.is_real" class="variable">is_real</dd>
+                <dd id="cos.is_commutative" class="variable">is_commutative</dd>
                 <dd id="cos.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="cos.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="cos.is_polar" class="variable">is_polar</dd>
+                <dd id="cos.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="cos.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="cos.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -16331,14 +16828,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="tan">
-                                <div class="attr class">
-        <a class="headerlink" href="#tan">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">tan</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.tan</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">tan</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.tan</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#tan"></a>
     
             <div class="docstring"><p>The tangent function.</p>
 
@@ -16346,7 +16843,7 @@ <h5>Inherited Members</h5>
 
 <h1 id="explanation">Explanation</h1>
 
-<p>See :class:<code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
+<p>See <code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -16367,29 +16864,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.14
-.. [3] http://functions.wolfram.com/ElementaryFunctions/Tan</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="tan.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#tan.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">tan</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">tan</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#tan.__init__"></a>
     
     
 
                             </div>
                             <div id="tan.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#tan.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#tan.default_assumptions"></a>
     
     
 
@@ -16550,37 +17051,37 @@ <h5>Inherited Members</h5>
                 <dd id="tan.simplify" class="function">simplify</dd>
                 <dd id="tan.refine" class="function">refine</dd>
                 <dd id="tan.rewrite" class="function">rewrite</dd>
-                <dd id="tan.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="tan.is_integer" class="variable">is_integer</dd>
-                <dd id="tan.is_complex" class="variable">is_complex</dd>
-                <dd id="tan.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="tan.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="tan.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="tan.is_irrational" class="variable">is_irrational</dd>
+                <dd id="tan.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="tan.is_positive" class="variable">is_positive</dd>
+                <dd id="tan.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="tan.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="tan.is_negative" class="variable">is_negative</dd>
+                <dd id="tan.is_composite" class="variable">is_composite</dd>
+                <dd id="tan.is_finite" class="variable">is_finite</dd>
                 <dd id="tan.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="tan.is_zero" class="variable">is_zero</dd>
                 <dd id="tan.is_even" class="variable">is_even</dd>
-                <dd id="tan.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="tan.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="tan.is_commutative" class="variable">is_commutative</dd>
-                <dd id="tan.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="tan.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="tan.is_real" class="variable">is_real</dd>
-                <dd id="tan.is_positive" class="variable">is_positive</dd>
-                <dd id="tan.is_finite" class="variable">is_finite</dd>
-                <dd id="tan.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="tan.is_infinite" class="variable">is_infinite</dd>
-                <dd id="tan.is_rational" class="variable">is_rational</dd>
-                <dd id="tan.is_composite" class="variable">is_composite</dd>
-                <dd id="tan.is_negative" class="variable">is_negative</dd>
-                <dd id="tan.is_polar" class="variable">is_polar</dd>
+                <dd id="tan.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="tan.is_odd" class="variable">is_odd</dd>
-                <dd id="tan.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="tan.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="tan.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="tan.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="tan.is_irrational" class="variable">is_irrational</dd>
+                <dd id="tan.is_integer" class="variable">is_integer</dd>
+                <dd id="tan.is_rational" class="variable">is_rational</dd>
+                <dd id="tan.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="tan.is_complex" class="variable">is_complex</dd>
                 <dd id="tan.is_prime" class="variable">is_prime</dd>
+                <dd id="tan.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="tan.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="tan.is_real" class="variable">is_real</dd>
+                <dd id="tan.is_commutative" class="variable">is_commutative</dd>
                 <dd id="tan.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="tan.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="tan.is_polar" class="variable">is_polar</dd>
+                <dd id="tan.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="tan.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="tan.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -16592,14 +17093,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="sec">
-                                <div class="attr class">
-        <a class="headerlink" href="#sec">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">sec</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.sec</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">sec</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.sec</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#sec"></a>
     
             <div class="docstring"><p>The secant function.</p>
 
@@ -16607,7 +17108,7 @@ <h5>Inherited Members</h5>
 
 <h1 id="explanation">Explanation</h1>
 
-<p>See :class:<code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
+<p>See <code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -16626,29 +17127,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.14
-.. [3] http://functions.wolfram.com/ElementaryFunctions/Sec</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="sec.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#sec.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">sec</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">sec</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#sec.__init__"></a>
     
     
 
                             </div>
                             <div id="sec.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sec.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#sec.default_assumptions"></a>
     
     
 
@@ -16811,37 +17316,37 @@ <h5>Inherited Members</h5>
                 <dd id="sec.simplify" class="function">simplify</dd>
                 <dd id="sec.refine" class="function">refine</dd>
                 <dd id="sec.rewrite" class="function">rewrite</dd>
-                <dd id="sec.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="sec.is_integer" class="variable">is_integer</dd>
-                <dd id="sec.is_complex" class="variable">is_complex</dd>
-                <dd id="sec.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="sec.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="sec.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="sec.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sec.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sec.is_positive" class="variable">is_positive</dd>
+                <dd id="sec.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="sec.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="sec.is_negative" class="variable">is_negative</dd>
+                <dd id="sec.is_composite" class="variable">is_composite</dd>
+                <dd id="sec.is_finite" class="variable">is_finite</dd>
                 <dd id="sec.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="sec.is_zero" class="variable">is_zero</dd>
                 <dd id="sec.is_even" class="variable">is_even</dd>
-                <dd id="sec.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="sec.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="sec.is_commutative" class="variable">is_commutative</dd>
-                <dd id="sec.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="sec.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="sec.is_real" class="variable">is_real</dd>
-                <dd id="sec.is_positive" class="variable">is_positive</dd>
-                <dd id="sec.is_finite" class="variable">is_finite</dd>
-                <dd id="sec.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="sec.is_infinite" class="variable">is_infinite</dd>
-                <dd id="sec.is_rational" class="variable">is_rational</dd>
-                <dd id="sec.is_composite" class="variable">is_composite</dd>
-                <dd id="sec.is_negative" class="variable">is_negative</dd>
-                <dd id="sec.is_polar" class="variable">is_polar</dd>
+                <dd id="sec.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="sec.is_odd" class="variable">is_odd</dd>
-                <dd id="sec.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="sec.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sec.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="sec.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="sec.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sec.is_integer" class="variable">is_integer</dd>
+                <dd id="sec.is_rational" class="variable">is_rational</dd>
+                <dd id="sec.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="sec.is_complex" class="variable">is_complex</dd>
                 <dd id="sec.is_prime" class="variable">is_prime</dd>
+                <dd id="sec.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="sec.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="sec.is_real" class="variable">is_real</dd>
+                <dd id="sec.is_commutative" class="variable">is_commutative</dd>
                 <dd id="sec.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="sec.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="sec.is_polar" class="variable">is_polar</dd>
+                <dd id="sec.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="sec.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="sec.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -16853,14 +17358,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="csc">
-                                <div class="attr class">
-        <a class="headerlink" href="#csc">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">csc</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.csc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">csc</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.csc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#csc"></a>
     
             <div class="docstring"><p>The cosecant function.</p>
 
@@ -16868,7 +17373,7 @@ <h5>Inherited Members</h5>
 
 <h1 id="explanation">Explanation</h1>
 
-<p>See :func:<code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
+<p>See <code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -16887,29 +17392,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.14
-.. [3] http://functions.wolfram.com/ElementaryFunctions/Csc</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="csc.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#csc.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">csc</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">csc</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#csc.__init__"></a>
     
     
 
                             </div>
                             <div id="csc.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#csc.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#csc.default_assumptions"></a>
     
     
 
@@ -17072,37 +17581,37 @@ <h5>Inherited Members</h5>
                 <dd id="csc.simplify" class="function">simplify</dd>
                 <dd id="csc.refine" class="function">refine</dd>
                 <dd id="csc.rewrite" class="function">rewrite</dd>
-                <dd id="csc.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="csc.is_integer" class="variable">is_integer</dd>
-                <dd id="csc.is_complex" class="variable">is_complex</dd>
-                <dd id="csc.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="csc.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="csc.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="csc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="csc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="csc.is_positive" class="variable">is_positive</dd>
+                <dd id="csc.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="csc.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="csc.is_negative" class="variable">is_negative</dd>
+                <dd id="csc.is_composite" class="variable">is_composite</dd>
+                <dd id="csc.is_finite" class="variable">is_finite</dd>
                 <dd id="csc.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="csc.is_zero" class="variable">is_zero</dd>
                 <dd id="csc.is_even" class="variable">is_even</dd>
-                <dd id="csc.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="csc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="csc.is_commutative" class="variable">is_commutative</dd>
-                <dd id="csc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="csc.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="csc.is_real" class="variable">is_real</dd>
-                <dd id="csc.is_positive" class="variable">is_positive</dd>
-                <dd id="csc.is_finite" class="variable">is_finite</dd>
-                <dd id="csc.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="csc.is_infinite" class="variable">is_infinite</dd>
-                <dd id="csc.is_rational" class="variable">is_rational</dd>
-                <dd id="csc.is_composite" class="variable">is_composite</dd>
-                <dd id="csc.is_negative" class="variable">is_negative</dd>
-                <dd id="csc.is_polar" class="variable">is_polar</dd>
+                <dd id="csc.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="csc.is_odd" class="variable">is_odd</dd>
-                <dd id="csc.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="csc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="csc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="csc.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="csc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="csc.is_integer" class="variable">is_integer</dd>
+                <dd id="csc.is_rational" class="variable">is_rational</dd>
+                <dd id="csc.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="csc.is_complex" class="variable">is_complex</dd>
                 <dd id="csc.is_prime" class="variable">is_prime</dd>
+                <dd id="csc.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="csc.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="csc.is_real" class="variable">is_real</dd>
+                <dd id="csc.is_commutative" class="variable">is_commutative</dd>
                 <dd id="csc.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="csc.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="csc.is_polar" class="variable">is_polar</dd>
+                <dd id="csc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="csc.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="csc.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -17114,14 +17623,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="cot">
-                                <div class="attr class">
-        <a class="headerlink" href="#cot">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">cot</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.cot</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">cot</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.cot</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#cot"></a>
     
             <div class="docstring"><p>The cotangent function.</p>
 
@@ -17129,7 +17638,7 @@ <h5>Inherited Members</h5>
 
 <h1 id="explanation">Explanation</h1>
 
-<p>See :class:<code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
+<p>See <code><a href="#sin">sin</a></code> for notes about automatic evaluation.</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -17150,29 +17659,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.14
-.. [3] http://functions.wolfram.com/ElementaryFunctions/Cot</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="cot.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#cot.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">cot</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">cot</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#cot.__init__"></a>
     
     
 
                             </div>
                             <div id="cot.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#cot.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#cot.default_assumptions"></a>
     
     
 
@@ -17333,37 +17846,37 @@ <h5>Inherited Members</h5>
                 <dd id="cot.simplify" class="function">simplify</dd>
                 <dd id="cot.refine" class="function">refine</dd>
                 <dd id="cot.rewrite" class="function">rewrite</dd>
-                <dd id="cot.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="cot.is_integer" class="variable">is_integer</dd>
-                <dd id="cot.is_complex" class="variable">is_complex</dd>
-                <dd id="cot.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="cot.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="cot.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="cot.is_irrational" class="variable">is_irrational</dd>
+                <dd id="cot.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="cot.is_positive" class="variable">is_positive</dd>
+                <dd id="cot.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="cot.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="cot.is_negative" class="variable">is_negative</dd>
+                <dd id="cot.is_composite" class="variable">is_composite</dd>
+                <dd id="cot.is_finite" class="variable">is_finite</dd>
                 <dd id="cot.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="cot.is_zero" class="variable">is_zero</dd>
                 <dd id="cot.is_even" class="variable">is_even</dd>
-                <dd id="cot.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="cot.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="cot.is_commutative" class="variable">is_commutative</dd>
-                <dd id="cot.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="cot.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="cot.is_real" class="variable">is_real</dd>
-                <dd id="cot.is_positive" class="variable">is_positive</dd>
-                <dd id="cot.is_finite" class="variable">is_finite</dd>
-                <dd id="cot.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="cot.is_infinite" class="variable">is_infinite</dd>
-                <dd id="cot.is_rational" class="variable">is_rational</dd>
-                <dd id="cot.is_composite" class="variable">is_composite</dd>
-                <dd id="cot.is_negative" class="variable">is_negative</dd>
-                <dd id="cot.is_polar" class="variable">is_polar</dd>
+                <dd id="cot.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="cot.is_odd" class="variable">is_odd</dd>
-                <dd id="cot.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="cot.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="cot.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="cot.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="cot.is_irrational" class="variable">is_irrational</dd>
+                <dd id="cot.is_integer" class="variable">is_integer</dd>
+                <dd id="cot.is_rational" class="variable">is_rational</dd>
+                <dd id="cot.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="cot.is_complex" class="variable">is_complex</dd>
                 <dd id="cot.is_prime" class="variable">is_prime</dd>
+                <dd id="cot.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="cot.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="cot.is_real" class="variable">is_real</dd>
+                <dd id="cot.is_commutative" class="variable">is_commutative</dd>
                 <dd id="cot.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="cot.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="cot.is_polar" class="variable">is_polar</dd>
+                <dd id="cot.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="cot.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="cot.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -17375,25 +17888,22 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="sinc">
-                                <div class="attr class">
-        <a class="headerlink" href="#sinc">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">sinc</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.sinc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">sinc</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.sinc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#sinc"></a>
     
             <div class="docstring"><p>Represents an unnormalized sinc function:</p>
 
-<p>.. math::</p>
-
-<pre><code>\operatorname{sinc}(x) =
+<p>$$\operatorname{sinc}(x) =
 \begin{cases}
-  \frac{\sin x}{x} &amp; \qquad x \neq 0 \\
+  \frac{\sin x}{x} &amp; \qquad x \neq 0 \
   1 &amp; \qquad x = 0
-\end{cases}
-</code></pre>
+\end{cases}$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -17443,27 +17953,33 @@ <h1 id="see-also">See also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Sinc_function</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="sinc.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#sinc.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">sinc</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">sinc</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#sinc.__init__"></a>
     
     
 
                             </div>
                             <div id="sinc.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sinc.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#sinc.default_assumptions"></a>
     
     
 
@@ -17618,37 +18134,37 @@ <h5>Inherited Members</h5>
                 <dd id="sinc.simplify" class="function">simplify</dd>
                 <dd id="sinc.refine" class="function">refine</dd>
                 <dd id="sinc.rewrite" class="function">rewrite</dd>
-                <dd id="sinc.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="sinc.is_integer" class="variable">is_integer</dd>
-                <dd id="sinc.is_complex" class="variable">is_complex</dd>
-                <dd id="sinc.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="sinc.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="sinc.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="sinc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sinc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sinc.is_positive" class="variable">is_positive</dd>
+                <dd id="sinc.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="sinc.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="sinc.is_negative" class="variable">is_negative</dd>
+                <dd id="sinc.is_composite" class="variable">is_composite</dd>
+                <dd id="sinc.is_finite" class="variable">is_finite</dd>
                 <dd id="sinc.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="sinc.is_zero" class="variable">is_zero</dd>
                 <dd id="sinc.is_even" class="variable">is_even</dd>
-                <dd id="sinc.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="sinc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="sinc.is_commutative" class="variable">is_commutative</dd>
-                <dd id="sinc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="sinc.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="sinc.is_real" class="variable">is_real</dd>
-                <dd id="sinc.is_positive" class="variable">is_positive</dd>
-                <dd id="sinc.is_finite" class="variable">is_finite</dd>
-                <dd id="sinc.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="sinc.is_infinite" class="variable">is_infinite</dd>
-                <dd id="sinc.is_rational" class="variable">is_rational</dd>
-                <dd id="sinc.is_composite" class="variable">is_composite</dd>
-                <dd id="sinc.is_negative" class="variable">is_negative</dd>
-                <dd id="sinc.is_polar" class="variable">is_polar</dd>
+                <dd id="sinc.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="sinc.is_odd" class="variable">is_odd</dd>
-                <dd id="sinc.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="sinc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sinc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="sinc.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="sinc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sinc.is_integer" class="variable">is_integer</dd>
+                <dd id="sinc.is_rational" class="variable">is_rational</dd>
+                <dd id="sinc.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="sinc.is_complex" class="variable">is_complex</dd>
                 <dd id="sinc.is_prime" class="variable">is_prime</dd>
+                <dd id="sinc.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="sinc.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="sinc.is_real" class="variable">is_real</dd>
+                <dd id="sinc.is_commutative" class="variable">is_commutative</dd>
                 <dd id="sinc.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="sinc.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="sinc.is_polar" class="variable">is_polar</dd>
+                <dd id="sinc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="sinc.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="sinc.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -17660,14 +18176,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="asin">
-                                <div class="attr class">
-        <a class="headerlink" href="#asin">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">asin</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.asin</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">asin</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.asin</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#asin"></a>
     
             <div class="docstring"><p>The inverse sine function.</p>
 
@@ -17701,29 +18217,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.23
-.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSin</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="asin.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#asin.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">asin</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">asin</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#asin.__init__"></a>
     
     
 
                             </div>
                             <div id="asin.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#asin.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#asin.default_assumptions"></a>
     
     
 
@@ -17879,37 +18399,37 @@ <h5>Inherited Members</h5>
                 <dd id="asin.simplify" class="function">simplify</dd>
                 <dd id="asin.refine" class="function">refine</dd>
                 <dd id="asin.rewrite" class="function">rewrite</dd>
-                <dd id="asin.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="asin.is_integer" class="variable">is_integer</dd>
-                <dd id="asin.is_complex" class="variable">is_complex</dd>
-                <dd id="asin.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="asin.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="asin.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="asin.is_irrational" class="variable">is_irrational</dd>
+                <dd id="asin.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="asin.is_positive" class="variable">is_positive</dd>
+                <dd id="asin.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="asin.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="asin.is_negative" class="variable">is_negative</dd>
+                <dd id="asin.is_composite" class="variable">is_composite</dd>
+                <dd id="asin.is_finite" class="variable">is_finite</dd>
                 <dd id="asin.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="asin.is_zero" class="variable">is_zero</dd>
                 <dd id="asin.is_even" class="variable">is_even</dd>
-                <dd id="asin.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="asin.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="asin.is_commutative" class="variable">is_commutative</dd>
-                <dd id="asin.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="asin.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="asin.is_real" class="variable">is_real</dd>
-                <dd id="asin.is_positive" class="variable">is_positive</dd>
-                <dd id="asin.is_finite" class="variable">is_finite</dd>
-                <dd id="asin.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="asin.is_infinite" class="variable">is_infinite</dd>
-                <dd id="asin.is_rational" class="variable">is_rational</dd>
-                <dd id="asin.is_composite" class="variable">is_composite</dd>
-                <dd id="asin.is_negative" class="variable">is_negative</dd>
-                <dd id="asin.is_polar" class="variable">is_polar</dd>
+                <dd id="asin.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="asin.is_odd" class="variable">is_odd</dd>
-                <dd id="asin.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="asin.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="asin.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="asin.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="asin.is_irrational" class="variable">is_irrational</dd>
+                <dd id="asin.is_integer" class="variable">is_integer</dd>
+                <dd id="asin.is_rational" class="variable">is_rational</dd>
+                <dd id="asin.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="asin.is_complex" class="variable">is_complex</dd>
                 <dd id="asin.is_prime" class="variable">is_prime</dd>
+                <dd id="asin.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="asin.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="asin.is_real" class="variable">is_real</dd>
+                <dd id="asin.is_commutative" class="variable">is_commutative</dd>
                 <dd id="asin.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="asin.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="asin.is_polar" class="variable">is_polar</dd>
+                <dd id="asin.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="asin.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="asin.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -17921,14 +18441,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="acos">
-                                <div class="attr class">
-        <a class="headerlink" href="#acos">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">acos</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.acos</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">acos</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.acos</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#acos"></a>
     
             <div class="docstring"><p>The inverse cosine function.</p>
 
@@ -17941,7 +18461,7 @@ <h1 id="examples">Examples</h1>
 the result is a rational multiple of $\pi$ (see the eval class method).</p>
 
 <p><code>acos(zoo)</code> evaluates to <code>zoo</code>
-(see note in :class:<code>sympy.functions.elementary.trigonometric.asec</code>)</p>
+(see note in <code>sympy.functions.elementary.trigonometric.asec</code>)</p>
 
 <p>A purely imaginary argument will be rewritten to asinh.</p>
 
@@ -17963,29 +18483,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.23
-.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCos</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="acos.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#acos.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">acos</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">acos</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#acos.__init__"></a>
     
     
 
                             </div>
                             <div id="acos.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#acos.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#acos.default_assumptions"></a>
     
     
 
@@ -18141,37 +18665,37 @@ <h5>Inherited Members</h5>
                 <dd id="acos.simplify" class="function">simplify</dd>
                 <dd id="acos.refine" class="function">refine</dd>
                 <dd id="acos.rewrite" class="function">rewrite</dd>
-                <dd id="acos.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="acos.is_integer" class="variable">is_integer</dd>
-                <dd id="acos.is_complex" class="variable">is_complex</dd>
-                <dd id="acos.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="acos.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="acos.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="acos.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acos.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acos.is_positive" class="variable">is_positive</dd>
+                <dd id="acos.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="acos.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="acos.is_negative" class="variable">is_negative</dd>
+                <dd id="acos.is_composite" class="variable">is_composite</dd>
+                <dd id="acos.is_finite" class="variable">is_finite</dd>
                 <dd id="acos.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="acos.is_zero" class="variable">is_zero</dd>
                 <dd id="acos.is_even" class="variable">is_even</dd>
-                <dd id="acos.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="acos.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="acos.is_commutative" class="variable">is_commutative</dd>
-                <dd id="acos.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="acos.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="acos.is_real" class="variable">is_real</dd>
-                <dd id="acos.is_positive" class="variable">is_positive</dd>
-                <dd id="acos.is_finite" class="variable">is_finite</dd>
-                <dd id="acos.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="acos.is_infinite" class="variable">is_infinite</dd>
-                <dd id="acos.is_rational" class="variable">is_rational</dd>
-                <dd id="acos.is_composite" class="variable">is_composite</dd>
-                <dd id="acos.is_negative" class="variable">is_negative</dd>
-                <dd id="acos.is_polar" class="variable">is_polar</dd>
+                <dd id="acos.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="acos.is_odd" class="variable">is_odd</dd>
-                <dd id="acos.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="acos.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acos.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="acos.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="acos.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acos.is_integer" class="variable">is_integer</dd>
+                <dd id="acos.is_rational" class="variable">is_rational</dd>
+                <dd id="acos.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="acos.is_complex" class="variable">is_complex</dd>
                 <dd id="acos.is_prime" class="variable">is_prime</dd>
+                <dd id="acos.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="acos.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="acos.is_real" class="variable">is_real</dd>
+                <dd id="acos.is_commutative" class="variable">is_commutative</dd>
                 <dd id="acos.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="acos.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="acos.is_polar" class="variable">is_polar</dd>
+                <dd id="acos.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="acos.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="acos.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -18183,14 +18707,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="atan">
-                                <div class="attr class">
-        <a class="headerlink" href="#atan">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">atan</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.atan</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">atan</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.atan</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#atan"></a>
     
             <div class="docstring"><p>The inverse tangent function.</p>
 
@@ -18220,29 +18744,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.23
-.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="atan.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#atan.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">atan</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">atan</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#atan.__init__"></a>
     
     
 
                             </div>
                             <div id="atan.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#atan.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#atan.default_assumptions"></a>
     
     
 
@@ -18398,37 +18926,37 @@ <h5>Inherited Members</h5>
                 <dd id="atan.simplify" class="function">simplify</dd>
                 <dd id="atan.refine" class="function">refine</dd>
                 <dd id="atan.rewrite" class="function">rewrite</dd>
-                <dd id="atan.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="atan.is_integer" class="variable">is_integer</dd>
-                <dd id="atan.is_complex" class="variable">is_complex</dd>
-                <dd id="atan.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="atan.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="atan.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="atan.is_irrational" class="variable">is_irrational</dd>
+                <dd id="atan.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="atan.is_positive" class="variable">is_positive</dd>
+                <dd id="atan.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="atan.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="atan.is_negative" class="variable">is_negative</dd>
+                <dd id="atan.is_composite" class="variable">is_composite</dd>
+                <dd id="atan.is_finite" class="variable">is_finite</dd>
                 <dd id="atan.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="atan.is_zero" class="variable">is_zero</dd>
                 <dd id="atan.is_even" class="variable">is_even</dd>
-                <dd id="atan.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="atan.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="atan.is_commutative" class="variable">is_commutative</dd>
-                <dd id="atan.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="atan.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="atan.is_real" class="variable">is_real</dd>
-                <dd id="atan.is_positive" class="variable">is_positive</dd>
-                <dd id="atan.is_finite" class="variable">is_finite</dd>
-                <dd id="atan.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="atan.is_infinite" class="variable">is_infinite</dd>
-                <dd id="atan.is_rational" class="variable">is_rational</dd>
-                <dd id="atan.is_composite" class="variable">is_composite</dd>
-                <dd id="atan.is_negative" class="variable">is_negative</dd>
-                <dd id="atan.is_polar" class="variable">is_polar</dd>
+                <dd id="atan.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="atan.is_odd" class="variable">is_odd</dd>
-                <dd id="atan.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="atan.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="atan.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="atan.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="atan.is_irrational" class="variable">is_irrational</dd>
+                <dd id="atan.is_integer" class="variable">is_integer</dd>
+                <dd id="atan.is_rational" class="variable">is_rational</dd>
+                <dd id="atan.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="atan.is_complex" class="variable">is_complex</dd>
                 <dd id="atan.is_prime" class="variable">is_prime</dd>
+                <dd id="atan.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="atan.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="atan.is_real" class="variable">is_real</dd>
+                <dd id="atan.is_commutative" class="variable">is_commutative</dd>
                 <dd id="atan.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="atan.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="atan.is_polar" class="variable">is_polar</dd>
+                <dd id="atan.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="atan.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="atan.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -18440,14 +18968,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="asec">
-                                <div class="attr class">
-        <a class="headerlink" href="#asec">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">asec</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.asec</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">asec</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.asec</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#asec"></a>
     
             <div class="docstring"><p>The inverse secant function.</p>
 
@@ -18460,18 +18988,16 @@ <h1 id="explanation">Explanation</h1>
 result is a rational multiple of $\pi$ (see the eval class method).</p>
 
 <p><code>asec(x)</code> has branch cut in the interval $[-1, 1]$. For complex arguments,
-it can be defined [4]_ as</p>
+it can be defined <sup class="footnote-ref" id="fnref-4"><a href="#fn-4">1</a></sup> as</p>
 
-<p>.. math::
-    \operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z}</p>
+<p>$$\operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z}$$</p>
 
 <p>At <code>x = 0</code>, for positive branch cut, the limit evaluates to <code>zoo</code>. For
 negative branch cut, the limit</p>
 
-<p>.. math::
-    \lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z}</p>
+<p>$$\lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z}$$</p>
 
-<p>simplifies to :math:<code>-i\log\left(z/2 + O\left(z^3\right)\right)</code> which
+<p>simplifies to \( -i\log\left(z/2 + O\left(z^3\right)\right) \) which
 ultimately evaluates to <code>zoo</code>.</p>
 
 <p>As <code>acos(x) = asec(1/x)</code>, a similar argument can be given for
@@ -18497,30 +19023,36 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.23
-.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec
-.. [4] http://reference.wolfram.com/language/ref/ArcSec.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-4">
+<p>http://reference.wolfram.com/language/ref/ArcSec.html&#160;<a href="#fnref-4" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="asec.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#asec.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">asec</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">asec</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#asec.__init__"></a>
     
     
 
                             </div>
                             <div id="asec.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#asec.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#asec.default_assumptions"></a>
     
     
 
@@ -18676,37 +19208,37 @@ <h5>Inherited Members</h5>
                 <dd id="asec.simplify" class="function">simplify</dd>
                 <dd id="asec.refine" class="function">refine</dd>
                 <dd id="asec.rewrite" class="function">rewrite</dd>
-                <dd id="asec.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="asec.is_integer" class="variable">is_integer</dd>
-                <dd id="asec.is_complex" class="variable">is_complex</dd>
-                <dd id="asec.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="asec.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="asec.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="asec.is_irrational" class="variable">is_irrational</dd>
+                <dd id="asec.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="asec.is_positive" class="variable">is_positive</dd>
+                <dd id="asec.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="asec.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="asec.is_negative" class="variable">is_negative</dd>
+                <dd id="asec.is_composite" class="variable">is_composite</dd>
+                <dd id="asec.is_finite" class="variable">is_finite</dd>
                 <dd id="asec.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="asec.is_zero" class="variable">is_zero</dd>
                 <dd id="asec.is_even" class="variable">is_even</dd>
-                <dd id="asec.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="asec.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="asec.is_commutative" class="variable">is_commutative</dd>
-                <dd id="asec.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="asec.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="asec.is_real" class="variable">is_real</dd>
-                <dd id="asec.is_positive" class="variable">is_positive</dd>
-                <dd id="asec.is_finite" class="variable">is_finite</dd>
-                <dd id="asec.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="asec.is_infinite" class="variable">is_infinite</dd>
-                <dd id="asec.is_rational" class="variable">is_rational</dd>
-                <dd id="asec.is_composite" class="variable">is_composite</dd>
-                <dd id="asec.is_negative" class="variable">is_negative</dd>
-                <dd id="asec.is_polar" class="variable">is_polar</dd>
+                <dd id="asec.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="asec.is_odd" class="variable">is_odd</dd>
-                <dd id="asec.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="asec.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="asec.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="asec.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="asec.is_irrational" class="variable">is_irrational</dd>
+                <dd id="asec.is_integer" class="variable">is_integer</dd>
+                <dd id="asec.is_rational" class="variable">is_rational</dd>
+                <dd id="asec.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="asec.is_complex" class="variable">is_complex</dd>
                 <dd id="asec.is_prime" class="variable">is_prime</dd>
+                <dd id="asec.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="asec.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="asec.is_real" class="variable">is_real</dd>
+                <dd id="asec.is_commutative" class="variable">is_commutative</dd>
                 <dd id="asec.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="asec.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="asec.is_polar" class="variable">is_polar</dd>
+                <dd id="asec.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="asec.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="asec.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -18718,14 +19250,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="acsc">
-                                <div class="attr class">
-        <a class="headerlink" href="#acsc">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">acsc</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.acsc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">acsc</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.acsc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#acsc"></a>
     
             <div class="docstring"><p>The inverse cosecant function.</p>
 
@@ -18759,29 +19291,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
-.. [2] http://dlmf.nist.gov/4.23
-.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsc</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="acsc.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#acsc.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">acsc</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">acsc</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#acsc.__init__"></a>
     
     
 
                             </div>
                             <div id="acsc.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#acsc.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#acsc.default_assumptions"></a>
     
     
 
@@ -18937,37 +19473,37 @@ <h5>Inherited Members</h5>
                 <dd id="acsc.simplify" class="function">simplify</dd>
                 <dd id="acsc.refine" class="function">refine</dd>
                 <dd id="acsc.rewrite" class="function">rewrite</dd>
-                <dd id="acsc.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="acsc.is_integer" class="variable">is_integer</dd>
-                <dd id="acsc.is_complex" class="variable">is_complex</dd>
-                <dd id="acsc.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="acsc.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="acsc.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="acsc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acsc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acsc.is_positive" class="variable">is_positive</dd>
+                <dd id="acsc.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="acsc.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="acsc.is_negative" class="variable">is_negative</dd>
+                <dd id="acsc.is_composite" class="variable">is_composite</dd>
+                <dd id="acsc.is_finite" class="variable">is_finite</dd>
                 <dd id="acsc.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="acsc.is_zero" class="variable">is_zero</dd>
                 <dd id="acsc.is_even" class="variable">is_even</dd>
-                <dd id="acsc.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="acsc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="acsc.is_commutative" class="variable">is_commutative</dd>
-                <dd id="acsc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="acsc.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="acsc.is_real" class="variable">is_real</dd>
-                <dd id="acsc.is_positive" class="variable">is_positive</dd>
-                <dd id="acsc.is_finite" class="variable">is_finite</dd>
-                <dd id="acsc.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="acsc.is_infinite" class="variable">is_infinite</dd>
-                <dd id="acsc.is_rational" class="variable">is_rational</dd>
-                <dd id="acsc.is_composite" class="variable">is_composite</dd>
-                <dd id="acsc.is_negative" class="variable">is_negative</dd>
-                <dd id="acsc.is_polar" class="variable">is_polar</dd>
+                <dd id="acsc.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="acsc.is_odd" class="variable">is_odd</dd>
-                <dd id="acsc.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="acsc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acsc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="acsc.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="acsc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acsc.is_integer" class="variable">is_integer</dd>
+                <dd id="acsc.is_rational" class="variable">is_rational</dd>
+                <dd id="acsc.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="acsc.is_complex" class="variable">is_complex</dd>
                 <dd id="acsc.is_prime" class="variable">is_prime</dd>
+                <dd id="acsc.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="acsc.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="acsc.is_real" class="variable">is_real</dd>
+                <dd id="acsc.is_commutative" class="variable">is_commutative</dd>
                 <dd id="acsc.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="acsc.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="acsc.is_polar" class="variable">is_polar</dd>
+                <dd id="acsc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="acsc.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="acsc.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -18979,14 +19515,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="acot">
-                                <div class="attr class">
-        <a class="headerlink" href="#acot">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">acot</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.acot</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">acot</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.acot</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#acot"></a>
     
             <div class="docstring"><p>The inverse cotangent function.</p>
 
@@ -19022,28 +19558,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://dlmf.nist.gov/4.23
-.. [2] http://functions.wolfram.com/ElementaryFunctions/ArcCot</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="acot.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#acot.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">acot</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">acot</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#acot.__init__"></a>
     
     
 
                             </div>
                             <div id="acot.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#acot.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#acot.default_assumptions"></a>
     
     
 
@@ -19199,37 +19740,37 @@ <h5>Inherited Members</h5>
                 <dd id="acot.simplify" class="function">simplify</dd>
                 <dd id="acot.refine" class="function">refine</dd>
                 <dd id="acot.rewrite" class="function">rewrite</dd>
-                <dd id="acot.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="acot.is_integer" class="variable">is_integer</dd>
-                <dd id="acot.is_complex" class="variable">is_complex</dd>
-                <dd id="acot.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="acot.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="acot.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="acot.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acot.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acot.is_positive" class="variable">is_positive</dd>
+                <dd id="acot.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="acot.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="acot.is_negative" class="variable">is_negative</dd>
+                <dd id="acot.is_composite" class="variable">is_composite</dd>
+                <dd id="acot.is_finite" class="variable">is_finite</dd>
                 <dd id="acot.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="acot.is_zero" class="variable">is_zero</dd>
                 <dd id="acot.is_even" class="variable">is_even</dd>
-                <dd id="acot.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="acot.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="acot.is_commutative" class="variable">is_commutative</dd>
-                <dd id="acot.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="acot.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="acot.is_real" class="variable">is_real</dd>
-                <dd id="acot.is_positive" class="variable">is_positive</dd>
-                <dd id="acot.is_finite" class="variable">is_finite</dd>
-                <dd id="acot.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="acot.is_infinite" class="variable">is_infinite</dd>
-                <dd id="acot.is_rational" class="variable">is_rational</dd>
-                <dd id="acot.is_composite" class="variable">is_composite</dd>
-                <dd id="acot.is_negative" class="variable">is_negative</dd>
-                <dd id="acot.is_polar" class="variable">is_polar</dd>
+                <dd id="acot.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="acot.is_odd" class="variable">is_odd</dd>
-                <dd id="acot.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="acot.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acot.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="acot.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="acot.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acot.is_integer" class="variable">is_integer</dd>
+                <dd id="acot.is_rational" class="variable">is_rational</dd>
+                <dd id="acot.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="acot.is_complex" class="variable">is_complex</dd>
                 <dd id="acot.is_prime" class="variable">is_prime</dd>
+                <dd id="acot.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="acot.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="acot.is_real" class="variable">is_real</dd>
+                <dd id="acot.is_commutative" class="variable">is_commutative</dd>
                 <dd id="acot.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="acot.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="acot.is_polar" class="variable">is_polar</dd>
+                <dd id="acot.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="acot.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="acot.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -19241,43 +19782,37 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="atan2">
-                                <div class="attr class">
-        <a class="headerlink" href="#atan2">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">atan2</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.atan2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">atan2</span><wbr>(<span class="base">sympy.functions.elementary.trigonometric.atan2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#atan2"></a>
     
             <div class="docstring"><p>The function <code>atan2(y, x)</code> computes <code>\operatorname{atan}(y/x)</code> taking
 two arguments <code>y</code> and <code>x</code>.  Signs of both <code>y</code> and <code>x</code> are considered to
 determine the appropriate quadrant of <code>\operatorname{atan}(y/x)</code>.
 The range is <code>(-\pi, \pi]</code>. The complete definition reads as follows:</p>
 
-<p>.. math::</p>
-
-<pre><code>\operatorname{atan2}(y, x) =
+<p>$$\operatorname{atan2}(y, x) =
 \begin{cases}
-  \arctan\left(\frac y x\right) &amp; \qquad x &gt; 0 \\
-  \arctan\left(\frac y x\right) + \pi&amp; \qquad y \ge 0, x &lt; 0 \\
-  \arctan\left(\frac y x\right) - \pi&amp; \qquad y &lt; 0, x &lt; 0 \\
-  +\frac{\pi}{2} &amp; \qquad y &gt; 0, x = 0 \\
-  -\frac{\pi}{2} &amp; \qquad y &lt; 0, x = 0 \\
+  \arctan\left(\frac y x\right) &amp; \qquad x &gt; 0 \
+  \arctan\left(\frac y x\right) + \pi&amp; \qquad y \ge 0, x &lt; 0 \
+  \arctan\left(\frac y x\right) - \pi&amp; \qquad y &lt; 0, x &lt; 0 \
+  +\frac{\pi}{2} &amp; \qquad y &gt; 0, x = 0 \
+  -\frac{\pi}{2} &amp; \qquad y &lt; 0, x = 0 \
   \text{undefined} &amp; \qquad y = 0, x = 0
-\end{cases}
-</code></pre>
+\end{cases}$$</p>
 
 <p>Attention: Note the role reversal of both arguments. The <code>y</code>-coordinate
 is the first argument and the <code>x</code>-coordinate the second.</p>
 
 <p>If either <code>x</code> or <code>y</code> is complex:</p>
 
-<p>.. math::</p>
-
-<pre><code>\operatorname{atan2}(y, x) =
-    -i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right)
-</code></pre>
+<p>$$\operatorname{atan2}(y, x) =
+    -i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right)$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -19350,29 +19885,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
-.. [2] https://en.wikipedia.org/wiki/Atan2
-.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan2</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="atan2.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#atan2.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">atan2</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">atan2</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#atan2.__init__"></a>
     
     
 
                             </div>
                             <div id="atan2.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#atan2.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#atan2.default_assumptions"></a>
     
     
 
@@ -19527,37 +20066,37 @@ <h5>Inherited Members</h5>
                 <dd id="atan2.simplify" class="function">simplify</dd>
                 <dd id="atan2.refine" class="function">refine</dd>
                 <dd id="atan2.rewrite" class="function">rewrite</dd>
-                <dd id="atan2.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="atan2.is_integer" class="variable">is_integer</dd>
-                <dd id="atan2.is_complex" class="variable">is_complex</dd>
-                <dd id="atan2.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="atan2.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="atan2.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="atan2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="atan2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="atan2.is_positive" class="variable">is_positive</dd>
+                <dd id="atan2.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="atan2.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="atan2.is_negative" class="variable">is_negative</dd>
+                <dd id="atan2.is_composite" class="variable">is_composite</dd>
+                <dd id="atan2.is_finite" class="variable">is_finite</dd>
                 <dd id="atan2.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="atan2.is_zero" class="variable">is_zero</dd>
                 <dd id="atan2.is_even" class="variable">is_even</dd>
-                <dd id="atan2.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="atan2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="atan2.is_commutative" class="variable">is_commutative</dd>
-                <dd id="atan2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="atan2.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="atan2.is_real" class="variable">is_real</dd>
-                <dd id="atan2.is_positive" class="variable">is_positive</dd>
-                <dd id="atan2.is_finite" class="variable">is_finite</dd>
-                <dd id="atan2.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="atan2.is_infinite" class="variable">is_infinite</dd>
-                <dd id="atan2.is_rational" class="variable">is_rational</dd>
-                <dd id="atan2.is_composite" class="variable">is_composite</dd>
-                <dd id="atan2.is_negative" class="variable">is_negative</dd>
-                <dd id="atan2.is_polar" class="variable">is_polar</dd>
+                <dd id="atan2.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="atan2.is_odd" class="variable">is_odd</dd>
-                <dd id="atan2.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="atan2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="atan2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="atan2.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="atan2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="atan2.is_integer" class="variable">is_integer</dd>
+                <dd id="atan2.is_rational" class="variable">is_rational</dd>
+                <dd id="atan2.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="atan2.is_complex" class="variable">is_complex</dd>
                 <dd id="atan2.is_prime" class="variable">is_prime</dd>
+                <dd id="atan2.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="atan2.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="atan2.is_real" class="variable">is_real</dd>
+                <dd id="atan2.is_commutative" class="variable">is_commutative</dd>
                 <dd id="atan2.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="atan2.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="atan2.is_polar" class="variable">is_polar</dd>
+                <dd id="atan2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="atan2.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="atan2.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -19569,14 +20108,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="exp_polar">
-                                <div class="attr class">
-        <a class="headerlink" href="#exp_polar">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">exp_polar</span><wbr>(<span class="base">sympy.functions.elementary.exponential.exp_polar</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">exp_polar</span><wbr>(<span class="base">sympy.functions.elementary.exponential.exp_polar</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#exp_polar"></a>
     
             <div class="docstring"><p>Represent a <em>polar number</em> (see g-function Sphinx documentation).</p>
 
@@ -19616,32 +20155,35 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="exp_polar.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#exp_polar.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">exp_polar</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">exp_polar</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#exp_polar.__init__"></a>
     
     
 
                             </div>
                             <div id="exp_polar.is_polar" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#exp_polar.is_polar">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_polar</span><span class="default_value"> = True</span>
 
-        <span class="name">is_polar</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#exp_polar.is_polar"></a>
     
     
 
                             </div>
                             <div id="exp_polar.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#exp_polar.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;polar&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;polar&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#exp_polar.default_assumptions"></a>
     
     
 
@@ -19802,36 +20344,36 @@ <h5>Inherited Members</h5>
                 <dd id="exp_polar.simplify" class="function">simplify</dd>
                 <dd id="exp_polar.refine" class="function">refine</dd>
                 <dd id="exp_polar.rewrite" class="function">rewrite</dd>
-                <dd id="exp_polar.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="exp_polar.is_integer" class="variable">is_integer</dd>
-                <dd id="exp_polar.is_complex" class="variable">is_complex</dd>
-                <dd id="exp_polar.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="exp_polar.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="exp_polar.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="exp_polar.is_irrational" class="variable">is_irrational</dd>
+                <dd id="exp_polar.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="exp_polar.is_positive" class="variable">is_positive</dd>
+                <dd id="exp_polar.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="exp_polar.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="exp_polar.is_negative" class="variable">is_negative</dd>
+                <dd id="exp_polar.is_composite" class="variable">is_composite</dd>
+                <dd id="exp_polar.is_finite" class="variable">is_finite</dd>
                 <dd id="exp_polar.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="exp_polar.is_zero" class="variable">is_zero</dd>
                 <dd id="exp_polar.is_even" class="variable">is_even</dd>
-                <dd id="exp_polar.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="exp_polar.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="exp_polar.is_commutative" class="variable">is_commutative</dd>
-                <dd id="exp_polar.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="exp_polar.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="exp_polar.is_real" class="variable">is_real</dd>
-                <dd id="exp_polar.is_positive" class="variable">is_positive</dd>
-                <dd id="exp_polar.is_finite" class="variable">is_finite</dd>
-                <dd id="exp_polar.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="exp_polar.is_infinite" class="variable">is_infinite</dd>
-                <dd id="exp_polar.is_rational" class="variable">is_rational</dd>
-                <dd id="exp_polar.is_composite" class="variable">is_composite</dd>
-                <dd id="exp_polar.is_negative" class="variable">is_negative</dd>
+                <dd id="exp_polar.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="exp_polar.is_odd" class="variable">is_odd</dd>
-                <dd id="exp_polar.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="exp_polar.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="exp_polar.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="exp_polar.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="exp_polar.is_irrational" class="variable">is_irrational</dd>
+                <dd id="exp_polar.is_integer" class="variable">is_integer</dd>
+                <dd id="exp_polar.is_rational" class="variable">is_rational</dd>
+                <dd id="exp_polar.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="exp_polar.is_complex" class="variable">is_complex</dd>
                 <dd id="exp_polar.is_prime" class="variable">is_prime</dd>
+                <dd id="exp_polar.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="exp_polar.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="exp_polar.is_real" class="variable">is_real</dd>
+                <dd id="exp_polar.is_commutative" class="variable">is_commutative</dd>
                 <dd id="exp_polar.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="exp_polar.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="exp_polar.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="exp_polar.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="exp_polar.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -19843,16 +20385,16 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="exp">
-                                <div class="attr class">
-        <a class="headerlink" href="#exp">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">exp</span><wbr>(<span class="base">sympy.functions.elementary.exponential.exp</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">exp</span><wbr>(<span class="base">sympy.functions.elementary.exponential.exp</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#exp"></a>
     
-            <div class="docstring"><p>The exponential function, :math:<code>e^x</code>.</p>
+            <div class="docstring"><p>The exponential function, \( e^x \).</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -19877,22 +20419,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="exp.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#exp.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">exp</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">exp</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#exp.__init__"></a>
     
     
 
                             </div>
                             <div id="exp.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#exp.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#exp.default_assumptions"></a>
     
     
 
@@ -20054,37 +20598,37 @@ <h5>Inherited Members</h5>
                 <dd id="exp.simplify" class="function">simplify</dd>
                 <dd id="exp.refine" class="function">refine</dd>
                 <dd id="exp.rewrite" class="function">rewrite</dd>
-                <dd id="exp.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="exp.is_integer" class="variable">is_integer</dd>
-                <dd id="exp.is_complex" class="variable">is_complex</dd>
-                <dd id="exp.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="exp.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="exp.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="exp.is_irrational" class="variable">is_irrational</dd>
+                <dd id="exp.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="exp.is_positive" class="variable">is_positive</dd>
+                <dd id="exp.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="exp.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="exp.is_negative" class="variable">is_negative</dd>
+                <dd id="exp.is_composite" class="variable">is_composite</dd>
+                <dd id="exp.is_finite" class="variable">is_finite</dd>
                 <dd id="exp.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="exp.is_zero" class="variable">is_zero</dd>
                 <dd id="exp.is_even" class="variable">is_even</dd>
-                <dd id="exp.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="exp.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="exp.is_commutative" class="variable">is_commutative</dd>
-                <dd id="exp.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="exp.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="exp.is_real" class="variable">is_real</dd>
-                <dd id="exp.is_positive" class="variable">is_positive</dd>
-                <dd id="exp.is_finite" class="variable">is_finite</dd>
-                <dd id="exp.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="exp.is_infinite" class="variable">is_infinite</dd>
-                <dd id="exp.is_rational" class="variable">is_rational</dd>
-                <dd id="exp.is_composite" class="variable">is_composite</dd>
-                <dd id="exp.is_negative" class="variable">is_negative</dd>
-                <dd id="exp.is_polar" class="variable">is_polar</dd>
+                <dd id="exp.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="exp.is_odd" class="variable">is_odd</dd>
-                <dd id="exp.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="exp.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="exp.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="exp.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="exp.is_irrational" class="variable">is_irrational</dd>
+                <dd id="exp.is_integer" class="variable">is_integer</dd>
+                <dd id="exp.is_rational" class="variable">is_rational</dd>
+                <dd id="exp.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="exp.is_complex" class="variable">is_complex</dd>
                 <dd id="exp.is_prime" class="variable">is_prime</dd>
+                <dd id="exp.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="exp.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="exp.is_real" class="variable">is_real</dd>
+                <dd id="exp.is_commutative" class="variable">is_commutative</dd>
                 <dd id="exp.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="exp.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="exp.is_polar" class="variable">is_polar</dd>
+                <dd id="exp.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="exp.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="exp.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -20096,14 +20640,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="ln">
-                                <div class="attr class">
-        <a class="headerlink" href="#ln">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">ln</span><wbr>(<span class="base">sympy.functions.elementary.exponential.log</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">ln</span><wbr>(<span class="base">sympy.functions.elementary.exponential.log</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#ln"></a>
     
             <div class="docstring"><p>The natural logarithm function <code>\ln(x)</code> or <code>\log(x)</code>.</p>
 
@@ -20136,22 +20680,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="ln.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#ln.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">ln</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">ln</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#ln.__init__"></a>
     
     
 
                             </div>
                             <div id="ln.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#ln.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#ln.default_assumptions"></a>
     
     
 
@@ -20307,37 +20853,37 @@ <h5>Inherited Members</h5>
                 <dd id="ln.simplify" class="function">simplify</dd>
                 <dd id="ln.refine" class="function">refine</dd>
                 <dd id="ln.rewrite" class="function">rewrite</dd>
-                <dd id="ln.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="ln.is_integer" class="variable">is_integer</dd>
-                <dd id="ln.is_complex" class="variable">is_complex</dd>
-                <dd id="ln.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="ln.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="ln.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="ln.is_irrational" class="variable">is_irrational</dd>
+                <dd id="ln.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="ln.is_positive" class="variable">is_positive</dd>
+                <dd id="ln.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="ln.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="ln.is_negative" class="variable">is_negative</dd>
+                <dd id="ln.is_composite" class="variable">is_composite</dd>
+                <dd id="ln.is_finite" class="variable">is_finite</dd>
                 <dd id="ln.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="ln.is_zero" class="variable">is_zero</dd>
                 <dd id="ln.is_even" class="variable">is_even</dd>
-                <dd id="ln.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="ln.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="ln.is_commutative" class="variable">is_commutative</dd>
-                <dd id="ln.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="ln.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="ln.is_real" class="variable">is_real</dd>
-                <dd id="ln.is_positive" class="variable">is_positive</dd>
-                <dd id="ln.is_finite" class="variable">is_finite</dd>
-                <dd id="ln.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="ln.is_infinite" class="variable">is_infinite</dd>
-                <dd id="ln.is_rational" class="variable">is_rational</dd>
-                <dd id="ln.is_composite" class="variable">is_composite</dd>
-                <dd id="ln.is_negative" class="variable">is_negative</dd>
-                <dd id="ln.is_polar" class="variable">is_polar</dd>
+                <dd id="ln.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="ln.is_odd" class="variable">is_odd</dd>
-                <dd id="ln.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="ln.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="ln.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="ln.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="ln.is_irrational" class="variable">is_irrational</dd>
+                <dd id="ln.is_integer" class="variable">is_integer</dd>
+                <dd id="ln.is_rational" class="variable">is_rational</dd>
+                <dd id="ln.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="ln.is_complex" class="variable">is_complex</dd>
                 <dd id="ln.is_prime" class="variable">is_prime</dd>
+                <dd id="ln.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="ln.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="ln.is_real" class="variable">is_real</dd>
+                <dd id="ln.is_commutative" class="variable">is_commutative</dd>
                 <dd id="ln.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="ln.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="ln.is_polar" class="variable">is_polar</dd>
+                <dd id="ln.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="ln.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="ln.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -20349,14 +20895,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="log">
-                                <div class="attr class">
-        <a class="headerlink" href="#log">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">log</span><wbr>(<span class="base">sympy.functions.elementary.exponential.log</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">log</span><wbr>(<span class="base">sympy.functions.elementary.exponential.log</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#log"></a>
     
             <div class="docstring"><p>The natural logarithm function <code>\ln(x)</code> or <code>\log(x)</code>.</p>
 
@@ -20389,22 +20935,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="log.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#log.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">log</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">log</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#log.__init__"></a>
     
     
 
                             </div>
                             <div id="log.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#log.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#log.default_assumptions"></a>
     
     
 
@@ -20560,37 +21108,37 @@ <h5>Inherited Members</h5>
                 <dd id="log.simplify" class="function">simplify</dd>
                 <dd id="log.refine" class="function">refine</dd>
                 <dd id="log.rewrite" class="function">rewrite</dd>
-                <dd id="log.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="log.is_integer" class="variable">is_integer</dd>
-                <dd id="log.is_complex" class="variable">is_complex</dd>
-                <dd id="log.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="log.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="log.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="log.is_irrational" class="variable">is_irrational</dd>
+                <dd id="log.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="log.is_positive" class="variable">is_positive</dd>
+                <dd id="log.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="log.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="log.is_negative" class="variable">is_negative</dd>
+                <dd id="log.is_composite" class="variable">is_composite</dd>
+                <dd id="log.is_finite" class="variable">is_finite</dd>
                 <dd id="log.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="log.is_zero" class="variable">is_zero</dd>
                 <dd id="log.is_even" class="variable">is_even</dd>
-                <dd id="log.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="log.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="log.is_commutative" class="variable">is_commutative</dd>
-                <dd id="log.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="log.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="log.is_real" class="variable">is_real</dd>
-                <dd id="log.is_positive" class="variable">is_positive</dd>
-                <dd id="log.is_finite" class="variable">is_finite</dd>
-                <dd id="log.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="log.is_infinite" class="variable">is_infinite</dd>
-                <dd id="log.is_rational" class="variable">is_rational</dd>
-                <dd id="log.is_composite" class="variable">is_composite</dd>
-                <dd id="log.is_negative" class="variable">is_negative</dd>
-                <dd id="log.is_polar" class="variable">is_polar</dd>
+                <dd id="log.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="log.is_odd" class="variable">is_odd</dd>
-                <dd id="log.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="log.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="log.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="log.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="log.is_irrational" class="variable">is_irrational</dd>
+                <dd id="log.is_integer" class="variable">is_integer</dd>
+                <dd id="log.is_rational" class="variable">is_rational</dd>
+                <dd id="log.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="log.is_complex" class="variable">is_complex</dd>
                 <dd id="log.is_prime" class="variable">is_prime</dd>
+                <dd id="log.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="log.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="log.is_real" class="variable">is_real</dd>
+                <dd id="log.is_commutative" class="variable">is_commutative</dd>
                 <dd id="log.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="log.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="log.is_polar" class="variable">is_polar</dd>
+                <dd id="log.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="log.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="log.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -20602,17 +21150,17 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="LambertW">
-                                <div class="attr class">
-        <a class="headerlink" href="#LambertW">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">LambertW</span><wbr>(<span class="base">sympy.functions.elementary.exponential.LambertW</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">LambertW</span><wbr>(<span class="base">sympy.functions.elementary.exponential.LambertW</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#LambertW"></a>
     
             <div class="docstring"><p>The Lambert W function $W(z)$ is defined as the inverse
-function of $w \exp(w)$ [1]_.</p>
+function of $w \exp(w)$ <sup class="footnote-ref" id="fnref-1"><a href="#fn-1">1</a></sup>.</p>
 
 <h1 id="explanation">Explanation</h1>
 
@@ -20640,27 +21188,36 @@ <h1 id="examples">Examples</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Lambert_W_function</p>
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-1">
+<p>https://en.wikipedia.org/wiki/Lambert_W_function&#160;<a href="#fnref-1" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="LambertW.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#LambertW.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">LambertW</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">LambertW</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#LambertW.__init__"></a>
     
     
 
                             </div>
                             <div id="LambertW.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LambertW.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#LambertW.default_assumptions"></a>
     
     
 
@@ -20815,37 +21372,37 @@ <h5>Inherited Members</h5>
                 <dd id="LambertW.simplify" class="function">simplify</dd>
                 <dd id="LambertW.refine" class="function">refine</dd>
                 <dd id="LambertW.rewrite" class="function">rewrite</dd>
-                <dd id="LambertW.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="LambertW.is_integer" class="variable">is_integer</dd>
-                <dd id="LambertW.is_complex" class="variable">is_complex</dd>
-                <dd id="LambertW.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="LambertW.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="LambertW.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="LambertW.is_irrational" class="variable">is_irrational</dd>
+                <dd id="LambertW.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="LambertW.is_positive" class="variable">is_positive</dd>
+                <dd id="LambertW.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="LambertW.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="LambertW.is_negative" class="variable">is_negative</dd>
+                <dd id="LambertW.is_composite" class="variable">is_composite</dd>
+                <dd id="LambertW.is_finite" class="variable">is_finite</dd>
                 <dd id="LambertW.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="LambertW.is_zero" class="variable">is_zero</dd>
                 <dd id="LambertW.is_even" class="variable">is_even</dd>
-                <dd id="LambertW.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="LambertW.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="LambertW.is_commutative" class="variable">is_commutative</dd>
-                <dd id="LambertW.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="LambertW.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="LambertW.is_real" class="variable">is_real</dd>
-                <dd id="LambertW.is_positive" class="variable">is_positive</dd>
-                <dd id="LambertW.is_finite" class="variable">is_finite</dd>
-                <dd id="LambertW.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="LambertW.is_infinite" class="variable">is_infinite</dd>
-                <dd id="LambertW.is_rational" class="variable">is_rational</dd>
-                <dd id="LambertW.is_composite" class="variable">is_composite</dd>
-                <dd id="LambertW.is_negative" class="variable">is_negative</dd>
-                <dd id="LambertW.is_polar" class="variable">is_polar</dd>
+                <dd id="LambertW.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="LambertW.is_odd" class="variable">is_odd</dd>
-                <dd id="LambertW.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="LambertW.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="LambertW.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="LambertW.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="LambertW.is_irrational" class="variable">is_irrational</dd>
+                <dd id="LambertW.is_integer" class="variable">is_integer</dd>
+                <dd id="LambertW.is_rational" class="variable">is_rational</dd>
+                <dd id="LambertW.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="LambertW.is_complex" class="variable">is_complex</dd>
                 <dd id="LambertW.is_prime" class="variable">is_prime</dd>
+                <dd id="LambertW.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="LambertW.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="LambertW.is_real" class="variable">is_real</dd>
+                <dd id="LambertW.is_commutative" class="variable">is_commutative</dd>
                 <dd id="LambertW.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="LambertW.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="LambertW.is_polar" class="variable">is_polar</dd>
+                <dd id="LambertW.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="LambertW.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="LambertW.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -20857,14 +21414,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="sinh">
-                                <div class="attr class">
-        <a class="headerlink" href="#sinh">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">sinh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.sinh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">sinh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.sinh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#sinh"></a>
     
             <div class="docstring"><p><code>sinh(x)</code> is the hyperbolic sine of <code>x</code>.</p>
 
@@ -20885,22 +21442,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="sinh.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#sinh.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">sinh</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">sinh</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#sinh.__init__"></a>
     
     
 
                             </div>
                             <div id="sinh.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sinh.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#sinh.default_assumptions"></a>
     
     
 
@@ -21060,37 +21619,37 @@ <h5>Inherited Members</h5>
                 <dd id="sinh.simplify" class="function">simplify</dd>
                 <dd id="sinh.refine" class="function">refine</dd>
                 <dd id="sinh.rewrite" class="function">rewrite</dd>
-                <dd id="sinh.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="sinh.is_integer" class="variable">is_integer</dd>
-                <dd id="sinh.is_complex" class="variable">is_complex</dd>
-                <dd id="sinh.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="sinh.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="sinh.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="sinh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sinh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sinh.is_positive" class="variable">is_positive</dd>
+                <dd id="sinh.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="sinh.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="sinh.is_negative" class="variable">is_negative</dd>
+                <dd id="sinh.is_composite" class="variable">is_composite</dd>
+                <dd id="sinh.is_finite" class="variable">is_finite</dd>
                 <dd id="sinh.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="sinh.is_zero" class="variable">is_zero</dd>
                 <dd id="sinh.is_even" class="variable">is_even</dd>
-                <dd id="sinh.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="sinh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="sinh.is_commutative" class="variable">is_commutative</dd>
-                <dd id="sinh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="sinh.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="sinh.is_real" class="variable">is_real</dd>
-                <dd id="sinh.is_positive" class="variable">is_positive</dd>
-                <dd id="sinh.is_finite" class="variable">is_finite</dd>
-                <dd id="sinh.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="sinh.is_infinite" class="variable">is_infinite</dd>
-                <dd id="sinh.is_rational" class="variable">is_rational</dd>
-                <dd id="sinh.is_composite" class="variable">is_composite</dd>
-                <dd id="sinh.is_negative" class="variable">is_negative</dd>
-                <dd id="sinh.is_polar" class="variable">is_polar</dd>
+                <dd id="sinh.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="sinh.is_odd" class="variable">is_odd</dd>
-                <dd id="sinh.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="sinh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sinh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="sinh.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="sinh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sinh.is_integer" class="variable">is_integer</dd>
+                <dd id="sinh.is_rational" class="variable">is_rational</dd>
+                <dd id="sinh.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="sinh.is_complex" class="variable">is_complex</dd>
                 <dd id="sinh.is_prime" class="variable">is_prime</dd>
+                <dd id="sinh.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="sinh.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="sinh.is_real" class="variable">is_real</dd>
+                <dd id="sinh.is_commutative" class="variable">is_commutative</dd>
                 <dd id="sinh.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="sinh.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="sinh.is_polar" class="variable">is_polar</dd>
+                <dd id="sinh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="sinh.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="sinh.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -21102,14 +21661,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="cosh">
-                                <div class="attr class">
-        <a class="headerlink" href="#cosh">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">cosh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.cosh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">cosh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.cosh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#cosh"></a>
     
             <div class="docstring"><p><code>cosh(x)</code> is the hyperbolic cosine of <code>x</code>.</p>
 
@@ -21130,22 +21689,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="cosh.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#cosh.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">cosh</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">cosh</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#cosh.__init__"></a>
     
     
 
                             </div>
                             <div id="cosh.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#cosh.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#cosh.default_assumptions"></a>
     
     
 
@@ -21304,37 +21865,37 @@ <h5>Inherited Members</h5>
                 <dd id="cosh.simplify" class="function">simplify</dd>
                 <dd id="cosh.refine" class="function">refine</dd>
                 <dd id="cosh.rewrite" class="function">rewrite</dd>
-                <dd id="cosh.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="cosh.is_integer" class="variable">is_integer</dd>
-                <dd id="cosh.is_complex" class="variable">is_complex</dd>
-                <dd id="cosh.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="cosh.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="cosh.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="cosh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="cosh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="cosh.is_positive" class="variable">is_positive</dd>
+                <dd id="cosh.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="cosh.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="cosh.is_negative" class="variable">is_negative</dd>
+                <dd id="cosh.is_composite" class="variable">is_composite</dd>
+                <dd id="cosh.is_finite" class="variable">is_finite</dd>
                 <dd id="cosh.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="cosh.is_zero" class="variable">is_zero</dd>
                 <dd id="cosh.is_even" class="variable">is_even</dd>
-                <dd id="cosh.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="cosh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="cosh.is_commutative" class="variable">is_commutative</dd>
-                <dd id="cosh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="cosh.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="cosh.is_real" class="variable">is_real</dd>
-                <dd id="cosh.is_positive" class="variable">is_positive</dd>
-                <dd id="cosh.is_finite" class="variable">is_finite</dd>
-                <dd id="cosh.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="cosh.is_infinite" class="variable">is_infinite</dd>
-                <dd id="cosh.is_rational" class="variable">is_rational</dd>
-                <dd id="cosh.is_composite" class="variable">is_composite</dd>
-                <dd id="cosh.is_negative" class="variable">is_negative</dd>
-                <dd id="cosh.is_polar" class="variable">is_polar</dd>
+                <dd id="cosh.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="cosh.is_odd" class="variable">is_odd</dd>
-                <dd id="cosh.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="cosh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="cosh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="cosh.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="cosh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="cosh.is_integer" class="variable">is_integer</dd>
+                <dd id="cosh.is_rational" class="variable">is_rational</dd>
+                <dd id="cosh.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="cosh.is_complex" class="variable">is_complex</dd>
                 <dd id="cosh.is_prime" class="variable">is_prime</dd>
+                <dd id="cosh.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="cosh.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="cosh.is_real" class="variable">is_real</dd>
+                <dd id="cosh.is_commutative" class="variable">is_commutative</dd>
                 <dd id="cosh.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="cosh.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="cosh.is_polar" class="variable">is_polar</dd>
+                <dd id="cosh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="cosh.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="cosh.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -21346,14 +21907,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="tanh">
-                                <div class="attr class">
-        <a class="headerlink" href="#tanh">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">tanh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.tanh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">tanh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.tanh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#tanh"></a>
     
             <div class="docstring"><p><code>tanh(x)</code> is the hyperbolic tangent of <code>x</code>.</p>
 
@@ -21374,22 +21935,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="tanh.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#tanh.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">tanh</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">tanh</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#tanh.__init__"></a>
     
     
 
                             </div>
                             <div id="tanh.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#tanh.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#tanh.default_assumptions"></a>
     
     
 
@@ -21549,37 +22112,37 @@ <h5>Inherited Members</h5>
                 <dd id="tanh.simplify" class="function">simplify</dd>
                 <dd id="tanh.refine" class="function">refine</dd>
                 <dd id="tanh.rewrite" class="function">rewrite</dd>
-                <dd id="tanh.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="tanh.is_integer" class="variable">is_integer</dd>
-                <dd id="tanh.is_complex" class="variable">is_complex</dd>
-                <dd id="tanh.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="tanh.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="tanh.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="tanh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="tanh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="tanh.is_positive" class="variable">is_positive</dd>
+                <dd id="tanh.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="tanh.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="tanh.is_negative" class="variable">is_negative</dd>
+                <dd id="tanh.is_composite" class="variable">is_composite</dd>
+                <dd id="tanh.is_finite" class="variable">is_finite</dd>
                 <dd id="tanh.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="tanh.is_zero" class="variable">is_zero</dd>
                 <dd id="tanh.is_even" class="variable">is_even</dd>
-                <dd id="tanh.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="tanh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="tanh.is_commutative" class="variable">is_commutative</dd>
-                <dd id="tanh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="tanh.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="tanh.is_real" class="variable">is_real</dd>
-                <dd id="tanh.is_positive" class="variable">is_positive</dd>
-                <dd id="tanh.is_finite" class="variable">is_finite</dd>
-                <dd id="tanh.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="tanh.is_infinite" class="variable">is_infinite</dd>
-                <dd id="tanh.is_rational" class="variable">is_rational</dd>
-                <dd id="tanh.is_composite" class="variable">is_composite</dd>
-                <dd id="tanh.is_negative" class="variable">is_negative</dd>
-                <dd id="tanh.is_polar" class="variable">is_polar</dd>
+                <dd id="tanh.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="tanh.is_odd" class="variable">is_odd</dd>
-                <dd id="tanh.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="tanh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="tanh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="tanh.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="tanh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="tanh.is_integer" class="variable">is_integer</dd>
+                <dd id="tanh.is_rational" class="variable">is_rational</dd>
+                <dd id="tanh.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="tanh.is_complex" class="variable">is_complex</dd>
                 <dd id="tanh.is_prime" class="variable">is_prime</dd>
+                <dd id="tanh.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="tanh.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="tanh.is_real" class="variable">is_real</dd>
+                <dd id="tanh.is_commutative" class="variable">is_commutative</dd>
                 <dd id="tanh.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="tanh.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="tanh.is_polar" class="variable">is_polar</dd>
+                <dd id="tanh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="tanh.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="tanh.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -21591,14 +22154,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="coth">
-                                <div class="attr class">
-        <a class="headerlink" href="#coth">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">coth</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.coth</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">coth</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.coth</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#coth"></a>
     
             <div class="docstring"><p><code>coth(x)</code> is the hyperbolic cotangent of <code>x</code>.</p>
 
@@ -21619,22 +22182,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="coth.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#coth.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">coth</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">coth</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#coth.__init__"></a>
     
     
 
                             </div>
                             <div id="coth.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#coth.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#coth.default_assumptions"></a>
     
     
 
@@ -21794,37 +22359,37 @@ <h5>Inherited Members</h5>
                 <dd id="coth.simplify" class="function">simplify</dd>
                 <dd id="coth.refine" class="function">refine</dd>
                 <dd id="coth.rewrite" class="function">rewrite</dd>
-                <dd id="coth.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="coth.is_integer" class="variable">is_integer</dd>
-                <dd id="coth.is_complex" class="variable">is_complex</dd>
-                <dd id="coth.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="coth.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="coth.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="coth.is_irrational" class="variable">is_irrational</dd>
+                <dd id="coth.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="coth.is_positive" class="variable">is_positive</dd>
+                <dd id="coth.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="coth.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="coth.is_negative" class="variable">is_negative</dd>
+                <dd id="coth.is_composite" class="variable">is_composite</dd>
+                <dd id="coth.is_finite" class="variable">is_finite</dd>
                 <dd id="coth.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="coth.is_zero" class="variable">is_zero</dd>
                 <dd id="coth.is_even" class="variable">is_even</dd>
-                <dd id="coth.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="coth.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="coth.is_commutative" class="variable">is_commutative</dd>
-                <dd id="coth.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="coth.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="coth.is_real" class="variable">is_real</dd>
-                <dd id="coth.is_positive" class="variable">is_positive</dd>
-                <dd id="coth.is_finite" class="variable">is_finite</dd>
-                <dd id="coth.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="coth.is_infinite" class="variable">is_infinite</dd>
-                <dd id="coth.is_rational" class="variable">is_rational</dd>
-                <dd id="coth.is_composite" class="variable">is_composite</dd>
-                <dd id="coth.is_negative" class="variable">is_negative</dd>
-                <dd id="coth.is_polar" class="variable">is_polar</dd>
+                <dd id="coth.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="coth.is_odd" class="variable">is_odd</dd>
-                <dd id="coth.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="coth.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="coth.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="coth.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="coth.is_irrational" class="variable">is_irrational</dd>
+                <dd id="coth.is_integer" class="variable">is_integer</dd>
+                <dd id="coth.is_rational" class="variable">is_rational</dd>
+                <dd id="coth.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="coth.is_complex" class="variable">is_complex</dd>
                 <dd id="coth.is_prime" class="variable">is_prime</dd>
+                <dd id="coth.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="coth.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="coth.is_real" class="variable">is_real</dd>
+                <dd id="coth.is_commutative" class="variable">is_commutative</dd>
                 <dd id="coth.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="coth.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="coth.is_polar" class="variable">is_polar</dd>
+                <dd id="coth.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="coth.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="coth.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -21836,14 +22401,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="sech">
-                                <div class="attr class">
-        <a class="headerlink" href="#sech">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">sech</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.sech</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">sech</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.sech</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#sech"></a>
     
             <div class="docstring"><p><code>sech(x)</code> is the hyperbolic secant of <code>x</code>.</p>
 
@@ -21864,22 +22429,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="sech.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#sech.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">sech</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">sech</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#sech.__init__"></a>
     
     
 
                             </div>
                             <div id="sech.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#sech.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#sech.default_assumptions"></a>
     
     
 
@@ -22041,37 +22608,37 @@ <h5>Inherited Members</h5>
                 <dd id="sech.simplify" class="function">simplify</dd>
                 <dd id="sech.refine" class="function">refine</dd>
                 <dd id="sech.rewrite" class="function">rewrite</dd>
-                <dd id="sech.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="sech.is_integer" class="variable">is_integer</dd>
-                <dd id="sech.is_complex" class="variable">is_complex</dd>
-                <dd id="sech.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="sech.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="sech.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="sech.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sech.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sech.is_positive" class="variable">is_positive</dd>
+                <dd id="sech.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="sech.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="sech.is_negative" class="variable">is_negative</dd>
+                <dd id="sech.is_composite" class="variable">is_composite</dd>
+                <dd id="sech.is_finite" class="variable">is_finite</dd>
                 <dd id="sech.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="sech.is_zero" class="variable">is_zero</dd>
                 <dd id="sech.is_even" class="variable">is_even</dd>
-                <dd id="sech.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="sech.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="sech.is_commutative" class="variable">is_commutative</dd>
-                <dd id="sech.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="sech.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="sech.is_real" class="variable">is_real</dd>
-                <dd id="sech.is_positive" class="variable">is_positive</dd>
-                <dd id="sech.is_finite" class="variable">is_finite</dd>
-                <dd id="sech.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="sech.is_infinite" class="variable">is_infinite</dd>
-                <dd id="sech.is_rational" class="variable">is_rational</dd>
-                <dd id="sech.is_composite" class="variable">is_composite</dd>
-                <dd id="sech.is_negative" class="variable">is_negative</dd>
-                <dd id="sech.is_polar" class="variable">is_polar</dd>
+                <dd id="sech.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="sech.is_odd" class="variable">is_odd</dd>
-                <dd id="sech.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="sech.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="sech.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="sech.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="sech.is_irrational" class="variable">is_irrational</dd>
+                <dd id="sech.is_integer" class="variable">is_integer</dd>
+                <dd id="sech.is_rational" class="variable">is_rational</dd>
+                <dd id="sech.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="sech.is_complex" class="variable">is_complex</dd>
                 <dd id="sech.is_prime" class="variable">is_prime</dd>
+                <dd id="sech.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="sech.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="sech.is_real" class="variable">is_real</dd>
+                <dd id="sech.is_commutative" class="variable">is_commutative</dd>
                 <dd id="sech.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="sech.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="sech.is_polar" class="variable">is_polar</dd>
+                <dd id="sech.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="sech.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="sech.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -22083,14 +22650,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="csch">
-                                <div class="attr class">
-        <a class="headerlink" href="#csch">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">csch</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.csch</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">csch</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.csch</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#csch"></a>
     
             <div class="docstring"><p><code>csch(x)</code> is the hyperbolic cosecant of <code>x</code>.</p>
 
@@ -22111,22 +22678,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="csch.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#csch.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">csch</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">csch</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#csch.__init__"></a>
     
     
 
                             </div>
                             <div id="csch.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#csch.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#csch.default_assumptions"></a>
     
     
 
@@ -22288,37 +22857,37 @@ <h5>Inherited Members</h5>
                 <dd id="csch.simplify" class="function">simplify</dd>
                 <dd id="csch.refine" class="function">refine</dd>
                 <dd id="csch.rewrite" class="function">rewrite</dd>
-                <dd id="csch.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="csch.is_integer" class="variable">is_integer</dd>
-                <dd id="csch.is_complex" class="variable">is_complex</dd>
-                <dd id="csch.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="csch.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="csch.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="csch.is_irrational" class="variable">is_irrational</dd>
+                <dd id="csch.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="csch.is_positive" class="variable">is_positive</dd>
+                <dd id="csch.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="csch.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="csch.is_negative" class="variable">is_negative</dd>
+                <dd id="csch.is_composite" class="variable">is_composite</dd>
+                <dd id="csch.is_finite" class="variable">is_finite</dd>
                 <dd id="csch.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="csch.is_zero" class="variable">is_zero</dd>
                 <dd id="csch.is_even" class="variable">is_even</dd>
-                <dd id="csch.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="csch.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="csch.is_commutative" class="variable">is_commutative</dd>
-                <dd id="csch.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="csch.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="csch.is_real" class="variable">is_real</dd>
-                <dd id="csch.is_positive" class="variable">is_positive</dd>
-                <dd id="csch.is_finite" class="variable">is_finite</dd>
-                <dd id="csch.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="csch.is_infinite" class="variable">is_infinite</dd>
-                <dd id="csch.is_rational" class="variable">is_rational</dd>
-                <dd id="csch.is_composite" class="variable">is_composite</dd>
-                <dd id="csch.is_negative" class="variable">is_negative</dd>
-                <dd id="csch.is_polar" class="variable">is_polar</dd>
+                <dd id="csch.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="csch.is_odd" class="variable">is_odd</dd>
-                <dd id="csch.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="csch.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="csch.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="csch.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="csch.is_irrational" class="variable">is_irrational</dd>
+                <dd id="csch.is_integer" class="variable">is_integer</dd>
+                <dd id="csch.is_rational" class="variable">is_rational</dd>
+                <dd id="csch.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="csch.is_complex" class="variable">is_complex</dd>
                 <dd id="csch.is_prime" class="variable">is_prime</dd>
+                <dd id="csch.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="csch.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="csch.is_real" class="variable">is_real</dd>
+                <dd id="csch.is_commutative" class="variable">is_commutative</dd>
                 <dd id="csch.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="csch.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="csch.is_polar" class="variable">is_polar</dd>
+                <dd id="csch.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="csch.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="csch.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -22330,14 +22899,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="asinh">
-                                <div class="attr class">
-        <a class="headerlink" href="#asinh">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">asinh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.asinh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">asinh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.asinh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#asinh"></a>
     
             <div class="docstring"><p><code>asinh(x)</code> is the inverse hyperbolic sine of <code>x</code>.</p>
 
@@ -22360,22 +22929,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="asinh.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#asinh.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">asinh</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">asinh</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#asinh.__init__"></a>
     
     
 
                             </div>
                             <div id="asinh.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#asinh.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#asinh.default_assumptions"></a>
     
     
 
@@ -22531,37 +23102,37 @@ <h5>Inherited Members</h5>
                 <dd id="asinh.simplify" class="function">simplify</dd>
                 <dd id="asinh.refine" class="function">refine</dd>
                 <dd id="asinh.rewrite" class="function">rewrite</dd>
-                <dd id="asinh.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="asinh.is_integer" class="variable">is_integer</dd>
-                <dd id="asinh.is_complex" class="variable">is_complex</dd>
-                <dd id="asinh.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="asinh.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="asinh.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="asinh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="asinh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="asinh.is_positive" class="variable">is_positive</dd>
+                <dd id="asinh.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="asinh.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="asinh.is_negative" class="variable">is_negative</dd>
+                <dd id="asinh.is_composite" class="variable">is_composite</dd>
+                <dd id="asinh.is_finite" class="variable">is_finite</dd>
                 <dd id="asinh.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="asinh.is_zero" class="variable">is_zero</dd>
                 <dd id="asinh.is_even" class="variable">is_even</dd>
-                <dd id="asinh.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="asinh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="asinh.is_commutative" class="variable">is_commutative</dd>
-                <dd id="asinh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="asinh.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="asinh.is_real" class="variable">is_real</dd>
-                <dd id="asinh.is_positive" class="variable">is_positive</dd>
-                <dd id="asinh.is_finite" class="variable">is_finite</dd>
-                <dd id="asinh.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="asinh.is_infinite" class="variable">is_infinite</dd>
-                <dd id="asinh.is_rational" class="variable">is_rational</dd>
-                <dd id="asinh.is_composite" class="variable">is_composite</dd>
-                <dd id="asinh.is_negative" class="variable">is_negative</dd>
-                <dd id="asinh.is_polar" class="variable">is_polar</dd>
+                <dd id="asinh.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="asinh.is_odd" class="variable">is_odd</dd>
-                <dd id="asinh.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="asinh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="asinh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="asinh.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="asinh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="asinh.is_integer" class="variable">is_integer</dd>
+                <dd id="asinh.is_rational" class="variable">is_rational</dd>
+                <dd id="asinh.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="asinh.is_complex" class="variable">is_complex</dd>
                 <dd id="asinh.is_prime" class="variable">is_prime</dd>
+                <dd id="asinh.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="asinh.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="asinh.is_real" class="variable">is_real</dd>
+                <dd id="asinh.is_commutative" class="variable">is_commutative</dd>
                 <dd id="asinh.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="asinh.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="asinh.is_polar" class="variable">is_polar</dd>
+                <dd id="asinh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="asinh.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="asinh.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -22573,14 +23144,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="acosh">
-                                <div class="attr class">
-        <a class="headerlink" href="#acosh">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">acosh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.acosh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">acosh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.acosh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#acosh"></a>
     
             <div class="docstring"><p><code>acosh(x)</code> is the inverse hyperbolic cosine of <code>x</code>.</p>
 
@@ -22603,22 +23174,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="acosh.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#acosh.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">acosh</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">acosh</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#acosh.__init__"></a>
     
     
 
                             </div>
                             <div id="acosh.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#acosh.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#acosh.default_assumptions"></a>
     
     
 
@@ -22774,37 +23347,37 @@ <h5>Inherited Members</h5>
                 <dd id="acosh.simplify" class="function">simplify</dd>
                 <dd id="acosh.refine" class="function">refine</dd>
                 <dd id="acosh.rewrite" class="function">rewrite</dd>
-                <dd id="acosh.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="acosh.is_integer" class="variable">is_integer</dd>
-                <dd id="acosh.is_complex" class="variable">is_complex</dd>
-                <dd id="acosh.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="acosh.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="acosh.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="acosh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acosh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acosh.is_positive" class="variable">is_positive</dd>
+                <dd id="acosh.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="acosh.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="acosh.is_negative" class="variable">is_negative</dd>
+                <dd id="acosh.is_composite" class="variable">is_composite</dd>
+                <dd id="acosh.is_finite" class="variable">is_finite</dd>
                 <dd id="acosh.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="acosh.is_zero" class="variable">is_zero</dd>
                 <dd id="acosh.is_even" class="variable">is_even</dd>
-                <dd id="acosh.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="acosh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="acosh.is_commutative" class="variable">is_commutative</dd>
-                <dd id="acosh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="acosh.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="acosh.is_real" class="variable">is_real</dd>
-                <dd id="acosh.is_positive" class="variable">is_positive</dd>
-                <dd id="acosh.is_finite" class="variable">is_finite</dd>
-                <dd id="acosh.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="acosh.is_infinite" class="variable">is_infinite</dd>
-                <dd id="acosh.is_rational" class="variable">is_rational</dd>
-                <dd id="acosh.is_composite" class="variable">is_composite</dd>
-                <dd id="acosh.is_negative" class="variable">is_negative</dd>
-                <dd id="acosh.is_polar" class="variable">is_polar</dd>
+                <dd id="acosh.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="acosh.is_odd" class="variable">is_odd</dd>
-                <dd id="acosh.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="acosh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acosh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="acosh.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="acosh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acosh.is_integer" class="variable">is_integer</dd>
+                <dd id="acosh.is_rational" class="variable">is_rational</dd>
+                <dd id="acosh.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="acosh.is_complex" class="variable">is_complex</dd>
                 <dd id="acosh.is_prime" class="variable">is_prime</dd>
+                <dd id="acosh.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="acosh.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="acosh.is_real" class="variable">is_real</dd>
+                <dd id="acosh.is_commutative" class="variable">is_commutative</dd>
                 <dd id="acosh.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="acosh.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="acosh.is_polar" class="variable">is_polar</dd>
+                <dd id="acosh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="acosh.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="acosh.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -22816,14 +23389,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="atanh">
-                                <div class="attr class">
-        <a class="headerlink" href="#atanh">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">atanh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.atanh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">atanh</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.atanh</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#atanh"></a>
     
             <div class="docstring"><p><code>atanh(x)</code> is the inverse hyperbolic tangent of <code>x</code>.</p>
 
@@ -22844,22 +23417,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="atanh.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#atanh.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">atanh</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">atanh</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#atanh.__init__"></a>
     
     
 
                             </div>
                             <div id="atanh.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#atanh.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#atanh.default_assumptions"></a>
     
     
 
@@ -23015,37 +23590,37 @@ <h5>Inherited Members</h5>
                 <dd id="atanh.simplify" class="function">simplify</dd>
                 <dd id="atanh.refine" class="function">refine</dd>
                 <dd id="atanh.rewrite" class="function">rewrite</dd>
-                <dd id="atanh.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="atanh.is_integer" class="variable">is_integer</dd>
-                <dd id="atanh.is_complex" class="variable">is_complex</dd>
-                <dd id="atanh.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="atanh.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="atanh.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="atanh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="atanh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="atanh.is_positive" class="variable">is_positive</dd>
+                <dd id="atanh.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="atanh.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="atanh.is_negative" class="variable">is_negative</dd>
+                <dd id="atanh.is_composite" class="variable">is_composite</dd>
+                <dd id="atanh.is_finite" class="variable">is_finite</dd>
                 <dd id="atanh.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="atanh.is_zero" class="variable">is_zero</dd>
                 <dd id="atanh.is_even" class="variable">is_even</dd>
-                <dd id="atanh.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="atanh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="atanh.is_commutative" class="variable">is_commutative</dd>
-                <dd id="atanh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="atanh.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="atanh.is_real" class="variable">is_real</dd>
-                <dd id="atanh.is_positive" class="variable">is_positive</dd>
-                <dd id="atanh.is_finite" class="variable">is_finite</dd>
-                <dd id="atanh.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="atanh.is_infinite" class="variable">is_infinite</dd>
-                <dd id="atanh.is_rational" class="variable">is_rational</dd>
-                <dd id="atanh.is_composite" class="variable">is_composite</dd>
-                <dd id="atanh.is_negative" class="variable">is_negative</dd>
-                <dd id="atanh.is_polar" class="variable">is_polar</dd>
+                <dd id="atanh.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="atanh.is_odd" class="variable">is_odd</dd>
-                <dd id="atanh.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="atanh.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="atanh.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="atanh.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="atanh.is_irrational" class="variable">is_irrational</dd>
+                <dd id="atanh.is_integer" class="variable">is_integer</dd>
+                <dd id="atanh.is_rational" class="variable">is_rational</dd>
+                <dd id="atanh.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="atanh.is_complex" class="variable">is_complex</dd>
                 <dd id="atanh.is_prime" class="variable">is_prime</dd>
+                <dd id="atanh.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="atanh.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="atanh.is_real" class="variable">is_real</dd>
+                <dd id="atanh.is_commutative" class="variable">is_commutative</dd>
                 <dd id="atanh.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="atanh.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="atanh.is_polar" class="variable">is_polar</dd>
+                <dd id="atanh.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="atanh.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="atanh.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -23057,14 +23632,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="acoth">
-                                <div class="attr class">
-        <a class="headerlink" href="#acoth">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">acoth</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.acoth</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">acoth</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.acoth</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#acoth"></a>
     
             <div class="docstring"><p><code>acoth(x)</code> is the inverse hyperbolic cotangent of <code>x</code>.</p>
 
@@ -23085,22 +23660,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="acoth.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#acoth.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">acoth</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">acoth</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#acoth.__init__"></a>
     
     
 
                             </div>
                             <div id="acoth.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#acoth.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#acoth.default_assumptions"></a>
     
     
 
@@ -23256,37 +23833,37 @@ <h5>Inherited Members</h5>
                 <dd id="acoth.simplify" class="function">simplify</dd>
                 <dd id="acoth.refine" class="function">refine</dd>
                 <dd id="acoth.rewrite" class="function">rewrite</dd>
-                <dd id="acoth.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="acoth.is_integer" class="variable">is_integer</dd>
-                <dd id="acoth.is_complex" class="variable">is_complex</dd>
-                <dd id="acoth.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="acoth.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="acoth.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="acoth.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acoth.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acoth.is_positive" class="variable">is_positive</dd>
+                <dd id="acoth.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="acoth.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="acoth.is_negative" class="variable">is_negative</dd>
+                <dd id="acoth.is_composite" class="variable">is_composite</dd>
+                <dd id="acoth.is_finite" class="variable">is_finite</dd>
                 <dd id="acoth.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="acoth.is_zero" class="variable">is_zero</dd>
                 <dd id="acoth.is_even" class="variable">is_even</dd>
-                <dd id="acoth.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="acoth.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="acoth.is_commutative" class="variable">is_commutative</dd>
-                <dd id="acoth.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="acoth.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="acoth.is_real" class="variable">is_real</dd>
-                <dd id="acoth.is_positive" class="variable">is_positive</dd>
-                <dd id="acoth.is_finite" class="variable">is_finite</dd>
-                <dd id="acoth.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="acoth.is_infinite" class="variable">is_infinite</dd>
-                <dd id="acoth.is_rational" class="variable">is_rational</dd>
-                <dd id="acoth.is_composite" class="variable">is_composite</dd>
-                <dd id="acoth.is_negative" class="variable">is_negative</dd>
-                <dd id="acoth.is_polar" class="variable">is_polar</dd>
+                <dd id="acoth.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="acoth.is_odd" class="variable">is_odd</dd>
-                <dd id="acoth.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="acoth.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acoth.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="acoth.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="acoth.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acoth.is_integer" class="variable">is_integer</dd>
+                <dd id="acoth.is_rational" class="variable">is_rational</dd>
+                <dd id="acoth.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="acoth.is_complex" class="variable">is_complex</dd>
                 <dd id="acoth.is_prime" class="variable">is_prime</dd>
+                <dd id="acoth.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="acoth.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="acoth.is_real" class="variable">is_real</dd>
+                <dd id="acoth.is_commutative" class="variable">is_commutative</dd>
                 <dd id="acoth.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="acoth.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="acoth.is_polar" class="variable">is_polar</dd>
+                <dd id="acoth.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="acoth.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="acoth.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -23298,14 +23875,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="asech">
-                                <div class="attr class">
-        <a class="headerlink" href="#asech">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">asech</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.asech</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">asech</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.asech</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#asech"></a>
     
             <div class="docstring"><p><code>asech(x)</code> is the inverse hyperbolic secant of <code>x</code>.</p>
 
@@ -23335,29 +23912,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
-.. [2] http://dlmf.nist.gov/4.37
-.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSech/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="asech.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#asech.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">asech</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">asech</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#asech.__init__"></a>
     
     
 
                             </div>
                             <div id="asech.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#asech.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#asech.default_assumptions"></a>
     
     
 
@@ -23514,37 +24095,37 @@ <h5>Inherited Members</h5>
                 <dd id="asech.simplify" class="function">simplify</dd>
                 <dd id="asech.refine" class="function">refine</dd>
                 <dd id="asech.rewrite" class="function">rewrite</dd>
-                <dd id="asech.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="asech.is_integer" class="variable">is_integer</dd>
-                <dd id="asech.is_complex" class="variable">is_complex</dd>
-                <dd id="asech.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="asech.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="asech.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="asech.is_irrational" class="variable">is_irrational</dd>
+                <dd id="asech.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="asech.is_positive" class="variable">is_positive</dd>
+                <dd id="asech.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="asech.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="asech.is_negative" class="variable">is_negative</dd>
+                <dd id="asech.is_composite" class="variable">is_composite</dd>
+                <dd id="asech.is_finite" class="variable">is_finite</dd>
                 <dd id="asech.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="asech.is_zero" class="variable">is_zero</dd>
                 <dd id="asech.is_even" class="variable">is_even</dd>
-                <dd id="asech.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="asech.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="asech.is_commutative" class="variable">is_commutative</dd>
-                <dd id="asech.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="asech.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="asech.is_real" class="variable">is_real</dd>
-                <dd id="asech.is_positive" class="variable">is_positive</dd>
-                <dd id="asech.is_finite" class="variable">is_finite</dd>
-                <dd id="asech.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="asech.is_infinite" class="variable">is_infinite</dd>
-                <dd id="asech.is_rational" class="variable">is_rational</dd>
-                <dd id="asech.is_composite" class="variable">is_composite</dd>
-                <dd id="asech.is_negative" class="variable">is_negative</dd>
-                <dd id="asech.is_polar" class="variable">is_polar</dd>
+                <dd id="asech.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="asech.is_odd" class="variable">is_odd</dd>
-                <dd id="asech.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="asech.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="asech.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="asech.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="asech.is_irrational" class="variable">is_irrational</dd>
+                <dd id="asech.is_integer" class="variable">is_integer</dd>
+                <dd id="asech.is_rational" class="variable">is_rational</dd>
+                <dd id="asech.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="asech.is_complex" class="variable">is_complex</dd>
                 <dd id="asech.is_prime" class="variable">is_prime</dd>
+                <dd id="asech.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="asech.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="asech.is_real" class="variable">is_real</dd>
+                <dd id="asech.is_commutative" class="variable">is_commutative</dd>
                 <dd id="asech.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="asech.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="asech.is_polar" class="variable">is_polar</dd>
+                <dd id="asech.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="asech.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="asech.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -23556,14 +24137,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="acsch">
-                                <div class="attr class">
-        <a class="headerlink" href="#acsch">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">acsch</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.acsch</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">acsch</span><wbr>(<span class="base">sympy.functions.elementary.hyperbolic.acsch</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#acsch"></a>
     
             <div class="docstring"><p><code>acsch(x)</code> is the inverse hyperbolic cosecant of <code>x</code>.</p>
 
@@ -23593,29 +24174,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
-.. [2] http://dlmf.nist.gov/4.37
-.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsch/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="acsch.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#acsch.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">acsch</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">acsch</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#acsch.__init__"></a>
     
     
 
                             </div>
                             <div id="acsch.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#acsch.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#acsch.default_assumptions"></a>
     
     
 
@@ -23771,37 +24356,37 @@ <h5>Inherited Members</h5>
                 <dd id="acsch.simplify" class="function">simplify</dd>
                 <dd id="acsch.refine" class="function">refine</dd>
                 <dd id="acsch.rewrite" class="function">rewrite</dd>
-                <dd id="acsch.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="acsch.is_integer" class="variable">is_integer</dd>
-                <dd id="acsch.is_complex" class="variable">is_complex</dd>
-                <dd id="acsch.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="acsch.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="acsch.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="acsch.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acsch.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acsch.is_positive" class="variable">is_positive</dd>
+                <dd id="acsch.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="acsch.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="acsch.is_negative" class="variable">is_negative</dd>
+                <dd id="acsch.is_composite" class="variable">is_composite</dd>
+                <dd id="acsch.is_finite" class="variable">is_finite</dd>
                 <dd id="acsch.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="acsch.is_zero" class="variable">is_zero</dd>
                 <dd id="acsch.is_even" class="variable">is_even</dd>
-                <dd id="acsch.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="acsch.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="acsch.is_commutative" class="variable">is_commutative</dd>
-                <dd id="acsch.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="acsch.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="acsch.is_real" class="variable">is_real</dd>
-                <dd id="acsch.is_positive" class="variable">is_positive</dd>
-                <dd id="acsch.is_finite" class="variable">is_finite</dd>
-                <dd id="acsch.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="acsch.is_infinite" class="variable">is_infinite</dd>
-                <dd id="acsch.is_rational" class="variable">is_rational</dd>
-                <dd id="acsch.is_composite" class="variable">is_composite</dd>
-                <dd id="acsch.is_negative" class="variable">is_negative</dd>
-                <dd id="acsch.is_polar" class="variable">is_polar</dd>
+                <dd id="acsch.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="acsch.is_odd" class="variable">is_odd</dd>
-                <dd id="acsch.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="acsch.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="acsch.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="acsch.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="acsch.is_irrational" class="variable">is_irrational</dd>
+                <dd id="acsch.is_integer" class="variable">is_integer</dd>
+                <dd id="acsch.is_rational" class="variable">is_rational</dd>
+                <dd id="acsch.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="acsch.is_complex" class="variable">is_complex</dd>
                 <dd id="acsch.is_prime" class="variable">is_prime</dd>
+                <dd id="acsch.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="acsch.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="acsch.is_real" class="variable">is_real</dd>
+                <dd id="acsch.is_commutative" class="variable">is_commutative</dd>
                 <dd id="acsch.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="acsch.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="acsch.is_polar" class="variable">is_polar</dd>
+                <dd id="acsch.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="acsch.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="acsch.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -23813,14 +24398,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="floor">
-                                <div class="attr class">
-        <a class="headerlink" href="#floor">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">floor</span><wbr>(<span class="base">sympy.functions.elementary.integers.floor</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">floor</span><wbr>(<span class="base">sympy.functions.elementary.integers.floor</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#floor"></a>
     
             <div class="docstring"><p>Floor is a univariate function which returns the largest integer
 value not greater than its argument. This implementation
@@ -23850,28 +24435,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] "Concrete mathematics" by Graham, pp. 87
-.. [2] http://mathworld.wolfram.com/FloorFunction.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="floor.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#floor.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">floor</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">floor</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#floor.__init__"></a>
     
     
 
                             </div>
                             <div id="floor.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#floor.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#floor.default_assumptions"></a>
     
     
 
@@ -24026,37 +24616,37 @@ <h5>Inherited Members</h5>
                 <dd id="floor.simplify" class="function">simplify</dd>
                 <dd id="floor.refine" class="function">refine</dd>
                 <dd id="floor.rewrite" class="function">rewrite</dd>
-                <dd id="floor.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="floor.is_integer" class="variable">is_integer</dd>
-                <dd id="floor.is_complex" class="variable">is_complex</dd>
-                <dd id="floor.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="floor.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="floor.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="floor.is_irrational" class="variable">is_irrational</dd>
+                <dd id="floor.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="floor.is_positive" class="variable">is_positive</dd>
+                <dd id="floor.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="floor.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="floor.is_negative" class="variable">is_negative</dd>
+                <dd id="floor.is_composite" class="variable">is_composite</dd>
+                <dd id="floor.is_finite" class="variable">is_finite</dd>
                 <dd id="floor.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="floor.is_zero" class="variable">is_zero</dd>
                 <dd id="floor.is_even" class="variable">is_even</dd>
-                <dd id="floor.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="floor.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="floor.is_commutative" class="variable">is_commutative</dd>
-                <dd id="floor.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="floor.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="floor.is_real" class="variable">is_real</dd>
-                <dd id="floor.is_positive" class="variable">is_positive</dd>
-                <dd id="floor.is_finite" class="variable">is_finite</dd>
-                <dd id="floor.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="floor.is_infinite" class="variable">is_infinite</dd>
-                <dd id="floor.is_rational" class="variable">is_rational</dd>
-                <dd id="floor.is_composite" class="variable">is_composite</dd>
-                <dd id="floor.is_negative" class="variable">is_negative</dd>
-                <dd id="floor.is_polar" class="variable">is_polar</dd>
+                <dd id="floor.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="floor.is_odd" class="variable">is_odd</dd>
-                <dd id="floor.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="floor.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="floor.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="floor.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="floor.is_irrational" class="variable">is_irrational</dd>
+                <dd id="floor.is_integer" class="variable">is_integer</dd>
+                <dd id="floor.is_rational" class="variable">is_rational</dd>
+                <dd id="floor.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="floor.is_complex" class="variable">is_complex</dd>
                 <dd id="floor.is_prime" class="variable">is_prime</dd>
+                <dd id="floor.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="floor.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="floor.is_real" class="variable">is_real</dd>
+                <dd id="floor.is_commutative" class="variable">is_commutative</dd>
                 <dd id="floor.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="floor.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="floor.is_polar" class="variable">is_polar</dd>
+                <dd id="floor.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="floor.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="floor.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -24068,14 +24658,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="ceiling">
-                                <div class="attr class">
-        <a class="headerlink" href="#ceiling">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">ceiling</span><wbr>(<span class="base">sympy.functions.elementary.integers.ceiling</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">ceiling</span><wbr>(<span class="base">sympy.functions.elementary.integers.ceiling</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#ceiling"></a>
     
             <div class="docstring"><p>Ceiling is a univariate function which returns the smallest integer
 value not less than its argument. This implementation
@@ -24105,28 +24695,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] "Concrete mathematics" by Graham, pp. 87
-.. [2] http://mathworld.wolfram.com/CeilingFunction.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="ceiling.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#ceiling.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">ceiling</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">ceiling</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#ceiling.__init__"></a>
     
     
 
                             </div>
                             <div id="ceiling.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#ceiling.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#ceiling.default_assumptions"></a>
     
     
 
@@ -24281,37 +24876,37 @@ <h5>Inherited Members</h5>
                 <dd id="ceiling.simplify" class="function">simplify</dd>
                 <dd id="ceiling.refine" class="function">refine</dd>
                 <dd id="ceiling.rewrite" class="function">rewrite</dd>
-                <dd id="ceiling.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="ceiling.is_integer" class="variable">is_integer</dd>
-                <dd id="ceiling.is_complex" class="variable">is_complex</dd>
-                <dd id="ceiling.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="ceiling.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="ceiling.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="ceiling.is_irrational" class="variable">is_irrational</dd>
+                <dd id="ceiling.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="ceiling.is_positive" class="variable">is_positive</dd>
+                <dd id="ceiling.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="ceiling.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="ceiling.is_negative" class="variable">is_negative</dd>
+                <dd id="ceiling.is_composite" class="variable">is_composite</dd>
+                <dd id="ceiling.is_finite" class="variable">is_finite</dd>
                 <dd id="ceiling.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="ceiling.is_zero" class="variable">is_zero</dd>
                 <dd id="ceiling.is_even" class="variable">is_even</dd>
-                <dd id="ceiling.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="ceiling.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="ceiling.is_commutative" class="variable">is_commutative</dd>
-                <dd id="ceiling.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="ceiling.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="ceiling.is_real" class="variable">is_real</dd>
-                <dd id="ceiling.is_positive" class="variable">is_positive</dd>
-                <dd id="ceiling.is_finite" class="variable">is_finite</dd>
-                <dd id="ceiling.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="ceiling.is_infinite" class="variable">is_infinite</dd>
-                <dd id="ceiling.is_rational" class="variable">is_rational</dd>
-                <dd id="ceiling.is_composite" class="variable">is_composite</dd>
-                <dd id="ceiling.is_negative" class="variable">is_negative</dd>
-                <dd id="ceiling.is_polar" class="variable">is_polar</dd>
+                <dd id="ceiling.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="ceiling.is_odd" class="variable">is_odd</dd>
-                <dd id="ceiling.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="ceiling.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="ceiling.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="ceiling.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="ceiling.is_irrational" class="variable">is_irrational</dd>
+                <dd id="ceiling.is_integer" class="variable">is_integer</dd>
+                <dd id="ceiling.is_rational" class="variable">is_rational</dd>
+                <dd id="ceiling.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="ceiling.is_complex" class="variable">is_complex</dd>
                 <dd id="ceiling.is_prime" class="variable">is_prime</dd>
+                <dd id="ceiling.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="ceiling.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="ceiling.is_real" class="variable">is_real</dd>
+                <dd id="ceiling.is_commutative" class="variable">is_commutative</dd>
                 <dd id="ceiling.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="ceiling.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="ceiling.is_polar" class="variable">is_polar</dd>
+                <dd id="ceiling.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="ceiling.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="ceiling.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -24323,21 +24918,20 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="frac">
-                                <div class="attr class">
-        <a class="headerlink" href="#frac">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">frac</span><wbr>(<span class="base">sympy.functions.elementary.integers.frac</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">frac</span><wbr>(<span class="base">sympy.functions.elementary.integers.frac</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#frac"></a>
     
             <div class="docstring"><p>Represents the fractional part of x</p>
 
-<p>For real numbers it is defined [1]_ as</p>
+<p>For real numbers it is defined <sup class="footnote-ref" id="fnref-1"><a href="#fn-1">1</a></sup> as</p>
 
-<p>.. math::
-    x - \left\lfloor{x}\right\rfloor</p>
+<p>$$x - \left\lfloor{x}\right\rfloor$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -24377,28 +24971,36 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Fractional_part
-.. [2] http://mathworld.wolfram.com/FractionalPart.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+<li id="fn-1">
+<p>https://en.wikipedia.org/wiki/Fractional_part&#160;<a href="#fnref-1" class="footnoteBackLink" title="Jump back to footnote 1 in the text.">&#8617;</a></p>
+</li>
+</ol>
+</div>
 </div>
 
 
                             <div id="frac.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#frac.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">frac</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">frac</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#frac.__init__"></a>
     
     
 
                             </div>
                             <div id="frac.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#frac.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#frac.default_assumptions"></a>
     
     
 
@@ -24553,37 +25155,37 @@ <h5>Inherited Members</h5>
                 <dd id="frac.simplify" class="function">simplify</dd>
                 <dd id="frac.refine" class="function">refine</dd>
                 <dd id="frac.rewrite" class="function">rewrite</dd>
-                <dd id="frac.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="frac.is_integer" class="variable">is_integer</dd>
-                <dd id="frac.is_complex" class="variable">is_complex</dd>
-                <dd id="frac.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="frac.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="frac.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="frac.is_irrational" class="variable">is_irrational</dd>
+                <dd id="frac.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="frac.is_positive" class="variable">is_positive</dd>
+                <dd id="frac.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="frac.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="frac.is_negative" class="variable">is_negative</dd>
+                <dd id="frac.is_composite" class="variable">is_composite</dd>
+                <dd id="frac.is_finite" class="variable">is_finite</dd>
                 <dd id="frac.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="frac.is_zero" class="variable">is_zero</dd>
                 <dd id="frac.is_even" class="variable">is_even</dd>
-                <dd id="frac.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="frac.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="frac.is_commutative" class="variable">is_commutative</dd>
-                <dd id="frac.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="frac.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="frac.is_real" class="variable">is_real</dd>
-                <dd id="frac.is_positive" class="variable">is_positive</dd>
-                <dd id="frac.is_finite" class="variable">is_finite</dd>
-                <dd id="frac.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="frac.is_infinite" class="variable">is_infinite</dd>
-                <dd id="frac.is_rational" class="variable">is_rational</dd>
-                <dd id="frac.is_composite" class="variable">is_composite</dd>
-                <dd id="frac.is_negative" class="variable">is_negative</dd>
-                <dd id="frac.is_polar" class="variable">is_polar</dd>
+                <dd id="frac.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="frac.is_odd" class="variable">is_odd</dd>
-                <dd id="frac.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="frac.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="frac.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="frac.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="frac.is_irrational" class="variable">is_irrational</dd>
+                <dd id="frac.is_integer" class="variable">is_integer</dd>
+                <dd id="frac.is_rational" class="variable">is_rational</dd>
+                <dd id="frac.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="frac.is_complex" class="variable">is_complex</dd>
                 <dd id="frac.is_prime" class="variable">is_prime</dd>
+                <dd id="frac.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="frac.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="frac.is_real" class="variable">is_real</dd>
+                <dd id="frac.is_commutative" class="variable">is_commutative</dd>
                 <dd id="frac.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="frac.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="frac.is_polar" class="variable">is_polar</dd>
+                <dd id="frac.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="frac.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="frac.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -24595,14 +25197,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Piecewise">
-                                <div class="attr class">
-        <a class="headerlink" href="#Piecewise">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Piecewise</span><wbr>(<span class="base">sympy.functions.elementary.piecewise.Piecewise</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Piecewise</span><wbr>(<span class="base">sympy.functions.elementary.piecewise.Piecewise</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Piecewise"></a>
     
             <div class="docstring"><p>Represents a piecewise function.</p>
 
@@ -24648,7 +25250,7 @@ <h1 id="examples">Examples</h1>
 
 <p>When a Boolean containing Piecewise (like cond) or a Piecewise
 with Boolean expressions (like folded_cond) is used as a condition,
-it is converted to an equivalent :class:<code>~.ITE</code> object:</p>
+it is converted to an equivalent <code>~.ITE</code> object:</p>
 
 <div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="n">Piecewise</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="n">folded_cond</span><span class="p">))</span>
 <span class="go">Piecewise((1, ITE(x &lt; 0, y &gt; 2, y &gt; 3)))</span>
@@ -24668,22 +25270,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="Piecewise.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Piecewise.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Piecewise</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Piecewise</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Piecewise.__init__"></a>
     
     
 
                             </div>
                             <div id="Piecewise.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Piecewise.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Piecewise.default_assumptions"></a>
     
     
 
@@ -24841,37 +25445,37 @@ <h5>Inherited Members</h5>
                 <dd id="Piecewise.simplify" class="function">simplify</dd>
                 <dd id="Piecewise.refine" class="function">refine</dd>
                 <dd id="Piecewise.rewrite" class="function">rewrite</dd>
-                <dd id="Piecewise.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Piecewise.is_integer" class="variable">is_integer</dd>
-                <dd id="Piecewise.is_complex" class="variable">is_complex</dd>
-                <dd id="Piecewise.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Piecewise.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Piecewise.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Piecewise.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Piecewise.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Piecewise.is_positive" class="variable">is_positive</dd>
+                <dd id="Piecewise.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Piecewise.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Piecewise.is_negative" class="variable">is_negative</dd>
+                <dd id="Piecewise.is_composite" class="variable">is_composite</dd>
+                <dd id="Piecewise.is_finite" class="variable">is_finite</dd>
                 <dd id="Piecewise.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Piecewise.is_zero" class="variable">is_zero</dd>
                 <dd id="Piecewise.is_even" class="variable">is_even</dd>
-                <dd id="Piecewise.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Piecewise.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Piecewise.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Piecewise.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Piecewise.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Piecewise.is_real" class="variable">is_real</dd>
-                <dd id="Piecewise.is_positive" class="variable">is_positive</dd>
-                <dd id="Piecewise.is_finite" class="variable">is_finite</dd>
-                <dd id="Piecewise.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Piecewise.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Piecewise.is_rational" class="variable">is_rational</dd>
-                <dd id="Piecewise.is_composite" class="variable">is_composite</dd>
-                <dd id="Piecewise.is_negative" class="variable">is_negative</dd>
-                <dd id="Piecewise.is_polar" class="variable">is_polar</dd>
+                <dd id="Piecewise.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Piecewise.is_odd" class="variable">is_odd</dd>
-                <dd id="Piecewise.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Piecewise.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Piecewise.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Piecewise.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Piecewise.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Piecewise.is_integer" class="variable">is_integer</dd>
+                <dd id="Piecewise.is_rational" class="variable">is_rational</dd>
+                <dd id="Piecewise.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Piecewise.is_complex" class="variable">is_complex</dd>
                 <dd id="Piecewise.is_prime" class="variable">is_prime</dd>
+                <dd id="Piecewise.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Piecewise.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Piecewise.is_real" class="variable">is_real</dd>
+                <dd id="Piecewise.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Piecewise.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Piecewise.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Piecewise.is_polar" class="variable">is_polar</dd>
+                <dd id="Piecewise.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Piecewise.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Piecewise.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -24883,14 +25487,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="erf">
-                                <div class="attr class">
-        <a class="headerlink" href="#erf">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">erf</span><wbr>(<span class="base">sympy.functions.special.error_functions.erf</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">erf</span><wbr>(<span class="base">sympy.functions.special.error_functions.erf</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#erf"></a>
     
             <div class="docstring"><p>The Gauss error function.</p>
 
@@ -24963,30 +25567,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Error_function
-.. [2] http://dlmf.nist.gov/7
-.. [3] http://mathworld.wolfram.com/Erf.html
-.. [4] http://functions.wolfram.com/GammaBetaErf/Erf</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="erf.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#erf.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">erf</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">erf</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#erf.__init__"></a>
     
     
 
                             </div>
                             <div id="erf.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#erf.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#erf.default_assumptions"></a>
     
     
 
@@ -25143,37 +25750,37 @@ <h5>Inherited Members</h5>
                 <dd id="erf.simplify" class="function">simplify</dd>
                 <dd id="erf.refine" class="function">refine</dd>
                 <dd id="erf.rewrite" class="function">rewrite</dd>
-                <dd id="erf.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="erf.is_integer" class="variable">is_integer</dd>
-                <dd id="erf.is_complex" class="variable">is_complex</dd>
-                <dd id="erf.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="erf.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="erf.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="erf.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erf.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erf.is_positive" class="variable">is_positive</dd>
+                <dd id="erf.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="erf.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="erf.is_negative" class="variable">is_negative</dd>
+                <dd id="erf.is_composite" class="variable">is_composite</dd>
+                <dd id="erf.is_finite" class="variable">is_finite</dd>
                 <dd id="erf.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="erf.is_zero" class="variable">is_zero</dd>
                 <dd id="erf.is_even" class="variable">is_even</dd>
-                <dd id="erf.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="erf.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="erf.is_commutative" class="variable">is_commutative</dd>
-                <dd id="erf.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="erf.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="erf.is_real" class="variable">is_real</dd>
-                <dd id="erf.is_positive" class="variable">is_positive</dd>
-                <dd id="erf.is_finite" class="variable">is_finite</dd>
-                <dd id="erf.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="erf.is_infinite" class="variable">is_infinite</dd>
-                <dd id="erf.is_rational" class="variable">is_rational</dd>
-                <dd id="erf.is_composite" class="variable">is_composite</dd>
-                <dd id="erf.is_negative" class="variable">is_negative</dd>
-                <dd id="erf.is_polar" class="variable">is_polar</dd>
+                <dd id="erf.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="erf.is_odd" class="variable">is_odd</dd>
-                <dd id="erf.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="erf.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erf.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="erf.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="erf.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erf.is_integer" class="variable">is_integer</dd>
+                <dd id="erf.is_rational" class="variable">is_rational</dd>
+                <dd id="erf.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="erf.is_complex" class="variable">is_complex</dd>
                 <dd id="erf.is_prime" class="variable">is_prime</dd>
+                <dd id="erf.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="erf.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="erf.is_real" class="variable">is_real</dd>
+                <dd id="erf.is_commutative" class="variable">is_commutative</dd>
                 <dd id="erf.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="erf.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="erf.is_polar" class="variable">is_polar</dd>
+                <dd id="erf.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="erf.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="erf.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -25185,14 +25792,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="erfc">
-                                <div class="attr class">
-        <a class="headerlink" href="#erfc">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">erfc</span><wbr>(<span class="base">sympy.functions.special.error_functions.erfc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">erfc</span><wbr>(<span class="base">sympy.functions.special.error_functions.erfc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#erfc"></a>
     
             <div class="docstring"><p>Complementary Error Function.</p>
 
@@ -25265,30 +25872,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Error_function
-.. [2] http://dlmf.nist.gov/7
-.. [3] http://mathworld.wolfram.com/Erfc.html
-.. [4] http://functions.wolfram.com/GammaBetaErf/Erfc</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="erfc.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#erfc.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">erfc</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">erfc</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#erfc.__init__"></a>
     
     
 
                             </div>
                             <div id="erfc.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#erfc.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#erfc.default_assumptions"></a>
     
     
 
@@ -25445,37 +26055,37 @@ <h5>Inherited Members</h5>
                 <dd id="erfc.simplify" class="function">simplify</dd>
                 <dd id="erfc.refine" class="function">refine</dd>
                 <dd id="erfc.rewrite" class="function">rewrite</dd>
-                <dd id="erfc.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="erfc.is_integer" class="variable">is_integer</dd>
-                <dd id="erfc.is_complex" class="variable">is_complex</dd>
-                <dd id="erfc.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="erfc.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="erfc.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="erfc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erfc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erfc.is_positive" class="variable">is_positive</dd>
+                <dd id="erfc.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="erfc.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="erfc.is_negative" class="variable">is_negative</dd>
+                <dd id="erfc.is_composite" class="variable">is_composite</dd>
+                <dd id="erfc.is_finite" class="variable">is_finite</dd>
                 <dd id="erfc.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="erfc.is_zero" class="variable">is_zero</dd>
                 <dd id="erfc.is_even" class="variable">is_even</dd>
-                <dd id="erfc.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="erfc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="erfc.is_commutative" class="variable">is_commutative</dd>
-                <dd id="erfc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="erfc.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="erfc.is_real" class="variable">is_real</dd>
-                <dd id="erfc.is_positive" class="variable">is_positive</dd>
-                <dd id="erfc.is_finite" class="variable">is_finite</dd>
-                <dd id="erfc.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="erfc.is_infinite" class="variable">is_infinite</dd>
-                <dd id="erfc.is_rational" class="variable">is_rational</dd>
-                <dd id="erfc.is_composite" class="variable">is_composite</dd>
-                <dd id="erfc.is_negative" class="variable">is_negative</dd>
-                <dd id="erfc.is_polar" class="variable">is_polar</dd>
+                <dd id="erfc.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="erfc.is_odd" class="variable">is_odd</dd>
-                <dd id="erfc.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="erfc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erfc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="erfc.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="erfc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erfc.is_integer" class="variable">is_integer</dd>
+                <dd id="erfc.is_rational" class="variable">is_rational</dd>
+                <dd id="erfc.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="erfc.is_complex" class="variable">is_complex</dd>
                 <dd id="erfc.is_prime" class="variable">is_prime</dd>
+                <dd id="erfc.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="erfc.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="erfc.is_real" class="variable">is_real</dd>
+                <dd id="erfc.is_commutative" class="variable">is_commutative</dd>
                 <dd id="erfc.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="erfc.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="erfc.is_polar" class="variable">is_polar</dd>
+                <dd id="erfc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="erfc.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="erfc.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -25487,14 +26097,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="erfi">
-                                <div class="attr class">
-        <a class="headerlink" href="#erfi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">erfi</span><wbr>(<span class="base">sympy.functions.special.error_functions.erfi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">erfi</span><wbr>(<span class="base">sympy.functions.special.error_functions.erfi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#erfi"></a>
     
             <div class="docstring"><p>Imaginary error function.</p>
 
@@ -25565,29 +26175,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Error_function
-.. [2] http://mathworld.wolfram.com/Erfi.html
-.. [3] http://functions.wolfram.com/GammaBetaErf/Erfi</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="erfi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#erfi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">erfi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">erfi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#erfi.__init__"></a>
     
     
 
                             </div>
                             <div id="erfi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#erfi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#erfi.default_assumptions"></a>
     
     
 
@@ -25743,37 +26357,37 @@ <h5>Inherited Members</h5>
                 <dd id="erfi.simplify" class="function">simplify</dd>
                 <dd id="erfi.refine" class="function">refine</dd>
                 <dd id="erfi.rewrite" class="function">rewrite</dd>
-                <dd id="erfi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="erfi.is_integer" class="variable">is_integer</dd>
-                <dd id="erfi.is_complex" class="variable">is_complex</dd>
-                <dd id="erfi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="erfi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="erfi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="erfi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erfi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erfi.is_positive" class="variable">is_positive</dd>
+                <dd id="erfi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="erfi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="erfi.is_negative" class="variable">is_negative</dd>
+                <dd id="erfi.is_composite" class="variable">is_composite</dd>
+                <dd id="erfi.is_finite" class="variable">is_finite</dd>
                 <dd id="erfi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="erfi.is_zero" class="variable">is_zero</dd>
                 <dd id="erfi.is_even" class="variable">is_even</dd>
-                <dd id="erfi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="erfi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="erfi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="erfi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="erfi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="erfi.is_real" class="variable">is_real</dd>
-                <dd id="erfi.is_positive" class="variable">is_positive</dd>
-                <dd id="erfi.is_finite" class="variable">is_finite</dd>
-                <dd id="erfi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="erfi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="erfi.is_rational" class="variable">is_rational</dd>
-                <dd id="erfi.is_composite" class="variable">is_composite</dd>
-                <dd id="erfi.is_negative" class="variable">is_negative</dd>
-                <dd id="erfi.is_polar" class="variable">is_polar</dd>
+                <dd id="erfi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="erfi.is_odd" class="variable">is_odd</dd>
-                <dd id="erfi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="erfi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erfi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="erfi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="erfi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erfi.is_integer" class="variable">is_integer</dd>
+                <dd id="erfi.is_rational" class="variable">is_rational</dd>
+                <dd id="erfi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="erfi.is_complex" class="variable">is_complex</dd>
                 <dd id="erfi.is_prime" class="variable">is_prime</dd>
+                <dd id="erfi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="erfi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="erfi.is_real" class="variable">is_real</dd>
+                <dd id="erfi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="erfi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="erfi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="erfi.is_polar" class="variable">is_polar</dd>
+                <dd id="erfi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="erfi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="erfi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -25785,14 +26399,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="erf2">
-                                <div class="attr class">
-        <a class="headerlink" href="#erf2">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">erf2</span><wbr>(<span class="base">sympy.functions.special.error_functions.erf2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">erf2</span><wbr>(<span class="base">sympy.functions.special.error_functions.erf2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#erf2"></a>
     
             <div class="docstring"><p>Two-argument error function.</p>
 
@@ -25858,27 +26472,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://functions.wolfram.com/GammaBetaErf/Erf2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="erf2.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#erf2.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">erf2</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">erf2</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#erf2.__init__"></a>
     
     
 
                             </div>
                             <div id="erf2.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#erf2.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#erf2.default_assumptions"></a>
     
     
 
@@ -26033,37 +26653,37 @@ <h5>Inherited Members</h5>
                 <dd id="erf2.simplify" class="function">simplify</dd>
                 <dd id="erf2.refine" class="function">refine</dd>
                 <dd id="erf2.rewrite" class="function">rewrite</dd>
-                <dd id="erf2.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="erf2.is_integer" class="variable">is_integer</dd>
-                <dd id="erf2.is_complex" class="variable">is_complex</dd>
-                <dd id="erf2.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="erf2.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="erf2.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="erf2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erf2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erf2.is_positive" class="variable">is_positive</dd>
+                <dd id="erf2.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="erf2.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="erf2.is_negative" class="variable">is_negative</dd>
+                <dd id="erf2.is_composite" class="variable">is_composite</dd>
+                <dd id="erf2.is_finite" class="variable">is_finite</dd>
                 <dd id="erf2.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="erf2.is_zero" class="variable">is_zero</dd>
                 <dd id="erf2.is_even" class="variable">is_even</dd>
-                <dd id="erf2.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="erf2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="erf2.is_commutative" class="variable">is_commutative</dd>
-                <dd id="erf2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="erf2.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="erf2.is_real" class="variable">is_real</dd>
-                <dd id="erf2.is_positive" class="variable">is_positive</dd>
-                <dd id="erf2.is_finite" class="variable">is_finite</dd>
-                <dd id="erf2.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="erf2.is_infinite" class="variable">is_infinite</dd>
-                <dd id="erf2.is_rational" class="variable">is_rational</dd>
-                <dd id="erf2.is_composite" class="variable">is_composite</dd>
-                <dd id="erf2.is_negative" class="variable">is_negative</dd>
-                <dd id="erf2.is_polar" class="variable">is_polar</dd>
+                <dd id="erf2.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="erf2.is_odd" class="variable">is_odd</dd>
-                <dd id="erf2.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="erf2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erf2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="erf2.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="erf2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erf2.is_integer" class="variable">is_integer</dd>
+                <dd id="erf2.is_rational" class="variable">is_rational</dd>
+                <dd id="erf2.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="erf2.is_complex" class="variable">is_complex</dd>
                 <dd id="erf2.is_prime" class="variable">is_prime</dd>
+                <dd id="erf2.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="erf2.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="erf2.is_real" class="variable">is_real</dd>
+                <dd id="erf2.is_commutative" class="variable">is_commutative</dd>
                 <dd id="erf2.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="erf2.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="erf2.is_polar" class="variable">is_polar</dd>
+                <dd id="erf2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="erf2.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="erf2.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -26075,14 +26695,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="erfinv">
-                                <div class="attr class">
-        <a class="headerlink" href="#erfinv">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">erfinv</span><wbr>(<span class="base">sympy.functions.special.error_functions.erfinv</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">erfinv</span><wbr>(<span class="base">sympy.functions.special.error_functions.erfinv</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#erfinv"></a>
     
             <div class="docstring"><p>Inverse Error Function. The erfinv function is defined as:</p>
 
@@ -26128,28 +26748,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions
-.. [2] http://functions.wolfram.com/GammaBetaErf/InverseErf/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="erfinv.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#erfinv.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">erfinv</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">erfinv</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#erfinv.__init__"></a>
     
     
 
                             </div>
                             <div id="erfinv.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#erfinv.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#erfinv.default_assumptions"></a>
     
     
 
@@ -26305,37 +26930,37 @@ <h5>Inherited Members</h5>
                 <dd id="erfinv.simplify" class="function">simplify</dd>
                 <dd id="erfinv.refine" class="function">refine</dd>
                 <dd id="erfinv.rewrite" class="function">rewrite</dd>
-                <dd id="erfinv.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="erfinv.is_integer" class="variable">is_integer</dd>
-                <dd id="erfinv.is_complex" class="variable">is_complex</dd>
-                <dd id="erfinv.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="erfinv.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="erfinv.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="erfinv.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erfinv.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erfinv.is_positive" class="variable">is_positive</dd>
+                <dd id="erfinv.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="erfinv.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="erfinv.is_negative" class="variable">is_negative</dd>
+                <dd id="erfinv.is_composite" class="variable">is_composite</dd>
+                <dd id="erfinv.is_finite" class="variable">is_finite</dd>
                 <dd id="erfinv.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="erfinv.is_zero" class="variable">is_zero</dd>
                 <dd id="erfinv.is_even" class="variable">is_even</dd>
-                <dd id="erfinv.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="erfinv.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="erfinv.is_commutative" class="variable">is_commutative</dd>
-                <dd id="erfinv.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="erfinv.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="erfinv.is_real" class="variable">is_real</dd>
-                <dd id="erfinv.is_positive" class="variable">is_positive</dd>
-                <dd id="erfinv.is_finite" class="variable">is_finite</dd>
-                <dd id="erfinv.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="erfinv.is_infinite" class="variable">is_infinite</dd>
-                <dd id="erfinv.is_rational" class="variable">is_rational</dd>
-                <dd id="erfinv.is_composite" class="variable">is_composite</dd>
-                <dd id="erfinv.is_negative" class="variable">is_negative</dd>
-                <dd id="erfinv.is_polar" class="variable">is_polar</dd>
+                <dd id="erfinv.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="erfinv.is_odd" class="variable">is_odd</dd>
-                <dd id="erfinv.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="erfinv.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erfinv.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="erfinv.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="erfinv.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erfinv.is_integer" class="variable">is_integer</dd>
+                <dd id="erfinv.is_rational" class="variable">is_rational</dd>
+                <dd id="erfinv.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="erfinv.is_complex" class="variable">is_complex</dd>
                 <dd id="erfinv.is_prime" class="variable">is_prime</dd>
+                <dd id="erfinv.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="erfinv.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="erfinv.is_real" class="variable">is_real</dd>
+                <dd id="erfinv.is_commutative" class="variable">is_commutative</dd>
                 <dd id="erfinv.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="erfinv.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="erfinv.is_polar" class="variable">is_polar</dd>
+                <dd id="erfinv.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="erfinv.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="erfinv.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -26347,14 +26972,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="erfcinv">
-                                <div class="attr class">
-        <a class="headerlink" href="#erfcinv">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">erfcinv</span><wbr>(<span class="base">sympy.functions.special.error_functions.erfcinv</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">erfcinv</span><wbr>(<span class="base">sympy.functions.special.error_functions.erfcinv</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#erfcinv"></a>
     
             <div class="docstring"><p>Inverse Complementary Error Function. The erfcinv function is defined as:</p>
 
@@ -26393,28 +27018,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions
-.. [2] http://functions.wolfram.com/GammaBetaErf/InverseErfc/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="erfcinv.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#erfcinv.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">erfcinv</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">erfcinv</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#erfcinv.__init__"></a>
     
     
 
                             </div>
                             <div id="erfcinv.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#erfcinv.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#erfcinv.default_assumptions"></a>
     
     
 
@@ -26570,37 +27200,37 @@ <h5>Inherited Members</h5>
                 <dd id="erfcinv.simplify" class="function">simplify</dd>
                 <dd id="erfcinv.refine" class="function">refine</dd>
                 <dd id="erfcinv.rewrite" class="function">rewrite</dd>
-                <dd id="erfcinv.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="erfcinv.is_integer" class="variable">is_integer</dd>
-                <dd id="erfcinv.is_complex" class="variable">is_complex</dd>
-                <dd id="erfcinv.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="erfcinv.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="erfcinv.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="erfcinv.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erfcinv.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erfcinv.is_positive" class="variable">is_positive</dd>
+                <dd id="erfcinv.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="erfcinv.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="erfcinv.is_negative" class="variable">is_negative</dd>
+                <dd id="erfcinv.is_composite" class="variable">is_composite</dd>
+                <dd id="erfcinv.is_finite" class="variable">is_finite</dd>
                 <dd id="erfcinv.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="erfcinv.is_zero" class="variable">is_zero</dd>
                 <dd id="erfcinv.is_even" class="variable">is_even</dd>
-                <dd id="erfcinv.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="erfcinv.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="erfcinv.is_commutative" class="variable">is_commutative</dd>
-                <dd id="erfcinv.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="erfcinv.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="erfcinv.is_real" class="variable">is_real</dd>
-                <dd id="erfcinv.is_positive" class="variable">is_positive</dd>
-                <dd id="erfcinv.is_finite" class="variable">is_finite</dd>
-                <dd id="erfcinv.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="erfcinv.is_infinite" class="variable">is_infinite</dd>
-                <dd id="erfcinv.is_rational" class="variable">is_rational</dd>
-                <dd id="erfcinv.is_composite" class="variable">is_composite</dd>
-                <dd id="erfcinv.is_negative" class="variable">is_negative</dd>
-                <dd id="erfcinv.is_polar" class="variable">is_polar</dd>
+                <dd id="erfcinv.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="erfcinv.is_odd" class="variable">is_odd</dd>
-                <dd id="erfcinv.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="erfcinv.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erfcinv.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="erfcinv.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="erfcinv.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erfcinv.is_integer" class="variable">is_integer</dd>
+                <dd id="erfcinv.is_rational" class="variable">is_rational</dd>
+                <dd id="erfcinv.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="erfcinv.is_complex" class="variable">is_complex</dd>
                 <dd id="erfcinv.is_prime" class="variable">is_prime</dd>
+                <dd id="erfcinv.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="erfcinv.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="erfcinv.is_real" class="variable">is_real</dd>
+                <dd id="erfcinv.is_commutative" class="variable">is_commutative</dd>
                 <dd id="erfcinv.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="erfcinv.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="erfcinv.is_polar" class="variable">is_polar</dd>
+                <dd id="erfcinv.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="erfcinv.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="erfcinv.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -26612,14 +27242,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="erf2inv">
-                                <div class="attr class">
-        <a class="headerlink" href="#erf2inv">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">erf2inv</span><wbr>(<span class="base">sympy.functions.special.error_functions.erf2inv</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">erf2inv</span><wbr>(<span class="base">sympy.functions.special.error_functions.erf2inv</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#erf2inv"></a>
     
             <div class="docstring"><p>Two-argument Inverse error function. The erf2inv function is defined as:</p>
 
@@ -26666,27 +27296,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://functions.wolfram.com/GammaBetaErf/InverseErf2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="erf2inv.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#erf2inv.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">erf2inv</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">erf2inv</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#erf2inv.__init__"></a>
     
     
 
                             </div>
                             <div id="erf2inv.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#erf2inv.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#erf2inv.default_assumptions"></a>
     
     
 
@@ -26841,37 +27477,37 @@ <h5>Inherited Members</h5>
                 <dd id="erf2inv.simplify" class="function">simplify</dd>
                 <dd id="erf2inv.refine" class="function">refine</dd>
                 <dd id="erf2inv.rewrite" class="function">rewrite</dd>
-                <dd id="erf2inv.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="erf2inv.is_integer" class="variable">is_integer</dd>
-                <dd id="erf2inv.is_complex" class="variable">is_complex</dd>
-                <dd id="erf2inv.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="erf2inv.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="erf2inv.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="erf2inv.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erf2inv.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erf2inv.is_positive" class="variable">is_positive</dd>
+                <dd id="erf2inv.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="erf2inv.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="erf2inv.is_negative" class="variable">is_negative</dd>
+                <dd id="erf2inv.is_composite" class="variable">is_composite</dd>
+                <dd id="erf2inv.is_finite" class="variable">is_finite</dd>
                 <dd id="erf2inv.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="erf2inv.is_zero" class="variable">is_zero</dd>
                 <dd id="erf2inv.is_even" class="variable">is_even</dd>
-                <dd id="erf2inv.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="erf2inv.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="erf2inv.is_commutative" class="variable">is_commutative</dd>
-                <dd id="erf2inv.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="erf2inv.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="erf2inv.is_real" class="variable">is_real</dd>
-                <dd id="erf2inv.is_positive" class="variable">is_positive</dd>
-                <dd id="erf2inv.is_finite" class="variable">is_finite</dd>
-                <dd id="erf2inv.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="erf2inv.is_infinite" class="variable">is_infinite</dd>
-                <dd id="erf2inv.is_rational" class="variable">is_rational</dd>
-                <dd id="erf2inv.is_composite" class="variable">is_composite</dd>
-                <dd id="erf2inv.is_negative" class="variable">is_negative</dd>
-                <dd id="erf2inv.is_polar" class="variable">is_polar</dd>
+                <dd id="erf2inv.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="erf2inv.is_odd" class="variable">is_odd</dd>
-                <dd id="erf2inv.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="erf2inv.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="erf2inv.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="erf2inv.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="erf2inv.is_irrational" class="variable">is_irrational</dd>
+                <dd id="erf2inv.is_integer" class="variable">is_integer</dd>
+                <dd id="erf2inv.is_rational" class="variable">is_rational</dd>
+                <dd id="erf2inv.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="erf2inv.is_complex" class="variable">is_complex</dd>
                 <dd id="erf2inv.is_prime" class="variable">is_prime</dd>
+                <dd id="erf2inv.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="erf2inv.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="erf2inv.is_real" class="variable">is_real</dd>
+                <dd id="erf2inv.is_commutative" class="variable">is_commutative</dd>
                 <dd id="erf2inv.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="erf2inv.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="erf2inv.is_polar" class="variable">is_polar</dd>
+                <dd id="erf2inv.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="erf2inv.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="erf2inv.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -26883,14 +27519,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Ei">
-                                <div class="attr class">
-        <a class="headerlink" href="#Ei">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Ei</span><wbr>(<span class="base">sympy.functions.special.error_functions.Ei</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Ei</span><wbr>(<span class="base">sympy.functions.special.error_functions.Ei</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Ei"></a>
     
             <div class="docstring"><p>The classical exponential integral.</p>
 
@@ -26898,8 +27534,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>For use in SymPy, this function is defined as</p>
 
-<p>.. math:: \operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!}
-                                 + \log(x) + \gamma,</p>
+<p>$$\operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!}+ \log(x) + \gamma,$$</p>
 
 <p>where $\gamma$ is the Euler-Mascheroni constant.</p>
 
@@ -26911,7 +27546,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>The name exponential integral comes from the following statement:</p>
 
-<p>.. math:: \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t</p>
+<p>$$\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t$$</p>
 
 <p>If the integral is interpreted as a Cauchy principal value, this statement
 holds for $x &gt; 0$ and $\operatorname{Ei}(x)$ as defined above.</p>
@@ -26974,29 +27609,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://dlmf.nist.gov/6.6
-.. [2] https://en.wikipedia.org/wiki/Exponential_integral
-.. [3] Abramowitz &amp; Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Ei.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Ei.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Ei</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Ei</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Ei.__init__"></a>
     
     
 
                             </div>
                             <div id="Ei.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Ei.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Ei.default_assumptions"></a>
     
     
 
@@ -27151,37 +27790,37 @@ <h5>Inherited Members</h5>
                 <dd id="Ei.simplify" class="function">simplify</dd>
                 <dd id="Ei.refine" class="function">refine</dd>
                 <dd id="Ei.rewrite" class="function">rewrite</dd>
-                <dd id="Ei.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Ei.is_integer" class="variable">is_integer</dd>
-                <dd id="Ei.is_complex" class="variable">is_complex</dd>
-                <dd id="Ei.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Ei.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Ei.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Ei.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Ei.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Ei.is_positive" class="variable">is_positive</dd>
+                <dd id="Ei.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Ei.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Ei.is_negative" class="variable">is_negative</dd>
+                <dd id="Ei.is_composite" class="variable">is_composite</dd>
+                <dd id="Ei.is_finite" class="variable">is_finite</dd>
                 <dd id="Ei.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Ei.is_zero" class="variable">is_zero</dd>
                 <dd id="Ei.is_even" class="variable">is_even</dd>
-                <dd id="Ei.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Ei.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Ei.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Ei.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Ei.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Ei.is_real" class="variable">is_real</dd>
-                <dd id="Ei.is_positive" class="variable">is_positive</dd>
-                <dd id="Ei.is_finite" class="variable">is_finite</dd>
-                <dd id="Ei.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Ei.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Ei.is_rational" class="variable">is_rational</dd>
-                <dd id="Ei.is_composite" class="variable">is_composite</dd>
-                <dd id="Ei.is_negative" class="variable">is_negative</dd>
-                <dd id="Ei.is_polar" class="variable">is_polar</dd>
+                <dd id="Ei.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Ei.is_odd" class="variable">is_odd</dd>
-                <dd id="Ei.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Ei.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Ei.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Ei.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Ei.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Ei.is_integer" class="variable">is_integer</dd>
+                <dd id="Ei.is_rational" class="variable">is_rational</dd>
+                <dd id="Ei.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Ei.is_complex" class="variable">is_complex</dd>
                 <dd id="Ei.is_prime" class="variable">is_prime</dd>
+                <dd id="Ei.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Ei.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Ei.is_real" class="variable">is_real</dd>
+                <dd id="Ei.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Ei.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Ei.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Ei.is_polar" class="variable">is_polar</dd>
+                <dd id="Ei.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Ei.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Ei.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -27193,14 +27832,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="expint">
-                                <div class="attr class">
-        <a class="headerlink" href="#expint">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">expint</span><wbr>(<span class="base">sympy.functions.special.error_functions.expint</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">expint</span><wbr>(<span class="base">sympy.functions.special.error_functions.expint</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#expint"></a>
     
             <div class="docstring"><p>Generalized exponential integral.</p>
 
@@ -27208,15 +27847,14 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined as</p>
 
-<p>.. math:: \operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),</p>
+<p>$$\operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),$$</p>
 
 <p>where $\Gamma(1 - \nu, z)$ is the upper incomplete gamma function
 (<code><a href="#uppergamma">uppergamma</a></code>).</p>
 
 <p>Hence for $z$ with positive real part we have</p>
 
-<p>.. math:: \operatorname{E}_\nu(z)
-          =   \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t,</p>
+<p>$$\operatorname{E}_\nu(z)=   \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t,$$</p>
 
 <p>which explains the name.</p>
 
@@ -27301,29 +27939,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://dlmf.nist.gov/8.19
-.. [2] http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
-.. [3] https://en.wikipedia.org/wiki/Exponential_integral</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="expint.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#expint.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">expint</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">expint</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#expint.__init__"></a>
     
     
 
                             </div>
                             <div id="expint.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#expint.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#expint.default_assumptions"></a>
     
     
 
@@ -27478,37 +28120,37 @@ <h5>Inherited Members</h5>
                 <dd id="expint.simplify" class="function">simplify</dd>
                 <dd id="expint.refine" class="function">refine</dd>
                 <dd id="expint.rewrite" class="function">rewrite</dd>
-                <dd id="expint.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="expint.is_integer" class="variable">is_integer</dd>
-                <dd id="expint.is_complex" class="variable">is_complex</dd>
-                <dd id="expint.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="expint.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="expint.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="expint.is_irrational" class="variable">is_irrational</dd>
+                <dd id="expint.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="expint.is_positive" class="variable">is_positive</dd>
+                <dd id="expint.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="expint.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="expint.is_negative" class="variable">is_negative</dd>
+                <dd id="expint.is_composite" class="variable">is_composite</dd>
+                <dd id="expint.is_finite" class="variable">is_finite</dd>
                 <dd id="expint.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="expint.is_zero" class="variable">is_zero</dd>
                 <dd id="expint.is_even" class="variable">is_even</dd>
-                <dd id="expint.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="expint.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="expint.is_commutative" class="variable">is_commutative</dd>
-                <dd id="expint.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="expint.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="expint.is_real" class="variable">is_real</dd>
-                <dd id="expint.is_positive" class="variable">is_positive</dd>
-                <dd id="expint.is_finite" class="variable">is_finite</dd>
-                <dd id="expint.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="expint.is_infinite" class="variable">is_infinite</dd>
-                <dd id="expint.is_rational" class="variable">is_rational</dd>
-                <dd id="expint.is_composite" class="variable">is_composite</dd>
-                <dd id="expint.is_negative" class="variable">is_negative</dd>
-                <dd id="expint.is_polar" class="variable">is_polar</dd>
+                <dd id="expint.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="expint.is_odd" class="variable">is_odd</dd>
-                <dd id="expint.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="expint.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="expint.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="expint.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="expint.is_irrational" class="variable">is_irrational</dd>
+                <dd id="expint.is_integer" class="variable">is_integer</dd>
+                <dd id="expint.is_rational" class="variable">is_rational</dd>
+                <dd id="expint.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="expint.is_complex" class="variable">is_complex</dd>
                 <dd id="expint.is_prime" class="variable">is_prime</dd>
+                <dd id="expint.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="expint.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="expint.is_real" class="variable">is_real</dd>
+                <dd id="expint.is_commutative" class="variable">is_commutative</dd>
                 <dd id="expint.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="expint.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="expint.is_polar" class="variable">is_polar</dd>
+                <dd id="expint.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="expint.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="expint.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -27520,14 +28162,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="li">
-                                <div class="attr class">
-        <a class="headerlink" href="#li">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">li</span><wbr>(<span class="base">sympy.functions.special.error_functions.li</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">li</span><wbr>(<span class="base">sympy.functions.special.error_functions.li</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#li"></a>
     
             <div class="docstring"><p>The classical logarithmic integral.</p>
 
@@ -27535,7 +28177,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>For use in SymPy, this function is defined as</p>
 
-<p>.. math:: \operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.</p>
+<p>$$\operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -27625,30 +28267,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Logarithmic_integral
-.. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html
-.. [3] http://dlmf.nist.gov/6
-.. [4] http://mathworld.wolfram.com/SoldnersConstant.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="li.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#li.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">li</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">li</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#li.__init__"></a>
     
     
 
                             </div>
                             <div id="li.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#li.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#li.default_assumptions"></a>
     
     
 
@@ -27803,37 +28448,37 @@ <h5>Inherited Members</h5>
                 <dd id="li.simplify" class="function">simplify</dd>
                 <dd id="li.refine" class="function">refine</dd>
                 <dd id="li.rewrite" class="function">rewrite</dd>
-                <dd id="li.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="li.is_integer" class="variable">is_integer</dd>
-                <dd id="li.is_complex" class="variable">is_complex</dd>
-                <dd id="li.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="li.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="li.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="li.is_irrational" class="variable">is_irrational</dd>
+                <dd id="li.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="li.is_positive" class="variable">is_positive</dd>
+                <dd id="li.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="li.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="li.is_negative" class="variable">is_negative</dd>
+                <dd id="li.is_composite" class="variable">is_composite</dd>
+                <dd id="li.is_finite" class="variable">is_finite</dd>
                 <dd id="li.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="li.is_zero" class="variable">is_zero</dd>
                 <dd id="li.is_even" class="variable">is_even</dd>
-                <dd id="li.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="li.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="li.is_commutative" class="variable">is_commutative</dd>
-                <dd id="li.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="li.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="li.is_real" class="variable">is_real</dd>
-                <dd id="li.is_positive" class="variable">is_positive</dd>
-                <dd id="li.is_finite" class="variable">is_finite</dd>
-                <dd id="li.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="li.is_infinite" class="variable">is_infinite</dd>
-                <dd id="li.is_rational" class="variable">is_rational</dd>
-                <dd id="li.is_composite" class="variable">is_composite</dd>
-                <dd id="li.is_negative" class="variable">is_negative</dd>
-                <dd id="li.is_polar" class="variable">is_polar</dd>
+                <dd id="li.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="li.is_odd" class="variable">is_odd</dd>
-                <dd id="li.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="li.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="li.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="li.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="li.is_irrational" class="variable">is_irrational</dd>
+                <dd id="li.is_integer" class="variable">is_integer</dd>
+                <dd id="li.is_rational" class="variable">is_rational</dd>
+                <dd id="li.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="li.is_complex" class="variable">is_complex</dd>
                 <dd id="li.is_prime" class="variable">is_prime</dd>
+                <dd id="li.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="li.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="li.is_real" class="variable">is_real</dd>
+                <dd id="li.is_commutative" class="variable">is_commutative</dd>
                 <dd id="li.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="li.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="li.is_polar" class="variable">is_polar</dd>
+                <dd id="li.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="li.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="li.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -27845,14 +28490,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Li">
-                                <div class="attr class">
-        <a class="headerlink" href="#Li">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Li</span><wbr>(<span class="base">sympy.functions.special.error_functions.Li</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Li</span><wbr>(<span class="base">sympy.functions.special.error_functions.Li</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Li"></a>
     
             <div class="docstring"><p>The offset logarithmic integral.</p>
 
@@ -27860,7 +28505,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>For use in SymPy, this function is defined as</p>
 
-<p>.. math:: \operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)</p>
+<p>$$\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -27912,29 +28557,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Logarithmic_integral
-.. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html
-.. [3] http://dlmf.nist.gov/6</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Li.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Li.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Li</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Li</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Li.__init__"></a>
     
     
 
                             </div>
                             <div id="Li.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Li.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Li.default_assumptions"></a>
     
     
 
@@ -28089,37 +28738,37 @@ <h5>Inherited Members</h5>
                 <dd id="Li.simplify" class="function">simplify</dd>
                 <dd id="Li.refine" class="function">refine</dd>
                 <dd id="Li.rewrite" class="function">rewrite</dd>
-                <dd id="Li.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Li.is_integer" class="variable">is_integer</dd>
-                <dd id="Li.is_complex" class="variable">is_complex</dd>
-                <dd id="Li.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Li.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Li.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Li.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Li.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Li.is_positive" class="variable">is_positive</dd>
+                <dd id="Li.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Li.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Li.is_negative" class="variable">is_negative</dd>
+                <dd id="Li.is_composite" class="variable">is_composite</dd>
+                <dd id="Li.is_finite" class="variable">is_finite</dd>
                 <dd id="Li.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Li.is_zero" class="variable">is_zero</dd>
                 <dd id="Li.is_even" class="variable">is_even</dd>
-                <dd id="Li.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Li.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Li.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Li.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Li.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Li.is_real" class="variable">is_real</dd>
-                <dd id="Li.is_positive" class="variable">is_positive</dd>
-                <dd id="Li.is_finite" class="variable">is_finite</dd>
-                <dd id="Li.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Li.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Li.is_rational" class="variable">is_rational</dd>
-                <dd id="Li.is_composite" class="variable">is_composite</dd>
-                <dd id="Li.is_negative" class="variable">is_negative</dd>
-                <dd id="Li.is_polar" class="variable">is_polar</dd>
+                <dd id="Li.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Li.is_odd" class="variable">is_odd</dd>
-                <dd id="Li.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Li.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Li.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Li.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Li.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Li.is_integer" class="variable">is_integer</dd>
+                <dd id="Li.is_rational" class="variable">is_rational</dd>
+                <dd id="Li.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Li.is_complex" class="variable">is_complex</dd>
                 <dd id="Li.is_prime" class="variable">is_prime</dd>
+                <dd id="Li.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Li.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Li.is_real" class="variable">is_real</dd>
+                <dd id="Li.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Li.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Li.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Li.is_polar" class="variable">is_polar</dd>
+                <dd id="Li.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Li.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Li.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -28131,14 +28780,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Si">
-                                <div class="attr class">
-        <a class="headerlink" href="#Si">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Si</span><wbr>(<span class="base">sympy.functions.special.error_functions.Si</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Si</span><wbr>(<span class="base">sympy.functions.special.error_functions.Si</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Si"></a>
     
             <div class="docstring"><p>Sine integral.</p>
 
@@ -28146,7 +28795,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined by</p>
 
-<p>.. math:: \operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.</p>
+<p>$$\operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.$$</p>
 
 <p>It is an entire function.</p>
 
@@ -28207,27 +28856,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Si.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Si.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Si</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Si</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Si.__init__"></a>
     
     
 
                             </div>
                             <div id="Si.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Si.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Si.default_assumptions"></a>
     
     
 
@@ -28382,37 +29037,37 @@ <h5>Inherited Members</h5>
                 <dd id="Si.simplify" class="function">simplify</dd>
                 <dd id="Si.refine" class="function">refine</dd>
                 <dd id="Si.rewrite" class="function">rewrite</dd>
-                <dd id="Si.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Si.is_integer" class="variable">is_integer</dd>
-                <dd id="Si.is_complex" class="variable">is_complex</dd>
-                <dd id="Si.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Si.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Si.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Si.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Si.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Si.is_positive" class="variable">is_positive</dd>
+                <dd id="Si.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Si.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Si.is_negative" class="variable">is_negative</dd>
+                <dd id="Si.is_composite" class="variable">is_composite</dd>
+                <dd id="Si.is_finite" class="variable">is_finite</dd>
                 <dd id="Si.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Si.is_zero" class="variable">is_zero</dd>
                 <dd id="Si.is_even" class="variable">is_even</dd>
-                <dd id="Si.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Si.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Si.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Si.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Si.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Si.is_real" class="variable">is_real</dd>
-                <dd id="Si.is_positive" class="variable">is_positive</dd>
-                <dd id="Si.is_finite" class="variable">is_finite</dd>
-                <dd id="Si.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Si.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Si.is_rational" class="variable">is_rational</dd>
-                <dd id="Si.is_composite" class="variable">is_composite</dd>
-                <dd id="Si.is_negative" class="variable">is_negative</dd>
-                <dd id="Si.is_polar" class="variable">is_polar</dd>
+                <dd id="Si.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Si.is_odd" class="variable">is_odd</dd>
-                <dd id="Si.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Si.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Si.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Si.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Si.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Si.is_integer" class="variable">is_integer</dd>
+                <dd id="Si.is_rational" class="variable">is_rational</dd>
+                <dd id="Si.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Si.is_complex" class="variable">is_complex</dd>
                 <dd id="Si.is_prime" class="variable">is_prime</dd>
+                <dd id="Si.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Si.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Si.is_real" class="variable">is_real</dd>
+                <dd id="Si.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Si.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Si.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Si.is_polar" class="variable">is_polar</dd>
+                <dd id="Si.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Si.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Si.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -28424,14 +29079,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Ci">
-                                <div class="attr class">
-        <a class="headerlink" href="#Ci">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Ci</span><wbr>(<span class="base">sympy.functions.special.error_functions.Ci</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Ci</span><wbr>(<span class="base">sympy.functions.special.error_functions.Ci</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Ci"></a>
     
             <div class="docstring"><p>Cosine integral.</p>
 
@@ -28439,17 +29094,15 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined for positive $x$ by</p>
 
-<p>.. math:: \operatorname{Ci}(x) = \gamma + \log{x}
-                     + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t
-       = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,</p>
+<p>$$\operatorname{Ci}(x) = \gamma + \log{x}+ \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t
+= -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,$$</p>
 
 <p>where $\gamma$ is the Euler-Mascheroni constant.</p>
 
 <p>We have</p>
 
-<p>.. math:: \operatorname{Ci}(z) =
-    -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right)
-           + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}</p>
+<p>$$\operatorname{Ci}(z) =-\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right)
+       + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}$$</p>
 
 <p>which holds for all polar $z$ and thus provides an analytic
 continuation to the Riemann surface of the logarithm.</p>
@@ -28508,27 +29161,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Ci.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Ci.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Ci</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Ci</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Ci.__init__"></a>
     
     
 
                             </div>
                             <div id="Ci.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Ci.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Ci.default_assumptions"></a>
     
     
 
@@ -28683,37 +29342,37 @@ <h5>Inherited Members</h5>
                 <dd id="Ci.simplify" class="function">simplify</dd>
                 <dd id="Ci.refine" class="function">refine</dd>
                 <dd id="Ci.rewrite" class="function">rewrite</dd>
-                <dd id="Ci.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Ci.is_integer" class="variable">is_integer</dd>
-                <dd id="Ci.is_complex" class="variable">is_complex</dd>
-                <dd id="Ci.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Ci.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Ci.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Ci.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Ci.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Ci.is_positive" class="variable">is_positive</dd>
+                <dd id="Ci.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Ci.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Ci.is_negative" class="variable">is_negative</dd>
+                <dd id="Ci.is_composite" class="variable">is_composite</dd>
+                <dd id="Ci.is_finite" class="variable">is_finite</dd>
                 <dd id="Ci.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Ci.is_zero" class="variable">is_zero</dd>
                 <dd id="Ci.is_even" class="variable">is_even</dd>
-                <dd id="Ci.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Ci.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Ci.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Ci.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Ci.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Ci.is_real" class="variable">is_real</dd>
-                <dd id="Ci.is_positive" class="variable">is_positive</dd>
-                <dd id="Ci.is_finite" class="variable">is_finite</dd>
-                <dd id="Ci.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Ci.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Ci.is_rational" class="variable">is_rational</dd>
-                <dd id="Ci.is_composite" class="variable">is_composite</dd>
-                <dd id="Ci.is_negative" class="variable">is_negative</dd>
-                <dd id="Ci.is_polar" class="variable">is_polar</dd>
+                <dd id="Ci.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Ci.is_odd" class="variable">is_odd</dd>
-                <dd id="Ci.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Ci.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Ci.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Ci.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Ci.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Ci.is_integer" class="variable">is_integer</dd>
+                <dd id="Ci.is_rational" class="variable">is_rational</dd>
+                <dd id="Ci.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Ci.is_complex" class="variable">is_complex</dd>
                 <dd id="Ci.is_prime" class="variable">is_prime</dd>
+                <dd id="Ci.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Ci.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Ci.is_real" class="variable">is_real</dd>
+                <dd id="Ci.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Ci.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Ci.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Ci.is_polar" class="variable">is_polar</dd>
+                <dd id="Ci.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Ci.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Ci.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -28725,14 +29384,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Shi">
-                                <div class="attr class">
-        <a class="headerlink" href="#Shi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Shi</span><wbr>(<span class="base">sympy.functions.special.error_functions.Shi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Shi</span><wbr>(<span class="base">sympy.functions.special.error_functions.Shi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Shi"></a>
     
             <div class="docstring"><p>Sinh integral.</p>
 
@@ -28740,7 +29399,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined by</p>
 
-<p>.. math:: \operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.</p>
+<p>$$\operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.$$</p>
 
 <p>It is an entire function.</p>
 
@@ -28793,27 +29452,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Shi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Shi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Shi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Shi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Shi.__init__"></a>
     
     
 
                             </div>
                             <div id="Shi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Shi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Shi.default_assumptions"></a>
     
     
 
@@ -28968,37 +29633,37 @@ <h5>Inherited Members</h5>
                 <dd id="Shi.simplify" class="function">simplify</dd>
                 <dd id="Shi.refine" class="function">refine</dd>
                 <dd id="Shi.rewrite" class="function">rewrite</dd>
-                <dd id="Shi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Shi.is_integer" class="variable">is_integer</dd>
-                <dd id="Shi.is_complex" class="variable">is_complex</dd>
-                <dd id="Shi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Shi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Shi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Shi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Shi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Shi.is_positive" class="variable">is_positive</dd>
+                <dd id="Shi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Shi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Shi.is_negative" class="variable">is_negative</dd>
+                <dd id="Shi.is_composite" class="variable">is_composite</dd>
+                <dd id="Shi.is_finite" class="variable">is_finite</dd>
                 <dd id="Shi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Shi.is_zero" class="variable">is_zero</dd>
                 <dd id="Shi.is_even" class="variable">is_even</dd>
-                <dd id="Shi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Shi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Shi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Shi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Shi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Shi.is_real" class="variable">is_real</dd>
-                <dd id="Shi.is_positive" class="variable">is_positive</dd>
-                <dd id="Shi.is_finite" class="variable">is_finite</dd>
-                <dd id="Shi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Shi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Shi.is_rational" class="variable">is_rational</dd>
-                <dd id="Shi.is_composite" class="variable">is_composite</dd>
-                <dd id="Shi.is_negative" class="variable">is_negative</dd>
-                <dd id="Shi.is_polar" class="variable">is_polar</dd>
+                <dd id="Shi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Shi.is_odd" class="variable">is_odd</dd>
-                <dd id="Shi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Shi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Shi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Shi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Shi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Shi.is_integer" class="variable">is_integer</dd>
+                <dd id="Shi.is_rational" class="variable">is_rational</dd>
+                <dd id="Shi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Shi.is_complex" class="variable">is_complex</dd>
                 <dd id="Shi.is_prime" class="variable">is_prime</dd>
+                <dd id="Shi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Shi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Shi.is_real" class="variable">is_real</dd>
+                <dd id="Shi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Shi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Shi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Shi.is_polar" class="variable">is_polar</dd>
+                <dd id="Shi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Shi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Shi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -29010,14 +29675,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Chi">
-                                <div class="attr class">
-        <a class="headerlink" href="#Chi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Chi</span><wbr>(<span class="base">sympy.functions.special.error_functions.Chi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Chi</span><wbr>(<span class="base">sympy.functions.special.error_functions.Chi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Chi"></a>
     
             <div class="docstring"><p>Cosh integral.</p>
 
@@ -29025,15 +29690,13 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined for positive $x$ by</p>
 
-<p>.. math:: \operatorname{Chi}(x) = \gamma + \log{x}
-                     + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,</p>
+<p>$$\operatorname{Chi}(x) = \gamma + \log{x}+ \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,$$</p>
 
 <p>where $\gamma$ is the Euler-Mascheroni constant.</p>
 
 <p>We have</p>
 
-<p>.. math:: \operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right)
-                     - i\frac{\pi}{2},</p>
+<p>$$\operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right)- i\frac{\pi}{2},$$</p>
 
 <p>which holds for all polar $z$ and thus provides an analytic
 continuation to the Riemann surface of the logarithm.
@@ -29089,27 +29752,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Chi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Chi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Chi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Chi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Chi.__init__"></a>
     
     
 
                             </div>
                             <div id="Chi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Chi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Chi.default_assumptions"></a>
     
     
 
@@ -29264,37 +29933,37 @@ <h5>Inherited Members</h5>
                 <dd id="Chi.simplify" class="function">simplify</dd>
                 <dd id="Chi.refine" class="function">refine</dd>
                 <dd id="Chi.rewrite" class="function">rewrite</dd>
-                <dd id="Chi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Chi.is_integer" class="variable">is_integer</dd>
-                <dd id="Chi.is_complex" class="variable">is_complex</dd>
-                <dd id="Chi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Chi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Chi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Chi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Chi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Chi.is_positive" class="variable">is_positive</dd>
+                <dd id="Chi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Chi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Chi.is_negative" class="variable">is_negative</dd>
+                <dd id="Chi.is_composite" class="variable">is_composite</dd>
+                <dd id="Chi.is_finite" class="variable">is_finite</dd>
                 <dd id="Chi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Chi.is_zero" class="variable">is_zero</dd>
                 <dd id="Chi.is_even" class="variable">is_even</dd>
-                <dd id="Chi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Chi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Chi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Chi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Chi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Chi.is_real" class="variable">is_real</dd>
-                <dd id="Chi.is_positive" class="variable">is_positive</dd>
-                <dd id="Chi.is_finite" class="variable">is_finite</dd>
-                <dd id="Chi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Chi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Chi.is_rational" class="variable">is_rational</dd>
-                <dd id="Chi.is_composite" class="variable">is_composite</dd>
-                <dd id="Chi.is_negative" class="variable">is_negative</dd>
-                <dd id="Chi.is_polar" class="variable">is_polar</dd>
+                <dd id="Chi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Chi.is_odd" class="variable">is_odd</dd>
-                <dd id="Chi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Chi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Chi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Chi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Chi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Chi.is_integer" class="variable">is_integer</dd>
+                <dd id="Chi.is_rational" class="variable">is_rational</dd>
+                <dd id="Chi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Chi.is_complex" class="variable">is_complex</dd>
                 <dd id="Chi.is_prime" class="variable">is_prime</dd>
+                <dd id="Chi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Chi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Chi.is_real" class="variable">is_real</dd>
+                <dd id="Chi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Chi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Chi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Chi.is_polar" class="variable">is_polar</dd>
+                <dd id="Chi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Chi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Chi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -29306,14 +29975,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="fresnels">
-                                <div class="attr class">
-        <a class="headerlink" href="#fresnels">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">fresnels</span><wbr>(<span class="base">sympy.functions.special.error_functions.fresnels</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">fresnels</span><wbr>(<span class="base">sympy.functions.special.error_functions.fresnels</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#fresnels"></a>
     
             <div class="docstring"><p>Fresnel integral S.</p>
 
@@ -29321,7 +29990,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined by</p>
 
-<p>.. math:: \operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.</p>
+<p>$$\operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.$$</p>
 
 <p>It is an entire function.</p>
 
@@ -29394,32 +30063,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Fresnel_integral
-.. [2] http://dlmf.nist.gov/7
-.. [3] http://mathworld.wolfram.com/FresnelIntegrals.html
-.. [4] http://functions.wolfram.com/GammaBetaErf/FresnelS
-.. [5] The converging factors for the fresnel integrals
-        by John W. Wrench Jr. and Vicki Alley</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="fresnels.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#fresnels.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">fresnels</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">fresnels</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#fresnels.__init__"></a>
     
     
 
                             </div>
                             <div id="fresnels.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#fresnels.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#fresnels.default_assumptions"></a>
     
     
 
@@ -29578,37 +30248,37 @@ <h5>Inherited Members</h5>
                 <dd id="fresnels.simplify" class="function">simplify</dd>
                 <dd id="fresnels.refine" class="function">refine</dd>
                 <dd id="fresnels.rewrite" class="function">rewrite</dd>
-                <dd id="fresnels.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="fresnels.is_integer" class="variable">is_integer</dd>
-                <dd id="fresnels.is_complex" class="variable">is_complex</dd>
-                <dd id="fresnels.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="fresnels.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="fresnels.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="fresnels.is_irrational" class="variable">is_irrational</dd>
+                <dd id="fresnels.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="fresnels.is_positive" class="variable">is_positive</dd>
+                <dd id="fresnels.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="fresnels.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="fresnels.is_negative" class="variable">is_negative</dd>
+                <dd id="fresnels.is_composite" class="variable">is_composite</dd>
+                <dd id="fresnels.is_finite" class="variable">is_finite</dd>
                 <dd id="fresnels.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="fresnels.is_zero" class="variable">is_zero</dd>
                 <dd id="fresnels.is_even" class="variable">is_even</dd>
-                <dd id="fresnels.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="fresnels.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="fresnels.is_commutative" class="variable">is_commutative</dd>
-                <dd id="fresnels.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="fresnels.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="fresnels.is_real" class="variable">is_real</dd>
-                <dd id="fresnels.is_positive" class="variable">is_positive</dd>
-                <dd id="fresnels.is_finite" class="variable">is_finite</dd>
-                <dd id="fresnels.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="fresnels.is_infinite" class="variable">is_infinite</dd>
-                <dd id="fresnels.is_rational" class="variable">is_rational</dd>
-                <dd id="fresnels.is_composite" class="variable">is_composite</dd>
-                <dd id="fresnels.is_negative" class="variable">is_negative</dd>
-                <dd id="fresnels.is_polar" class="variable">is_polar</dd>
+                <dd id="fresnels.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="fresnels.is_odd" class="variable">is_odd</dd>
-                <dd id="fresnels.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="fresnels.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="fresnels.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="fresnels.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="fresnels.is_irrational" class="variable">is_irrational</dd>
+                <dd id="fresnels.is_integer" class="variable">is_integer</dd>
+                <dd id="fresnels.is_rational" class="variable">is_rational</dd>
+                <dd id="fresnels.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="fresnels.is_complex" class="variable">is_complex</dd>
                 <dd id="fresnels.is_prime" class="variable">is_prime</dd>
+                <dd id="fresnels.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="fresnels.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="fresnels.is_real" class="variable">is_real</dd>
+                <dd id="fresnels.is_commutative" class="variable">is_commutative</dd>
                 <dd id="fresnels.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="fresnels.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="fresnels.is_polar" class="variable">is_polar</dd>
+                <dd id="fresnels.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="fresnels.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="fresnels.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -29620,14 +30290,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="fresnelc">
-                                <div class="attr class">
-        <a class="headerlink" href="#fresnelc">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">fresnelc</span><wbr>(<span class="base">sympy.functions.special.error_functions.fresnelc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">fresnelc</span><wbr>(<span class="base">sympy.functions.special.error_functions.fresnelc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#fresnelc"></a>
     
             <div class="docstring"><p>Fresnel integral C.</p>
 
@@ -29635,7 +30305,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined by</p>
 
-<p>.. math:: \operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.</p>
+<p>$$\operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.$$</p>
 
 <p>It is an entire function.</p>
 
@@ -29708,32 +30378,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Fresnel_integral
-.. [2] http://dlmf.nist.gov/7
-.. [3] http://mathworld.wolfram.com/FresnelIntegrals.html
-.. [4] http://functions.wolfram.com/GammaBetaErf/FresnelC
-.. [5] The converging factors for the fresnel integrals
-        by John W. Wrench Jr. and Vicki Alley</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="fresnelc.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#fresnelc.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">fresnelc</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">fresnelc</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#fresnelc.__init__"></a>
     
     
 
                             </div>
                             <div id="fresnelc.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#fresnelc.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#fresnelc.default_assumptions"></a>
     
     
 
@@ -29892,37 +30563,37 @@ <h5>Inherited Members</h5>
                 <dd id="fresnelc.simplify" class="function">simplify</dd>
                 <dd id="fresnelc.refine" class="function">refine</dd>
                 <dd id="fresnelc.rewrite" class="function">rewrite</dd>
-                <dd id="fresnelc.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="fresnelc.is_integer" class="variable">is_integer</dd>
-                <dd id="fresnelc.is_complex" class="variable">is_complex</dd>
-                <dd id="fresnelc.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="fresnelc.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="fresnelc.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="fresnelc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="fresnelc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="fresnelc.is_positive" class="variable">is_positive</dd>
+                <dd id="fresnelc.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="fresnelc.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="fresnelc.is_negative" class="variable">is_negative</dd>
+                <dd id="fresnelc.is_composite" class="variable">is_composite</dd>
+                <dd id="fresnelc.is_finite" class="variable">is_finite</dd>
                 <dd id="fresnelc.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="fresnelc.is_zero" class="variable">is_zero</dd>
                 <dd id="fresnelc.is_even" class="variable">is_even</dd>
-                <dd id="fresnelc.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="fresnelc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="fresnelc.is_commutative" class="variable">is_commutative</dd>
-                <dd id="fresnelc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="fresnelc.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="fresnelc.is_real" class="variable">is_real</dd>
-                <dd id="fresnelc.is_positive" class="variable">is_positive</dd>
-                <dd id="fresnelc.is_finite" class="variable">is_finite</dd>
-                <dd id="fresnelc.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="fresnelc.is_infinite" class="variable">is_infinite</dd>
-                <dd id="fresnelc.is_rational" class="variable">is_rational</dd>
-                <dd id="fresnelc.is_composite" class="variable">is_composite</dd>
-                <dd id="fresnelc.is_negative" class="variable">is_negative</dd>
-                <dd id="fresnelc.is_polar" class="variable">is_polar</dd>
+                <dd id="fresnelc.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="fresnelc.is_odd" class="variable">is_odd</dd>
-                <dd id="fresnelc.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="fresnelc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="fresnelc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="fresnelc.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="fresnelc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="fresnelc.is_integer" class="variable">is_integer</dd>
+                <dd id="fresnelc.is_rational" class="variable">is_rational</dd>
+                <dd id="fresnelc.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="fresnelc.is_complex" class="variable">is_complex</dd>
                 <dd id="fresnelc.is_prime" class="variable">is_prime</dd>
+                <dd id="fresnelc.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="fresnelc.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="fresnelc.is_real" class="variable">is_real</dd>
+                <dd id="fresnelc.is_commutative" class="variable">is_commutative</dd>
                 <dd id="fresnelc.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="fresnelc.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="fresnelc.is_polar" class="variable">is_polar</dd>
+                <dd id="fresnelc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="fresnelc.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="fresnelc.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -29934,19 +30605,18 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="gamma">
-                                <div class="attr class">
-        <a class="headerlink" href="#gamma">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">gamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.gamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">gamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.gamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#gamma"></a>
     
             <div class="docstring"><p>The gamma function</p>
 
-<p>.. math::
-    \Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{-t} \mathrm{d}t.</p>
+<p>$$\Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{-t} \mathrm{d}t.$$</p>
 
 <h1 id="explanation">Explanation</h1>
 
@@ -30013,30 +30683,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Gamma_function
-.. [2] http://dlmf.nist.gov/5
-.. [3] http://mathworld.wolfram.com/GammaFunction.html
-.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="gamma.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#gamma.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">gamma</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">gamma</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#gamma.__init__"></a>
     
     
 
                             </div>
                             <div id="gamma.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#gamma.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#gamma.default_assumptions"></a>
     
     
 
@@ -30192,37 +30865,37 @@ <h5>Inherited Members</h5>
                 <dd id="gamma.simplify" class="function">simplify</dd>
                 <dd id="gamma.refine" class="function">refine</dd>
                 <dd id="gamma.rewrite" class="function">rewrite</dd>
-                <dd id="gamma.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="gamma.is_integer" class="variable">is_integer</dd>
-                <dd id="gamma.is_complex" class="variable">is_complex</dd>
-                <dd id="gamma.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="gamma.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="gamma.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="gamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="gamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="gamma.is_positive" class="variable">is_positive</dd>
+                <dd id="gamma.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="gamma.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="gamma.is_negative" class="variable">is_negative</dd>
+                <dd id="gamma.is_composite" class="variable">is_composite</dd>
+                <dd id="gamma.is_finite" class="variable">is_finite</dd>
                 <dd id="gamma.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="gamma.is_zero" class="variable">is_zero</dd>
                 <dd id="gamma.is_even" class="variable">is_even</dd>
-                <dd id="gamma.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="gamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="gamma.is_commutative" class="variable">is_commutative</dd>
-                <dd id="gamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="gamma.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="gamma.is_real" class="variable">is_real</dd>
-                <dd id="gamma.is_positive" class="variable">is_positive</dd>
-                <dd id="gamma.is_finite" class="variable">is_finite</dd>
-                <dd id="gamma.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="gamma.is_infinite" class="variable">is_infinite</dd>
-                <dd id="gamma.is_rational" class="variable">is_rational</dd>
-                <dd id="gamma.is_composite" class="variable">is_composite</dd>
-                <dd id="gamma.is_negative" class="variable">is_negative</dd>
-                <dd id="gamma.is_polar" class="variable">is_polar</dd>
+                <dd id="gamma.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="gamma.is_odd" class="variable">is_odd</dd>
-                <dd id="gamma.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="gamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="gamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="gamma.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="gamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="gamma.is_integer" class="variable">is_integer</dd>
+                <dd id="gamma.is_rational" class="variable">is_rational</dd>
+                <dd id="gamma.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="gamma.is_complex" class="variable">is_complex</dd>
                 <dd id="gamma.is_prime" class="variable">is_prime</dd>
+                <dd id="gamma.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="gamma.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="gamma.is_real" class="variable">is_real</dd>
+                <dd id="gamma.is_commutative" class="variable">is_commutative</dd>
                 <dd id="gamma.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="gamma.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="gamma.is_polar" class="variable">is_polar</dd>
+                <dd id="gamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="gamma.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="gamma.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -30234,14 +30907,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="lowergamma">
-                                <div class="attr class">
-        <a class="headerlink" href="#lowergamma">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">lowergamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.lowergamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">lowergamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.lowergamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#lowergamma"></a>
     
             <div class="docstring"><p>The lower incomplete gamma function.</p>
 
@@ -30249,13 +30922,11 @@ <h1 id="explanation">Explanation</h1>
 
 <p>It can be defined as the meromorphic continuation of</p>
 
-<p>.. math::
-    \gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x).</p>
+<p>$$\gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x).$$</p>
 
 <p>This can be shown to be the same as</p>
 
-<p>.. math::
-    \gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),</p>
+<p>$$\gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),$$</p>
 
 <p>where ${}_1F_1$ is the (confluent) hypergeometric function.</p>
 
@@ -30283,33 +30954,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function
-.. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6,
-       Section 5, Handbook of Mathematical Functions with Formulas, Graphs,
-       and Mathematical Tables
-.. [3] http://dlmf.nist.gov/8
-.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma2/
-.. [5] http://functions.wolfram.com/GammaBetaErf/Gamma3/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="lowergamma.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#lowergamma.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">lowergamma</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">lowergamma</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#lowergamma.__init__"></a>
     
     
 
                             </div>
                             <div id="lowergamma.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#lowergamma.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#lowergamma.default_assumptions"></a>
     
     
 
@@ -30464,37 +31135,37 @@ <h5>Inherited Members</h5>
                 <dd id="lowergamma.simplify" class="function">simplify</dd>
                 <dd id="lowergamma.refine" class="function">refine</dd>
                 <dd id="lowergamma.rewrite" class="function">rewrite</dd>
-                <dd id="lowergamma.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="lowergamma.is_integer" class="variable">is_integer</dd>
-                <dd id="lowergamma.is_complex" class="variable">is_complex</dd>
-                <dd id="lowergamma.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="lowergamma.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="lowergamma.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="lowergamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="lowergamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="lowergamma.is_positive" class="variable">is_positive</dd>
+                <dd id="lowergamma.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="lowergamma.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="lowergamma.is_negative" class="variable">is_negative</dd>
+                <dd id="lowergamma.is_composite" class="variable">is_composite</dd>
+                <dd id="lowergamma.is_finite" class="variable">is_finite</dd>
                 <dd id="lowergamma.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="lowergamma.is_zero" class="variable">is_zero</dd>
                 <dd id="lowergamma.is_even" class="variable">is_even</dd>
-                <dd id="lowergamma.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="lowergamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="lowergamma.is_commutative" class="variable">is_commutative</dd>
-                <dd id="lowergamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="lowergamma.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="lowergamma.is_real" class="variable">is_real</dd>
-                <dd id="lowergamma.is_positive" class="variable">is_positive</dd>
-                <dd id="lowergamma.is_finite" class="variable">is_finite</dd>
-                <dd id="lowergamma.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="lowergamma.is_infinite" class="variable">is_infinite</dd>
-                <dd id="lowergamma.is_rational" class="variable">is_rational</dd>
-                <dd id="lowergamma.is_composite" class="variable">is_composite</dd>
-                <dd id="lowergamma.is_negative" class="variable">is_negative</dd>
-                <dd id="lowergamma.is_polar" class="variable">is_polar</dd>
+                <dd id="lowergamma.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="lowergamma.is_odd" class="variable">is_odd</dd>
-                <dd id="lowergamma.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="lowergamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="lowergamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="lowergamma.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="lowergamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="lowergamma.is_integer" class="variable">is_integer</dd>
+                <dd id="lowergamma.is_rational" class="variable">is_rational</dd>
+                <dd id="lowergamma.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="lowergamma.is_complex" class="variable">is_complex</dd>
                 <dd id="lowergamma.is_prime" class="variable">is_prime</dd>
+                <dd id="lowergamma.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="lowergamma.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="lowergamma.is_real" class="variable">is_real</dd>
+                <dd id="lowergamma.is_commutative" class="variable">is_commutative</dd>
                 <dd id="lowergamma.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="lowergamma.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="lowergamma.is_polar" class="variable">is_polar</dd>
+                <dd id="lowergamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="lowergamma.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="lowergamma.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -30506,14 +31177,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="uppergamma">
-                                <div class="attr class">
-        <a class="headerlink" href="#uppergamma">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">uppergamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.uppergamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">uppergamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.uppergamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#uppergamma"></a>
     
             <div class="docstring"><p>The upper incomplete gamma function.</p>
 
@@ -30521,22 +31192,19 @@ <h1 id="explanation">Explanation</h1>
 
 <p>It can be defined as the meromorphic continuation of</p>
 
-<p>.. math::
-    \Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x).</p>
+<p>$$\Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x).$$</p>
 
 <p>where $\gamma(s, x)$ is the lower incomplete gamma function,
-:class:<code><a href="#lowergamma">lowergamma</a></code>. This can be shown to be the same as</p>
+<code><a href="#lowergamma">lowergamma</a></code>. This can be shown to be the same as</p>
 
-<p>.. math::
-    \Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),</p>
+<p>$$\Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),$$</p>
 
 <p>where ${}_1F_1$ is the (confluent) hypergeometric function.</p>
 
 <p>The upper incomplete gamma function is also essentially equivalent to the
 generalized exponential integral:</p>
 
-<p>.. math::
-    \operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x).</p>
+<p>$$\operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x).$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -30564,34 +31232,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function
-.. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6,
-       Section 5, Handbook of Mathematical Functions with Formulas, Graphs,
-       and Mathematical Tables
-.. [3] http://dlmf.nist.gov/8
-.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma2/
-.. [5] http://functions.wolfram.com/GammaBetaErf/Gamma3/
-.. [6] https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="uppergamma.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#uppergamma.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">uppergamma</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">uppergamma</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#uppergamma.__init__"></a>
     
     
 
                             </div>
                             <div id="uppergamma.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#uppergamma.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#uppergamma.default_assumptions"></a>
     
     
 
@@ -30746,37 +31413,37 @@ <h5>Inherited Members</h5>
                 <dd id="uppergamma.simplify" class="function">simplify</dd>
                 <dd id="uppergamma.refine" class="function">refine</dd>
                 <dd id="uppergamma.rewrite" class="function">rewrite</dd>
-                <dd id="uppergamma.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="uppergamma.is_integer" class="variable">is_integer</dd>
-                <dd id="uppergamma.is_complex" class="variable">is_complex</dd>
-                <dd id="uppergamma.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="uppergamma.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="uppergamma.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="uppergamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="uppergamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="uppergamma.is_positive" class="variable">is_positive</dd>
+                <dd id="uppergamma.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="uppergamma.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="uppergamma.is_negative" class="variable">is_negative</dd>
+                <dd id="uppergamma.is_composite" class="variable">is_composite</dd>
+                <dd id="uppergamma.is_finite" class="variable">is_finite</dd>
                 <dd id="uppergamma.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="uppergamma.is_zero" class="variable">is_zero</dd>
                 <dd id="uppergamma.is_even" class="variable">is_even</dd>
-                <dd id="uppergamma.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="uppergamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="uppergamma.is_commutative" class="variable">is_commutative</dd>
-                <dd id="uppergamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="uppergamma.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="uppergamma.is_real" class="variable">is_real</dd>
-                <dd id="uppergamma.is_positive" class="variable">is_positive</dd>
-                <dd id="uppergamma.is_finite" class="variable">is_finite</dd>
-                <dd id="uppergamma.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="uppergamma.is_infinite" class="variable">is_infinite</dd>
-                <dd id="uppergamma.is_rational" class="variable">is_rational</dd>
-                <dd id="uppergamma.is_composite" class="variable">is_composite</dd>
-                <dd id="uppergamma.is_negative" class="variable">is_negative</dd>
-                <dd id="uppergamma.is_polar" class="variable">is_polar</dd>
+                <dd id="uppergamma.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="uppergamma.is_odd" class="variable">is_odd</dd>
-                <dd id="uppergamma.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="uppergamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="uppergamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="uppergamma.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="uppergamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="uppergamma.is_integer" class="variable">is_integer</dd>
+                <dd id="uppergamma.is_rational" class="variable">is_rational</dd>
+                <dd id="uppergamma.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="uppergamma.is_complex" class="variable">is_complex</dd>
                 <dd id="uppergamma.is_prime" class="variable">is_prime</dd>
+                <dd id="uppergamma.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="uppergamma.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="uppergamma.is_real" class="variable">is_real</dd>
+                <dd id="uppergamma.is_commutative" class="variable">is_commutative</dd>
                 <dd id="uppergamma.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="uppergamma.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="uppergamma.is_polar" class="variable">is_polar</dd>
+                <dd id="uppergamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="uppergamma.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="uppergamma.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -30788,14 +31455,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="polygamma">
-                                <div class="attr class">
-        <a class="headerlink" href="#polygamma">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">polygamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.polygamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">polygamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.polygamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#polygamma"></a>
     
             <div class="docstring"><p>The function <code>polygamma(n, z)</code> returns <code>log(gamma(z)).diff(n + 1)</code>.</p>
 
@@ -30804,8 +31471,7 @@ <h1 id="explanation">Explanation</h1>
 <p>It is a meromorphic function on $\mathbb{C}$ and defined as the $(n+1)$-th
 derivative of the logarithm of the gamma function:</p>
 
-<p>.. math::
-    \psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z).</p>
+<p>$$\psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z).$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -30888,30 +31554,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Polygamma_function
-.. [2] http://mathworld.wolfram.com/PolygammaFunction.html
-.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma/
-.. [4] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="polygamma.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#polygamma.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">polygamma</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">polygamma</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#polygamma.__init__"></a>
     
     
 
                             </div>
                             <div id="polygamma.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#polygamma.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#polygamma.default_assumptions"></a>
     
     
 
@@ -31066,37 +31735,37 @@ <h5>Inherited Members</h5>
                 <dd id="polygamma.simplify" class="function">simplify</dd>
                 <dd id="polygamma.refine" class="function">refine</dd>
                 <dd id="polygamma.rewrite" class="function">rewrite</dd>
-                <dd id="polygamma.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="polygamma.is_integer" class="variable">is_integer</dd>
-                <dd id="polygamma.is_complex" class="variable">is_complex</dd>
-                <dd id="polygamma.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="polygamma.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="polygamma.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="polygamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="polygamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="polygamma.is_positive" class="variable">is_positive</dd>
+                <dd id="polygamma.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="polygamma.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="polygamma.is_negative" class="variable">is_negative</dd>
+                <dd id="polygamma.is_composite" class="variable">is_composite</dd>
+                <dd id="polygamma.is_finite" class="variable">is_finite</dd>
                 <dd id="polygamma.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="polygamma.is_zero" class="variable">is_zero</dd>
                 <dd id="polygamma.is_even" class="variable">is_even</dd>
-                <dd id="polygamma.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="polygamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="polygamma.is_commutative" class="variable">is_commutative</dd>
-                <dd id="polygamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="polygamma.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="polygamma.is_real" class="variable">is_real</dd>
-                <dd id="polygamma.is_positive" class="variable">is_positive</dd>
-                <dd id="polygamma.is_finite" class="variable">is_finite</dd>
-                <dd id="polygamma.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="polygamma.is_infinite" class="variable">is_infinite</dd>
-                <dd id="polygamma.is_rational" class="variable">is_rational</dd>
-                <dd id="polygamma.is_composite" class="variable">is_composite</dd>
-                <dd id="polygamma.is_negative" class="variable">is_negative</dd>
-                <dd id="polygamma.is_polar" class="variable">is_polar</dd>
+                <dd id="polygamma.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="polygamma.is_odd" class="variable">is_odd</dd>
-                <dd id="polygamma.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="polygamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="polygamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="polygamma.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="polygamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="polygamma.is_integer" class="variable">is_integer</dd>
+                <dd id="polygamma.is_rational" class="variable">is_rational</dd>
+                <dd id="polygamma.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="polygamma.is_complex" class="variable">is_complex</dd>
                 <dd id="polygamma.is_prime" class="variable">is_prime</dd>
+                <dd id="polygamma.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="polygamma.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="polygamma.is_real" class="variable">is_real</dd>
+                <dd id="polygamma.is_commutative" class="variable">is_commutative</dd>
                 <dd id="polygamma.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="polygamma.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="polygamma.is_polar" class="variable">is_polar</dd>
+                <dd id="polygamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="polygamma.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="polygamma.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -31108,14 +31777,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="loggamma">
-                                <div class="attr class">
-        <a class="headerlink" href="#loggamma">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">loggamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.loggamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">loggamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.loggamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#loggamma"></a>
     
             <div class="docstring"><p>The <code><a href="#loggamma">loggamma</a></code> function implements the logarithm of the
 gamma function (i.e., $\log\Gamma(x)$).</p>
@@ -31225,30 +31894,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Gamma_function
-.. [2] http://dlmf.nist.gov/5
-.. [3] http://mathworld.wolfram.com/LogGammaFunction.html
-.. [4] http://functions.wolfram.com/GammaBetaErf/LogGamma/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="loggamma.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#loggamma.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">loggamma</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">loggamma</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#loggamma.__init__"></a>
     
     
 
                             </div>
                             <div id="loggamma.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#loggamma.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#loggamma.default_assumptions"></a>
     
     
 
@@ -31403,37 +32075,37 @@ <h5>Inherited Members</h5>
                 <dd id="loggamma.simplify" class="function">simplify</dd>
                 <dd id="loggamma.refine" class="function">refine</dd>
                 <dd id="loggamma.rewrite" class="function">rewrite</dd>
-                <dd id="loggamma.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="loggamma.is_integer" class="variable">is_integer</dd>
-                <dd id="loggamma.is_complex" class="variable">is_complex</dd>
-                <dd id="loggamma.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="loggamma.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="loggamma.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="loggamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="loggamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="loggamma.is_positive" class="variable">is_positive</dd>
+                <dd id="loggamma.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="loggamma.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="loggamma.is_negative" class="variable">is_negative</dd>
+                <dd id="loggamma.is_composite" class="variable">is_composite</dd>
+                <dd id="loggamma.is_finite" class="variable">is_finite</dd>
                 <dd id="loggamma.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="loggamma.is_zero" class="variable">is_zero</dd>
                 <dd id="loggamma.is_even" class="variable">is_even</dd>
-                <dd id="loggamma.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="loggamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="loggamma.is_commutative" class="variable">is_commutative</dd>
-                <dd id="loggamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="loggamma.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="loggamma.is_real" class="variable">is_real</dd>
-                <dd id="loggamma.is_positive" class="variable">is_positive</dd>
-                <dd id="loggamma.is_finite" class="variable">is_finite</dd>
-                <dd id="loggamma.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="loggamma.is_infinite" class="variable">is_infinite</dd>
-                <dd id="loggamma.is_rational" class="variable">is_rational</dd>
-                <dd id="loggamma.is_composite" class="variable">is_composite</dd>
-                <dd id="loggamma.is_negative" class="variable">is_negative</dd>
-                <dd id="loggamma.is_polar" class="variable">is_polar</dd>
+                <dd id="loggamma.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="loggamma.is_odd" class="variable">is_odd</dd>
-                <dd id="loggamma.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="loggamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="loggamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="loggamma.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="loggamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="loggamma.is_integer" class="variable">is_integer</dd>
+                <dd id="loggamma.is_rational" class="variable">is_rational</dd>
+                <dd id="loggamma.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="loggamma.is_complex" class="variable">is_complex</dd>
                 <dd id="loggamma.is_prime" class="variable">is_prime</dd>
+                <dd id="loggamma.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="loggamma.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="loggamma.is_real" class="variable">is_real</dd>
+                <dd id="loggamma.is_commutative" class="variable">is_commutative</dd>
                 <dd id="loggamma.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="loggamma.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="loggamma.is_polar" class="variable">is_polar</dd>
+                <dd id="loggamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="loggamma.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="loggamma.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -31445,21 +32117,20 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="digamma">
-                                <div class="attr class">
-        <a class="headerlink" href="#digamma">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">digamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.digamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">digamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.digamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#digamma"></a>
     
             <div class="docstring"><p>The <code><a href="#digamma">digamma</a></code> function is the first derivative of the <code><a href="#loggamma">loggamma</a></code>
 function</p>
 
-<p>.. math::
-    \psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z)
-            = \frac{\Gamma'(z)}{\Gamma(z) }.</p>
+<p>$$\psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z)
+        = \frac{\Gamma'(z)}{\Gamma(z) }.$$</p>
 
 <p>In this case, <code>digamma(z) = polygamma(0, z)</code>.</p>
 
@@ -31494,29 +32165,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Digamma_function
-.. [2] http://mathworld.wolfram.com/DigammaFunction.html
-.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="digamma.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#digamma.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">digamma</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">digamma</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#digamma.__init__"></a>
     
     
 
                             </div>
                             <div id="digamma.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#digamma.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#digamma.default_assumptions"></a>
     
     
 
@@ -31671,37 +32346,37 @@ <h5>Inherited Members</h5>
                 <dd id="digamma.simplify" class="function">simplify</dd>
                 <dd id="digamma.refine" class="function">refine</dd>
                 <dd id="digamma.rewrite" class="function">rewrite</dd>
-                <dd id="digamma.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="digamma.is_integer" class="variable">is_integer</dd>
-                <dd id="digamma.is_complex" class="variable">is_complex</dd>
-                <dd id="digamma.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="digamma.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="digamma.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="digamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="digamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="digamma.is_positive" class="variable">is_positive</dd>
+                <dd id="digamma.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="digamma.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="digamma.is_negative" class="variable">is_negative</dd>
+                <dd id="digamma.is_composite" class="variable">is_composite</dd>
+                <dd id="digamma.is_finite" class="variable">is_finite</dd>
                 <dd id="digamma.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="digamma.is_zero" class="variable">is_zero</dd>
                 <dd id="digamma.is_even" class="variable">is_even</dd>
-                <dd id="digamma.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="digamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="digamma.is_commutative" class="variable">is_commutative</dd>
-                <dd id="digamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="digamma.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="digamma.is_real" class="variable">is_real</dd>
-                <dd id="digamma.is_positive" class="variable">is_positive</dd>
-                <dd id="digamma.is_finite" class="variable">is_finite</dd>
-                <dd id="digamma.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="digamma.is_infinite" class="variable">is_infinite</dd>
-                <dd id="digamma.is_rational" class="variable">is_rational</dd>
-                <dd id="digamma.is_composite" class="variable">is_composite</dd>
-                <dd id="digamma.is_negative" class="variable">is_negative</dd>
-                <dd id="digamma.is_polar" class="variable">is_polar</dd>
+                <dd id="digamma.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="digamma.is_odd" class="variable">is_odd</dd>
-                <dd id="digamma.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="digamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="digamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="digamma.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="digamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="digamma.is_integer" class="variable">is_integer</dd>
+                <dd id="digamma.is_rational" class="variable">is_rational</dd>
+                <dd id="digamma.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="digamma.is_complex" class="variable">is_complex</dd>
                 <dd id="digamma.is_prime" class="variable">is_prime</dd>
+                <dd id="digamma.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="digamma.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="digamma.is_real" class="variable">is_real</dd>
+                <dd id="digamma.is_commutative" class="variable">is_commutative</dd>
                 <dd id="digamma.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="digamma.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="digamma.is_polar" class="variable">is_polar</dd>
+                <dd id="digamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="digamma.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="digamma.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -31713,20 +32388,19 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="trigamma">
-                                <div class="attr class">
-        <a class="headerlink" href="#trigamma">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">trigamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.trigamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">trigamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.trigamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#trigamma"></a>
     
             <div class="docstring"><p>The <code><a href="#trigamma">trigamma</a></code> function is the second derivative of the <code><a href="#loggamma">loggamma</a></code>
 function</p>
 
-<p>.. math::
-    \psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z).</p>
+<p>$$\psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z).$$</p>
 
 <p>In this case, <code>trigamma(z) = polygamma(1, z)</code>.</p>
 
@@ -31761,29 +32435,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Trigamma_function
-.. [2] http://mathworld.wolfram.com/TrigammaFunction.html
-.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="trigamma.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#trigamma.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">trigamma</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">trigamma</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#trigamma.__init__"></a>
     
     
 
                             </div>
                             <div id="trigamma.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#trigamma.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#trigamma.default_assumptions"></a>
     
     
 
@@ -31938,37 +32616,37 @@ <h5>Inherited Members</h5>
                 <dd id="trigamma.simplify" class="function">simplify</dd>
                 <dd id="trigamma.refine" class="function">refine</dd>
                 <dd id="trigamma.rewrite" class="function">rewrite</dd>
-                <dd id="trigamma.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="trigamma.is_integer" class="variable">is_integer</dd>
-                <dd id="trigamma.is_complex" class="variable">is_complex</dd>
-                <dd id="trigamma.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="trigamma.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="trigamma.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="trigamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="trigamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="trigamma.is_positive" class="variable">is_positive</dd>
+                <dd id="trigamma.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="trigamma.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="trigamma.is_negative" class="variable">is_negative</dd>
+                <dd id="trigamma.is_composite" class="variable">is_composite</dd>
+                <dd id="trigamma.is_finite" class="variable">is_finite</dd>
                 <dd id="trigamma.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="trigamma.is_zero" class="variable">is_zero</dd>
                 <dd id="trigamma.is_even" class="variable">is_even</dd>
-                <dd id="trigamma.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="trigamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="trigamma.is_commutative" class="variable">is_commutative</dd>
-                <dd id="trigamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="trigamma.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="trigamma.is_real" class="variable">is_real</dd>
-                <dd id="trigamma.is_positive" class="variable">is_positive</dd>
-                <dd id="trigamma.is_finite" class="variable">is_finite</dd>
-                <dd id="trigamma.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="trigamma.is_infinite" class="variable">is_infinite</dd>
-                <dd id="trigamma.is_rational" class="variable">is_rational</dd>
-                <dd id="trigamma.is_composite" class="variable">is_composite</dd>
-                <dd id="trigamma.is_negative" class="variable">is_negative</dd>
-                <dd id="trigamma.is_polar" class="variable">is_polar</dd>
+                <dd id="trigamma.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="trigamma.is_odd" class="variable">is_odd</dd>
-                <dd id="trigamma.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="trigamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="trigamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="trigamma.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="trigamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="trigamma.is_integer" class="variable">is_integer</dd>
+                <dd id="trigamma.is_rational" class="variable">is_rational</dd>
+                <dd id="trigamma.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="trigamma.is_complex" class="variable">is_complex</dd>
                 <dd id="trigamma.is_prime" class="variable">is_prime</dd>
+                <dd id="trigamma.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="trigamma.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="trigamma.is_real" class="variable">is_real</dd>
+                <dd id="trigamma.is_commutative" class="variable">is_commutative</dd>
                 <dd id="trigamma.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="trigamma.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="trigamma.is_polar" class="variable">is_polar</dd>
+                <dd id="trigamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="trigamma.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="trigamma.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -31980,19 +32658,18 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="multigamma">
-                                <div class="attr class">
-        <a class="headerlink" href="#multigamma">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">multigamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.multigamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">multigamma</span><wbr>(<span class="base">sympy.functions.special.gamma_functions.multigamma</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#multigamma"></a>
     
             <div class="docstring"><p>The multivariate gamma function is a generalization of the gamma function</p>
 
-<p>.. math::
-    \Gamma_p(z) = \pi^{p(p-1)/4}\prod_{k=1}^p \Gamma[z + (1 - k)/2].</p>
+<p>$$\Gamma_p(z) = \pi^{p(p-1)/4}\prod_{k=1}^p \Gamma[z + (1 - k)/2].$$</p>
 
 <p>In a special case, <code>multigamma(x, 1) = gamma(x)</code>.</p>
 
@@ -32043,27 +32720,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Multivariate_gamma_function</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="multigamma.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#multigamma.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">multigamma</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">multigamma</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#multigamma.__init__"></a>
     
     
 
                             </div>
                             <div id="multigamma.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#multigamma.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#multigamma.default_assumptions"></a>
     
     
 
@@ -32219,37 +32902,37 @@ <h5>Inherited Members</h5>
                 <dd id="multigamma.simplify" class="function">simplify</dd>
                 <dd id="multigamma.refine" class="function">refine</dd>
                 <dd id="multigamma.rewrite" class="function">rewrite</dd>
-                <dd id="multigamma.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="multigamma.is_integer" class="variable">is_integer</dd>
-                <dd id="multigamma.is_complex" class="variable">is_complex</dd>
-                <dd id="multigamma.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="multigamma.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="multigamma.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="multigamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="multigamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="multigamma.is_positive" class="variable">is_positive</dd>
+                <dd id="multigamma.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="multigamma.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="multigamma.is_negative" class="variable">is_negative</dd>
+                <dd id="multigamma.is_composite" class="variable">is_composite</dd>
+                <dd id="multigamma.is_finite" class="variable">is_finite</dd>
                 <dd id="multigamma.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="multigamma.is_zero" class="variable">is_zero</dd>
                 <dd id="multigamma.is_even" class="variable">is_even</dd>
-                <dd id="multigamma.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="multigamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="multigamma.is_commutative" class="variable">is_commutative</dd>
-                <dd id="multigamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="multigamma.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="multigamma.is_real" class="variable">is_real</dd>
-                <dd id="multigamma.is_positive" class="variable">is_positive</dd>
-                <dd id="multigamma.is_finite" class="variable">is_finite</dd>
-                <dd id="multigamma.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="multigamma.is_infinite" class="variable">is_infinite</dd>
-                <dd id="multigamma.is_rational" class="variable">is_rational</dd>
-                <dd id="multigamma.is_composite" class="variable">is_composite</dd>
-                <dd id="multigamma.is_negative" class="variable">is_negative</dd>
-                <dd id="multigamma.is_polar" class="variable">is_polar</dd>
+                <dd id="multigamma.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="multigamma.is_odd" class="variable">is_odd</dd>
-                <dd id="multigamma.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="multigamma.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="multigamma.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="multigamma.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="multigamma.is_irrational" class="variable">is_irrational</dd>
+                <dd id="multigamma.is_integer" class="variable">is_integer</dd>
+                <dd id="multigamma.is_rational" class="variable">is_rational</dd>
+                <dd id="multigamma.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="multigamma.is_complex" class="variable">is_complex</dd>
                 <dd id="multigamma.is_prime" class="variable">is_prime</dd>
+                <dd id="multigamma.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="multigamma.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="multigamma.is_real" class="variable">is_real</dd>
+                <dd id="multigamma.is_commutative" class="variable">is_commutative</dd>
                 <dd id="multigamma.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="multigamma.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="multigamma.is_polar" class="variable">is_polar</dd>
+                <dd id="multigamma.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="multigamma.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="multigamma.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -32261,14 +32944,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="dirichlet_eta">
-                                <div class="attr class">
-        <a class="headerlink" href="#dirichlet_eta">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">dirichlet_eta</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.dirichlet_eta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">dirichlet_eta</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.dirichlet_eta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#dirichlet_eta"></a>
     
             <div class="docstring"><p>Dirichlet eta function.</p>
 
@@ -32276,7 +32959,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>For $\operatorname{Re}(s) &gt; 0$, this function is defined as</p>
 
-<p>.. math:: \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}.</p>
+<p>$$\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}.$$</p>
 
 <p>It admits a unique analytic continuation to all of $\mathbb{C}$.
 It is an entire, unbranched function.</p>
@@ -32297,27 +32980,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Dirichlet_eta_function</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="dirichlet_eta.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#dirichlet_eta.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">dirichlet_eta</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">dirichlet_eta</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#dirichlet_eta.__init__"></a>
     
     
 
                             </div>
                             <div id="dirichlet_eta.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#dirichlet_eta.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#dirichlet_eta.default_assumptions"></a>
     
     
 
@@ -32472,37 +33161,37 @@ <h5>Inherited Members</h5>
                 <dd id="dirichlet_eta.simplify" class="function">simplify</dd>
                 <dd id="dirichlet_eta.refine" class="function">refine</dd>
                 <dd id="dirichlet_eta.rewrite" class="function">rewrite</dd>
-                <dd id="dirichlet_eta.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="dirichlet_eta.is_integer" class="variable">is_integer</dd>
-                <dd id="dirichlet_eta.is_complex" class="variable">is_complex</dd>
-                <dd id="dirichlet_eta.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="dirichlet_eta.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="dirichlet_eta.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="dirichlet_eta.is_irrational" class="variable">is_irrational</dd>
+                <dd id="dirichlet_eta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="dirichlet_eta.is_positive" class="variable">is_positive</dd>
+                <dd id="dirichlet_eta.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="dirichlet_eta.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="dirichlet_eta.is_negative" class="variable">is_negative</dd>
+                <dd id="dirichlet_eta.is_composite" class="variable">is_composite</dd>
+                <dd id="dirichlet_eta.is_finite" class="variable">is_finite</dd>
                 <dd id="dirichlet_eta.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="dirichlet_eta.is_zero" class="variable">is_zero</dd>
                 <dd id="dirichlet_eta.is_even" class="variable">is_even</dd>
-                <dd id="dirichlet_eta.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="dirichlet_eta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="dirichlet_eta.is_commutative" class="variable">is_commutative</dd>
-                <dd id="dirichlet_eta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="dirichlet_eta.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="dirichlet_eta.is_real" class="variable">is_real</dd>
-                <dd id="dirichlet_eta.is_positive" class="variable">is_positive</dd>
-                <dd id="dirichlet_eta.is_finite" class="variable">is_finite</dd>
-                <dd id="dirichlet_eta.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="dirichlet_eta.is_infinite" class="variable">is_infinite</dd>
-                <dd id="dirichlet_eta.is_rational" class="variable">is_rational</dd>
-                <dd id="dirichlet_eta.is_composite" class="variable">is_composite</dd>
-                <dd id="dirichlet_eta.is_negative" class="variable">is_negative</dd>
-                <dd id="dirichlet_eta.is_polar" class="variable">is_polar</dd>
+                <dd id="dirichlet_eta.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="dirichlet_eta.is_odd" class="variable">is_odd</dd>
-                <dd id="dirichlet_eta.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="dirichlet_eta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="dirichlet_eta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="dirichlet_eta.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="dirichlet_eta.is_irrational" class="variable">is_irrational</dd>
+                <dd id="dirichlet_eta.is_integer" class="variable">is_integer</dd>
+                <dd id="dirichlet_eta.is_rational" class="variable">is_rational</dd>
+                <dd id="dirichlet_eta.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="dirichlet_eta.is_complex" class="variable">is_complex</dd>
                 <dd id="dirichlet_eta.is_prime" class="variable">is_prime</dd>
+                <dd id="dirichlet_eta.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="dirichlet_eta.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="dirichlet_eta.is_real" class="variable">is_real</dd>
+                <dd id="dirichlet_eta.is_commutative" class="variable">is_commutative</dd>
                 <dd id="dirichlet_eta.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="dirichlet_eta.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="dirichlet_eta.is_polar" class="variable">is_polar</dd>
+                <dd id="dirichlet_eta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="dirichlet_eta.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="dirichlet_eta.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -32514,14 +33203,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="zeta">
-                                <div class="attr class">
-        <a class="headerlink" href="#zeta">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">zeta</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.zeta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">zeta</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.zeta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#zeta"></a>
     
             <div class="docstring"><p>Hurwitz zeta function (or Riemann zeta function).</p>
 
@@ -32530,7 +33219,7 @@ <h1 id="explanation">Explanation</h1>
 <p>For $\operatorname{Re}(a) &gt; 0$ and $\operatorname{Re}(s) &gt; 1$, this
 function is defined as</p>
 
-<p>.. math:: \zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},</p>
+<p>$$\zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},$$</p>
 
 <p>where the standard choice of argument for $n + a$ is used. For fixed
 $a$ with $\operatorname{Re}(a) &gt; 0$ the Hurwitz zeta function admits a
@@ -32542,11 +33231,11 @@ <h1 id="explanation">Explanation</h1>
 
 <p>The Hurwitz zeta function is a special case of the Lerch transcendent:</p>
 
-<p>.. math:: \zeta(s, a) = \Phi(1, s, a).</p>
+<p>$$\zeta(s, a) = \Phi(1, s, a).$$</p>
 
 <p>This formula defines an analytic continuation for all possible values of
 $s$ and $a$ (also $\operatorname{Re}(a) &lt; 0$), see the documentation of
-:class:<code><a href="#lerchphi">lerchphi</a></code> for a description of the branching behavior.</p>
+<code><a href="#lerchphi">lerchphi</a></code> for a description of the branching behavior.</p>
 
 <p>If no value is passed for $a$, by this function assumes a default value
 of $a = 1$, yielding the Riemann zeta function.</p>
@@ -32556,7 +33245,7 @@ <h1 id="examples">Examples</h1>
 <p>For $a = 1$ the Hurwitz zeta function reduces to the famous Riemann
 zeta function:</p>
 
-<p>.. math:: \zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.</p>
+<p>$$\zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.$$</p>
 
 <div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
 <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span>
@@ -32587,8 +33276,9 @@ <h1 id="examples">Examples</h1>
 
 <p>The specific formulae are:</p>
 
-<p>.. math:: \zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}
-.. math:: \zeta(-n) = -\frac{B_{n+1}}{n+1}</p>
+<p>$$\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}$$</p>
+
+<p>$$\zeta(-n) = -\frac{B_{n+1}}{n+1}$$</p>
 
 <p>At negative even integers the Riemann zeta function is zero:</p>
 
@@ -32618,7 +33308,7 @@ <h1 id="examples">Examples</h1>
 </code></pre></div>
 
 <p>The Hurwitz zeta function can be expressed in terms of the Lerch
-transcendent, :class:<code>~.lerchphi</code>:</p>
+transcendent, <code>~.lerchphi</code>:</p>
 
 <div class="pdoc-code codehilite"><pre><span></span><code><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lerchphi</span>
 <span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">)</span>
@@ -32631,28 +33321,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://dlmf.nist.gov/25.11
-.. [2] https://en.wikipedia.org/wiki/Hurwitz_zeta_function</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="zeta.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#zeta.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">zeta</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">zeta</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#zeta.__init__"></a>
     
     
 
                             </div>
                             <div id="zeta.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#zeta.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#zeta.default_assumptions"></a>
     
     
 
@@ -32807,37 +33502,37 @@ <h5>Inherited Members</h5>
                 <dd id="zeta.simplify" class="function">simplify</dd>
                 <dd id="zeta.refine" class="function">refine</dd>
                 <dd id="zeta.rewrite" class="function">rewrite</dd>
-                <dd id="zeta.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="zeta.is_integer" class="variable">is_integer</dd>
-                <dd id="zeta.is_complex" class="variable">is_complex</dd>
-                <dd id="zeta.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="zeta.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="zeta.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="zeta.is_irrational" class="variable">is_irrational</dd>
+                <dd id="zeta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="zeta.is_positive" class="variable">is_positive</dd>
+                <dd id="zeta.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="zeta.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="zeta.is_negative" class="variable">is_negative</dd>
+                <dd id="zeta.is_composite" class="variable">is_composite</dd>
+                <dd id="zeta.is_finite" class="variable">is_finite</dd>
                 <dd id="zeta.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="zeta.is_zero" class="variable">is_zero</dd>
                 <dd id="zeta.is_even" class="variable">is_even</dd>
-                <dd id="zeta.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="zeta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="zeta.is_commutative" class="variable">is_commutative</dd>
-                <dd id="zeta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="zeta.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="zeta.is_real" class="variable">is_real</dd>
-                <dd id="zeta.is_positive" class="variable">is_positive</dd>
-                <dd id="zeta.is_finite" class="variable">is_finite</dd>
-                <dd id="zeta.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="zeta.is_infinite" class="variable">is_infinite</dd>
-                <dd id="zeta.is_rational" class="variable">is_rational</dd>
-                <dd id="zeta.is_composite" class="variable">is_composite</dd>
-                <dd id="zeta.is_negative" class="variable">is_negative</dd>
-                <dd id="zeta.is_polar" class="variable">is_polar</dd>
+                <dd id="zeta.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="zeta.is_odd" class="variable">is_odd</dd>
-                <dd id="zeta.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="zeta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="zeta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="zeta.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="zeta.is_irrational" class="variable">is_irrational</dd>
+                <dd id="zeta.is_integer" class="variable">is_integer</dd>
+                <dd id="zeta.is_rational" class="variable">is_rational</dd>
+                <dd id="zeta.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="zeta.is_complex" class="variable">is_complex</dd>
                 <dd id="zeta.is_prime" class="variable">is_prime</dd>
+                <dd id="zeta.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="zeta.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="zeta.is_real" class="variable">is_real</dd>
+                <dd id="zeta.is_commutative" class="variable">is_commutative</dd>
                 <dd id="zeta.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="zeta.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="zeta.is_polar" class="variable">is_polar</dd>
+                <dd id="zeta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="zeta.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="zeta.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -32849,14 +33544,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="lerchphi">
-                                <div class="attr class">
-        <a class="headerlink" href="#lerchphi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">lerchphi</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.lerchphi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">lerchphi</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.lerchphi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#lerchphi"></a>
     
             <div class="docstring"><p>Lerch transcendent (Lerch phi function).</p>
 
@@ -32872,27 +33567,25 @@ <h1 id="explanation">Explanation</h1>
 
 <p>A commonly used related function is the Lerch zeta function, defined by</p>
 
-<p>.. math:: L(q, s, a) = \Phi(e^{2\pi i q}, s, a).</p>
+<p>$$L(q, s, a) = \Phi(e^{2\pi i q}, s, a).$$</p>
 
 <p><strong>Analytic Continuation and Branching Behavior</strong></p>
 
 <p>It can be shown that</p>
 
-<p>.. math:: \Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.</p>
+<p>$$\Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.$$</p>
 
 <p>This provides the analytic continuation to $\operatorname{Re}(a) \le 0$.</p>
 
 <p>Assume now $\operatorname{Re}(a) &gt; 0$. The integral representation</p>
 
-<p>.. math:: \Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}}
-                            \frac{\mathrm{d}t}{\Gamma(s)}</p>
+<p>$$\Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}}\frac{\mathrm{d}t}{\Gamma(s)}$$</p>
 
 <p>provides an analytic continuation to $\mathbb{C} - [1, \infty)$.
 Finally, for $x \in (1, \infty)$ we find</p>
 
-<p>.. math:: \lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a)
-         -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a)
-         = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},</p>
+<p>$$\lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a)-\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a)
+= \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},$$</p>
 
 <p>using the standard branch for both $\log{x}$ and
 $\log{\log{x}}$ (a branch of $\log{\log{x}}$ is needed to
@@ -32956,30 +33649,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions,
-       Vol. I, New York: McGraw-Hill. Section 1.11.
-.. [2] http://dlmf.nist.gov/25.14
-.. [3] https://en.wikipedia.org/wiki/Lerch_transcendent</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="lerchphi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#lerchphi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">lerchphi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">lerchphi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#lerchphi.__init__"></a>
     
     
 
                             </div>
                             <div id="lerchphi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#lerchphi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#lerchphi.default_assumptions"></a>
     
     
 
@@ -33134,37 +33830,37 @@ <h5>Inherited Members</h5>
                 <dd id="lerchphi.simplify" class="function">simplify</dd>
                 <dd id="lerchphi.refine" class="function">refine</dd>
                 <dd id="lerchphi.rewrite" class="function">rewrite</dd>
-                <dd id="lerchphi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="lerchphi.is_integer" class="variable">is_integer</dd>
-                <dd id="lerchphi.is_complex" class="variable">is_complex</dd>
-                <dd id="lerchphi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="lerchphi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="lerchphi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="lerchphi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="lerchphi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="lerchphi.is_positive" class="variable">is_positive</dd>
+                <dd id="lerchphi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="lerchphi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="lerchphi.is_negative" class="variable">is_negative</dd>
+                <dd id="lerchphi.is_composite" class="variable">is_composite</dd>
+                <dd id="lerchphi.is_finite" class="variable">is_finite</dd>
                 <dd id="lerchphi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="lerchphi.is_zero" class="variable">is_zero</dd>
                 <dd id="lerchphi.is_even" class="variable">is_even</dd>
-                <dd id="lerchphi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="lerchphi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="lerchphi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="lerchphi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="lerchphi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="lerchphi.is_real" class="variable">is_real</dd>
-                <dd id="lerchphi.is_positive" class="variable">is_positive</dd>
-                <dd id="lerchphi.is_finite" class="variable">is_finite</dd>
-                <dd id="lerchphi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="lerchphi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="lerchphi.is_rational" class="variable">is_rational</dd>
-                <dd id="lerchphi.is_composite" class="variable">is_composite</dd>
-                <dd id="lerchphi.is_negative" class="variable">is_negative</dd>
-                <dd id="lerchphi.is_polar" class="variable">is_polar</dd>
+                <dd id="lerchphi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="lerchphi.is_odd" class="variable">is_odd</dd>
-                <dd id="lerchphi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="lerchphi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="lerchphi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="lerchphi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="lerchphi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="lerchphi.is_integer" class="variable">is_integer</dd>
+                <dd id="lerchphi.is_rational" class="variable">is_rational</dd>
+                <dd id="lerchphi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="lerchphi.is_complex" class="variable">is_complex</dd>
                 <dd id="lerchphi.is_prime" class="variable">is_prime</dd>
+                <dd id="lerchphi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="lerchphi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="lerchphi.is_real" class="variable">is_real</dd>
+                <dd id="lerchphi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="lerchphi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="lerchphi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="lerchphi.is_polar" class="variable">is_polar</dd>
+                <dd id="lerchphi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="lerchphi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="lerchphi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -33176,14 +33872,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="polylog">
-                                <div class="attr class">
-        <a class="headerlink" href="#polylog">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">polylog</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.polylog</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">polylog</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.polylog</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#polylog"></a>
     
             <div class="docstring"><p>Polylogarithm function.</p>
 
@@ -33192,7 +33888,7 @@ <h1 id="explanation">Explanation</h1>
 <p>For $|z| &lt; 1$ and $s \in \mathbb{C}$, the polylogarithm is
 defined by</p>
 
-<p>.. math:: \operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},</p>
+<p>$$\operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},$$</p>
 
 <p>where the standard branch of the argument is used for $n$. It admits
 an analytic continuation which is branched at $z=1$ (notably not on the
@@ -33201,12 +33897,11 @@ <h1 id="explanation">Explanation</h1>
 <p>The name polylogarithm comes from the fact that for $s=1$, the
 polylogarithm is related to the ordinary logarithm (see examples), and that</p>
 
-<p>.. math:: \operatorname{Li}_{s+1}(z) =
-                \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.</p>
+<p>$$\operatorname{Li}_{s+1}(z) =\int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.$$</p>
 
 <p>The polylogarithm is a special case of the Lerch transcendent:</p>
 
-<p>.. math:: \operatorname{Li}_{s}(z) = z \Phi(z, s, 1).</p>
+<p>$$\operatorname{Li}_{s}(z) = z \Phi(z, s, 1).$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -33255,22 +33950,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="polylog.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#polylog.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">polylog</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">polylog</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#polylog.__init__"></a>
     
     
 
                             </div>
                             <div id="polylog.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#polylog.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#polylog.default_assumptions"></a>
     
     
 
@@ -33425,37 +34122,37 @@ <h5>Inherited Members</h5>
                 <dd id="polylog.simplify" class="function">simplify</dd>
                 <dd id="polylog.refine" class="function">refine</dd>
                 <dd id="polylog.rewrite" class="function">rewrite</dd>
-                <dd id="polylog.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="polylog.is_integer" class="variable">is_integer</dd>
-                <dd id="polylog.is_complex" class="variable">is_complex</dd>
-                <dd id="polylog.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="polylog.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="polylog.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="polylog.is_irrational" class="variable">is_irrational</dd>
+                <dd id="polylog.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="polylog.is_positive" class="variable">is_positive</dd>
+                <dd id="polylog.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="polylog.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="polylog.is_negative" class="variable">is_negative</dd>
+                <dd id="polylog.is_composite" class="variable">is_composite</dd>
+                <dd id="polylog.is_finite" class="variable">is_finite</dd>
                 <dd id="polylog.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="polylog.is_zero" class="variable">is_zero</dd>
                 <dd id="polylog.is_even" class="variable">is_even</dd>
-                <dd id="polylog.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="polylog.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="polylog.is_commutative" class="variable">is_commutative</dd>
-                <dd id="polylog.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="polylog.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="polylog.is_real" class="variable">is_real</dd>
-                <dd id="polylog.is_positive" class="variable">is_positive</dd>
-                <dd id="polylog.is_finite" class="variable">is_finite</dd>
-                <dd id="polylog.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="polylog.is_infinite" class="variable">is_infinite</dd>
-                <dd id="polylog.is_rational" class="variable">is_rational</dd>
-                <dd id="polylog.is_composite" class="variable">is_composite</dd>
-                <dd id="polylog.is_negative" class="variable">is_negative</dd>
-                <dd id="polylog.is_polar" class="variable">is_polar</dd>
+                <dd id="polylog.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="polylog.is_odd" class="variable">is_odd</dd>
-                <dd id="polylog.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="polylog.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="polylog.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="polylog.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="polylog.is_irrational" class="variable">is_irrational</dd>
+                <dd id="polylog.is_integer" class="variable">is_integer</dd>
+                <dd id="polylog.is_rational" class="variable">is_rational</dd>
+                <dd id="polylog.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="polylog.is_complex" class="variable">is_complex</dd>
                 <dd id="polylog.is_prime" class="variable">is_prime</dd>
+                <dd id="polylog.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="polylog.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="polylog.is_real" class="variable">is_real</dd>
+                <dd id="polylog.is_commutative" class="variable">is_commutative</dd>
                 <dd id="polylog.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="polylog.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="polylog.is_polar" class="variable">is_polar</dd>
+                <dd id="polylog.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="polylog.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="polylog.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -33467,14 +34164,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="stieltjes">
-                                <div class="attr class">
-        <a class="headerlink" href="#stieltjes">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">stieltjes</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.stieltjes</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">stieltjes</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.stieltjes</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#stieltjes"></a>
     
             <div class="docstring"><p>Represents Stieltjes constants, $\gamma_{k}$ that occur in
 Laurent Series expansion of the Riemann zeta function.</p>
@@ -33509,27 +34206,33 @@ <h1 id="examples">Examples</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Stieltjes_constants</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="stieltjes.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#stieltjes.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">stieltjes</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">stieltjes</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#stieltjes.__init__"></a>
     
     
 
                             </div>
                             <div id="stieltjes.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#stieltjes.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#stieltjes.default_assumptions"></a>
     
     
 
@@ -33684,37 +34387,37 @@ <h5>Inherited Members</h5>
                 <dd id="stieltjes.simplify" class="function">simplify</dd>
                 <dd id="stieltjes.refine" class="function">refine</dd>
                 <dd id="stieltjes.rewrite" class="function">rewrite</dd>
-                <dd id="stieltjes.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="stieltjes.is_integer" class="variable">is_integer</dd>
-                <dd id="stieltjes.is_complex" class="variable">is_complex</dd>
-                <dd id="stieltjes.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="stieltjes.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="stieltjes.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="stieltjes.is_irrational" class="variable">is_irrational</dd>
+                <dd id="stieltjes.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="stieltjes.is_positive" class="variable">is_positive</dd>
+                <dd id="stieltjes.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="stieltjes.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="stieltjes.is_negative" class="variable">is_negative</dd>
+                <dd id="stieltjes.is_composite" class="variable">is_composite</dd>
+                <dd id="stieltjes.is_finite" class="variable">is_finite</dd>
                 <dd id="stieltjes.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="stieltjes.is_zero" class="variable">is_zero</dd>
                 <dd id="stieltjes.is_even" class="variable">is_even</dd>
-                <dd id="stieltjes.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="stieltjes.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="stieltjes.is_commutative" class="variable">is_commutative</dd>
-                <dd id="stieltjes.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="stieltjes.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="stieltjes.is_real" class="variable">is_real</dd>
-                <dd id="stieltjes.is_positive" class="variable">is_positive</dd>
-                <dd id="stieltjes.is_finite" class="variable">is_finite</dd>
-                <dd id="stieltjes.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="stieltjes.is_infinite" class="variable">is_infinite</dd>
-                <dd id="stieltjes.is_rational" class="variable">is_rational</dd>
-                <dd id="stieltjes.is_composite" class="variable">is_composite</dd>
-                <dd id="stieltjes.is_negative" class="variable">is_negative</dd>
-                <dd id="stieltjes.is_polar" class="variable">is_polar</dd>
+                <dd id="stieltjes.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="stieltjes.is_odd" class="variable">is_odd</dd>
-                <dd id="stieltjes.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="stieltjes.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="stieltjes.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="stieltjes.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="stieltjes.is_irrational" class="variable">is_irrational</dd>
+                <dd id="stieltjes.is_integer" class="variable">is_integer</dd>
+                <dd id="stieltjes.is_rational" class="variable">is_rational</dd>
+                <dd id="stieltjes.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="stieltjes.is_complex" class="variable">is_complex</dd>
                 <dd id="stieltjes.is_prime" class="variable">is_prime</dd>
+                <dd id="stieltjes.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="stieltjes.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="stieltjes.is_real" class="variable">is_real</dd>
+                <dd id="stieltjes.is_commutative" class="variable">is_commutative</dd>
                 <dd id="stieltjes.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="stieltjes.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="stieltjes.is_polar" class="variable">is_polar</dd>
+                <dd id="stieltjes.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="stieltjes.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="stieltjes.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -33726,14 +34429,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="LeviCivita">
-                                <div class="attr class">
-        <a class="headerlink" href="#LeviCivita">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">LeviCivita</span><wbr>(<span class="base">sympy.functions.special.tensor_functions.LeviCivita</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">LeviCivita</span><wbr>(<span class="base">sympy.functions.special.tensor_functions.LeviCivita</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#LeviCivita"></a>
     
             <div class="docstring"><p>Represent the Levi-Civita symbol.</p>
 
@@ -33767,162 +34470,178 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="LeviCivita.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#LeviCivita.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">LeviCivita</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">LeviCivita</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#LeviCivita.__init__"></a>
     
     
 
                             </div>
                             <div id="LeviCivita.is_integer" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_integer">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_integer</span><span class="default_value"> = True</span>
 
-        <span class="name">is_integer</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_integer"></a>
     
     
 
                             </div>
                             <div id="LeviCivita.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;integer&#39;: True, &#39;transcendental&#39;: False, &#39;imaginary&#39;: False, &#39;commutative&#39;: True, &#39;infinite&#39;: False, &#39;irrational&#39;: False, &#39;real&#39;: True, &#39;hermitian&#39;: True, &#39;finite&#39;: True, &#39;extended_real&#39;: True, &#39;algebraic&#39;: True, &#39;complex&#39;: True, &#39;noninteger&#39;: False, &#39;rational&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;integer&#39;: True, &#39;hermitian&#39;: True, &#39;extended_real&#39;: True, &#39;commutative&#39;: True, &#39;complex&#39;: True, &#39;real&#39;: True, &#39;transcendental&#39;: False, &#39;infinite&#39;: False, &#39;imaginary&#39;: False, &#39;algebraic&#39;: True, &#39;finite&#39;: True, &#39;rational&#39;: True, &#39;noninteger&#39;: False, &#39;irrational&#39;: False}</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_hermitian" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_hermitian">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_transcendental" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_transcendental</span><span class="default_value"> = False</span>
 
-        <span class="name">is_hermitian</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_transcendental"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_extended_real">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_imaginary" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_imaginary"></a>
     
     
 
                             </div>
                             <div id="LeviCivita.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_commutative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_commutative"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_complex" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_complex">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_infinite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_infinite</span><span class="default_value"> = False</span>
 
-        <span class="name">is_complex</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_infinite"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_real">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_irrational" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_irrational</span><span class="default_value"> = False</span>
 
-        <span class="name">is_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_irrational"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_transcendental" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_transcendental">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_real" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_transcendental</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_real"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_infinite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_infinite">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_hermitian" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_hermitian</span><span class="default_value"> = True</span>
 
-        <span class="name">is_infinite</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_hermitian"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_imaginary">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_finite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_finite</span><span class="default_value"> = True</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_finite"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_algebraic" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_algebraic">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_extended_real" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_algebraic</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_extended_real"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_finite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_finite">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_algebraic" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_algebraic</span><span class="default_value"> = True</span>
 
-        <span class="name">is_finite</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_algebraic"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_rational" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_rational">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_complex" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_complex</span><span class="default_value"> = True</span>
 
-        <span class="name">is_rational</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_complex"></a>
     
     
 
                             </div>
                             <div id="LeviCivita.is_noninteger" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_noninteger">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_noninteger</span><span class="default_value"> = False</span>
 
-        <span class="name">is_noninteger</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_noninteger"></a>
     
     
 
                             </div>
-                            <div id="LeviCivita.is_irrational" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#LeviCivita.is_irrational">#&nbsp;&nbsp</a>
+                            <div id="LeviCivita.is_rational" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_rational</span><span class="default_value"> = True</span>
 
-        <span class="name">is_irrational</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#LeviCivita.is_rational"></a>
     
     
 
@@ -34077,23 +34796,23 @@ <h5>Inherited Members</h5>
                 <dd id="LeviCivita.simplify" class="function">simplify</dd>
                 <dd id="LeviCivita.refine" class="function">refine</dd>
                 <dd id="LeviCivita.rewrite" class="function">rewrite</dd>
-                <dd id="LeviCivita.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="LeviCivita.is_extended_positive" class="variable">is_extended_positive</dd>
+                <dd id="LeviCivita.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="LeviCivita.is_positive" class="variable">is_positive</dd>
+                <dd id="LeviCivita.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="LeviCivita.is_negative" class="variable">is_negative</dd>
+                <dd id="LeviCivita.is_composite" class="variable">is_composite</dd>
                 <dd id="LeviCivita.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="LeviCivita.is_zero" class="variable">is_zero</dd>
                 <dd id="LeviCivita.is_even" class="variable">is_even</dd>
-                <dd id="LeviCivita.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="LeviCivita.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="LeviCivita.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="LeviCivita.is_positive" class="variable">is_positive</dd>
-                <dd id="LeviCivita.is_extended_negative" class="variable">is_extended_negative</dd>
-                <dd id="LeviCivita.is_composite" class="variable">is_composite</dd>
-                <dd id="LeviCivita.is_negative" class="variable">is_negative</dd>
-                <dd id="LeviCivita.is_polar" class="variable">is_polar</dd>
                 <dd id="LeviCivita.is_odd" class="variable">is_odd</dd>
-                <dd id="LeviCivita.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="LeviCivita.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="LeviCivita.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="LeviCivita.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="LeviCivita.is_prime" class="variable">is_prime</dd>
+                <dd id="LeviCivita.is_polar" class="variable">is_polar</dd>
+                <dd id="LeviCivita.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="LeviCivita.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="LeviCivita.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -34105,14 +34824,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="KroneckerDelta">
-                                <div class="attr class">
-        <a class="headerlink" href="#KroneckerDelta">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">KroneckerDelta</span><wbr>(<span class="base">sympy.functions.special.tensor_functions.KroneckerDelta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">KroneckerDelta</span><wbr>(<span class="base">sympy.functions.special.tensor_functions.KroneckerDelta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta"></a>
     
             <div class="docstring"><p>The discrete, or Kronecker, delta function.</p>
 
@@ -34159,167 +34878,187 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Kronecker_delta</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="KroneckerDelta.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#KroneckerDelta.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">KroneckerDelta</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">KroneckerDelta</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.__init__"></a>
     
     
 
                             </div>
                             <div id="KroneckerDelta.is_integer" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_integer">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_integer</span><span class="default_value"> = True</span>
 
-        <span class="name">is_integer</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_integer"></a>
     
     
 
                             </div>
                             <div id="KroneckerDelta.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;integer&#39;: True, &#39;transcendental&#39;: False, &#39;imaginary&#39;: False, &#39;commutative&#39;: True, &#39;infinite&#39;: False, &#39;irrational&#39;: False, &#39;real&#39;: True, &#39;hermitian&#39;: True, &#39;finite&#39;: True, &#39;extended_real&#39;: True, &#39;algebraic&#39;: True, &#39;complex&#39;: True, &#39;noninteger&#39;: False, &#39;rational&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;integer&#39;: True, &#39;hermitian&#39;: True, &#39;extended_real&#39;: True, &#39;commutative&#39;: True, &#39;complex&#39;: True, &#39;real&#39;: True, &#39;transcendental&#39;: False, &#39;infinite&#39;: False, &#39;imaginary&#39;: False, &#39;algebraic&#39;: True, &#39;finite&#39;: True, &#39;rational&#39;: True, &#39;noninteger&#39;: False, &#39;irrational&#39;: False}</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_hermitian" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_hermitian">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_transcendental" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_transcendental</span><span class="default_value"> = False</span>
 
-        <span class="name">is_hermitian</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_transcendental"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_extended_real">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_imaginary" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_imaginary"></a>
     
     
 
                             </div>
                             <div id="KroneckerDelta.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_commutative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_commutative"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_complex" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_complex">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_infinite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_infinite</span><span class="default_value"> = False</span>
 
-        <span class="name">is_complex</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_infinite"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_real">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_irrational" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_irrational</span><span class="default_value"> = False</span>
 
-        <span class="name">is_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_irrational"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_transcendental" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_transcendental">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_real" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_transcendental</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_real"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_infinite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_infinite">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_hermitian" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_hermitian</span><span class="default_value"> = True</span>
 
-        <span class="name">is_infinite</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_hermitian"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_imaginary">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_finite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_finite</span><span class="default_value"> = True</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_finite"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_algebraic" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_algebraic">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_extended_real" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_algebraic</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_extended_real"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_finite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_finite">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_algebraic" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_algebraic</span><span class="default_value"> = True</span>
 
-        <span class="name">is_finite</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_algebraic"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_rational" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_rational">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_complex" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_complex</span><span class="default_value"> = True</span>
 
-        <span class="name">is_rational</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_complex"></a>
     
     
 
                             </div>
                             <div id="KroneckerDelta.is_noninteger" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_noninteger">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_noninteger</span><span class="default_value"> = False</span>
 
-        <span class="name">is_noninteger</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_noninteger"></a>
     
     
 
                             </div>
-                            <div id="KroneckerDelta.is_irrational" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#KroneckerDelta.is_irrational">#&nbsp;&nbsp</a>
+                            <div id="KroneckerDelta.is_rational" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_rational</span><span class="default_value"> = True</span>
 
-        <span class="name">is_irrational</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#KroneckerDelta.is_rational"></a>
     
     
 
@@ -34483,23 +35222,23 @@ <h5>Inherited Members</h5>
                 <dd id="KroneckerDelta.simplify" class="function">simplify</dd>
                 <dd id="KroneckerDelta.refine" class="function">refine</dd>
                 <dd id="KroneckerDelta.rewrite" class="function">rewrite</dd>
-                <dd id="KroneckerDelta.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="KroneckerDelta.is_extended_positive" class="variable">is_extended_positive</dd>
+                <dd id="KroneckerDelta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="KroneckerDelta.is_positive" class="variable">is_positive</dd>
+                <dd id="KroneckerDelta.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="KroneckerDelta.is_negative" class="variable">is_negative</dd>
+                <dd id="KroneckerDelta.is_composite" class="variable">is_composite</dd>
                 <dd id="KroneckerDelta.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="KroneckerDelta.is_zero" class="variable">is_zero</dd>
                 <dd id="KroneckerDelta.is_even" class="variable">is_even</dd>
-                <dd id="KroneckerDelta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="KroneckerDelta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="KroneckerDelta.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="KroneckerDelta.is_positive" class="variable">is_positive</dd>
-                <dd id="KroneckerDelta.is_extended_negative" class="variable">is_extended_negative</dd>
-                <dd id="KroneckerDelta.is_composite" class="variable">is_composite</dd>
-                <dd id="KroneckerDelta.is_negative" class="variable">is_negative</dd>
-                <dd id="KroneckerDelta.is_polar" class="variable">is_polar</dd>
                 <dd id="KroneckerDelta.is_odd" class="variable">is_odd</dd>
-                <dd id="KroneckerDelta.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="KroneckerDelta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="KroneckerDelta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="KroneckerDelta.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="KroneckerDelta.is_prime" class="variable">is_prime</dd>
+                <dd id="KroneckerDelta.is_polar" class="variable">is_polar</dd>
+                <dd id="KroneckerDelta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="KroneckerDelta.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="KroneckerDelta.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -34511,14 +35250,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="SingularityFunction">
-                                <div class="attr class">
-        <a class="headerlink" href="#SingularityFunction">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">SingularityFunction</span><wbr>(<span class="base">sympy.functions.special.singularity_functions.SingularityFunction</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">SingularityFunction</span><wbr>(<span class="base">sympy.functions.special.singularity_functions.SingularityFunction</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#SingularityFunction"></a>
     
             <div class="docstring"><p>Singularity functions are a class of discontinuous functions.</p>
 
@@ -34584,107 +35323,121 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Singularity_function</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="SingularityFunction.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#SingularityFunction.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">SingularityFunction</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">SingularityFunction</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.__init__"></a>
     
     
 
                             </div>
                             <div id="SingularityFunction.is_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.is_real">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.is_real"></a>
     
     
 
                             </div>
                             <div id="SingularityFunction.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;real&#39;: True, &#39;imaginary&#39;: False, &#39;commutative&#39;: True, &#39;infinite&#39;: False, &#39;finite&#39;: True, &#39;hermitian&#39;: True, &#39;extended_real&#39;: True, &#39;complex&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;real&#39;: True, &#39;hermitian&#39;: True, &#39;commutative&#39;: True, &#39;extended_real&#39;: True, &#39;complex&#39;: True, &#39;infinite&#39;: False, &#39;imaginary&#39;: False, &#39;finite&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="SingularityFunction.is_hermitian" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.is_hermitian">#&nbsp;&nbsp</a>
+                            <div id="SingularityFunction.is_imaginary" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_hermitian</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.is_imaginary"></a>
     
     
 
                             </div>
                             <div id="SingularityFunction.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.is_commutative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.is_commutative"></a>
     
     
 
                             </div>
-                            <div id="SingularityFunction.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.is_extended_real">#&nbsp;&nbsp</a>
+                            <div id="SingularityFunction.is_infinite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_infinite</span><span class="default_value"> = False</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.is_infinite"></a>
     
     
 
                             </div>
-                            <div id="SingularityFunction.is_complex" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.is_complex">#&nbsp;&nbsp</a>
+                            <div id="SingularityFunction.is_finite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_finite</span><span class="default_value"> = True</span>
 
-        <span class="name">is_complex</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.is_finite"></a>
     
     
 
                             </div>
-                            <div id="SingularityFunction.is_infinite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.is_infinite">#&nbsp;&nbsp</a>
+                            <div id="SingularityFunction.is_hermitian" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_hermitian</span><span class="default_value"> = True</span>
 
-        <span class="name">is_infinite</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.is_hermitian"></a>
     
     
 
                             </div>
-                            <div id="SingularityFunction.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.is_imaginary">#&nbsp;&nbsp</a>
+                            <div id="SingularityFunction.is_extended_real" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.is_extended_real"></a>
     
     
 
                             </div>
-                            <div id="SingularityFunction.is_finite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#SingularityFunction.is_finite">#&nbsp;&nbsp</a>
+                            <div id="SingularityFunction.is_complex" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_complex</span><span class="default_value"> = True</span>
 
-        <span class="name">is_finite</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#SingularityFunction.is_complex"></a>
     
     
 
@@ -34839,29 +35592,29 @@ <h5>Inherited Members</h5>
                 <dd id="SingularityFunction.simplify" class="function">simplify</dd>
                 <dd id="SingularityFunction.refine" class="function">refine</dd>
                 <dd id="SingularityFunction.rewrite" class="function">rewrite</dd>
-                <dd id="SingularityFunction.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="SingularityFunction.is_integer" class="variable">is_integer</dd>
-                <dd id="SingularityFunction.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="SingularityFunction.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="SingularityFunction.is_irrational" class="variable">is_irrational</dd>
+                <dd id="SingularityFunction.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="SingularityFunction.is_positive" class="variable">is_positive</dd>
+                <dd id="SingularityFunction.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="SingularityFunction.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="SingularityFunction.is_negative" class="variable">is_negative</dd>
+                <dd id="SingularityFunction.is_composite" class="variable">is_composite</dd>
                 <dd id="SingularityFunction.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="SingularityFunction.is_zero" class="variable">is_zero</dd>
                 <dd id="SingularityFunction.is_even" class="variable">is_even</dd>
-                <dd id="SingularityFunction.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="SingularityFunction.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="SingularityFunction.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="SingularityFunction.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="SingularityFunction.is_positive" class="variable">is_positive</dd>
+                <dd id="SingularityFunction.is_odd" class="variable">is_odd</dd>
+                <dd id="SingularityFunction.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="SingularityFunction.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="SingularityFunction.is_irrational" class="variable">is_irrational</dd>
+                <dd id="SingularityFunction.is_integer" class="variable">is_integer</dd>
                 <dd id="SingularityFunction.is_rational" class="variable">is_rational</dd>
-                <dd id="SingularityFunction.is_composite" class="variable">is_composite</dd>
-                <dd id="SingularityFunction.is_negative" class="variable">is_negative</dd>
-                <dd id="SingularityFunction.is_polar" class="variable">is_polar</dd>
-                <dd id="SingularityFunction.is_odd" class="variable">is_odd</dd>
-                <dd id="SingularityFunction.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="SingularityFunction.is_nonzero" class="variable">is_nonzero</dd>
                 <dd id="SingularityFunction.is_prime" class="variable">is_prime</dd>
+                <dd id="SingularityFunction.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="SingularityFunction.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="SingularityFunction.is_polar" class="variable">is_polar</dd>
+                <dd id="SingularityFunction.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="SingularityFunction.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="SingularityFunction.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -34873,14 +35626,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="DiracDelta">
-                                <div class="attr class">
-        <a class="headerlink" href="#DiracDelta">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">DiracDelta</span><wbr>(<span class="base">sympy.functions.special.delta_functions.DiracDelta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">DiracDelta</span><wbr>(<span class="base">sympy.functions.special.delta_functions.DiracDelta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#DiracDelta"></a>
     
             <div class="docstring"><p>The DiracDelta function and its derivatives.</p>
 
@@ -34958,107 +35711,121 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://mathworld.wolfram.com/DeltaFunction.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="DiracDelta.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#DiracDelta.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">DiracDelta</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">DiracDelta</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#DiracDelta.__init__"></a>
     
     
 
                             </div>
                             <div id="DiracDelta.is_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.is_real">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.is_real"></a>
     
     
 
                             </div>
                             <div id="DiracDelta.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;real&#39;: True, &#39;imaginary&#39;: False, &#39;commutative&#39;: True, &#39;infinite&#39;: False, &#39;finite&#39;: True, &#39;hermitian&#39;: True, &#39;extended_real&#39;: True, &#39;complex&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;real&#39;: True, &#39;hermitian&#39;: True, &#39;commutative&#39;: True, &#39;extended_real&#39;: True, &#39;complex&#39;: True, &#39;infinite&#39;: False, &#39;imaginary&#39;: False, &#39;finite&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="DiracDelta.is_hermitian" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.is_hermitian">#&nbsp;&nbsp</a>
+                            <div id="DiracDelta.is_imaginary" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_hermitian</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.is_imaginary"></a>
     
     
 
                             </div>
                             <div id="DiracDelta.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.is_commutative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.is_commutative"></a>
     
     
 
                             </div>
-                            <div id="DiracDelta.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.is_extended_real">#&nbsp;&nbsp</a>
+                            <div id="DiracDelta.is_infinite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_infinite</span><span class="default_value"> = False</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.is_infinite"></a>
     
     
 
                             </div>
-                            <div id="DiracDelta.is_complex" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.is_complex">#&nbsp;&nbsp</a>
+                            <div id="DiracDelta.is_finite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_finite</span><span class="default_value"> = True</span>
 
-        <span class="name">is_complex</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.is_finite"></a>
     
     
 
                             </div>
-                            <div id="DiracDelta.is_infinite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.is_infinite">#&nbsp;&nbsp</a>
+                            <div id="DiracDelta.is_hermitian" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_hermitian</span><span class="default_value"> = True</span>
 
-        <span class="name">is_infinite</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.is_hermitian"></a>
     
     
 
                             </div>
-                            <div id="DiracDelta.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.is_imaginary">#&nbsp;&nbsp</a>
+                            <div id="DiracDelta.is_extended_real" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.is_extended_real"></a>
     
     
 
                             </div>
-                            <div id="DiracDelta.is_finite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#DiracDelta.is_finite">#&nbsp;&nbsp</a>
+                            <div id="DiracDelta.is_complex" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_complex</span><span class="default_value"> = True</span>
 
-        <span class="name">is_finite</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#DiracDelta.is_complex"></a>
     
     
 
@@ -35214,29 +35981,29 @@ <h5>Inherited Members</h5>
                 <dd id="DiracDelta.simplify" class="function">simplify</dd>
                 <dd id="DiracDelta.refine" class="function">refine</dd>
                 <dd id="DiracDelta.rewrite" class="function">rewrite</dd>
-                <dd id="DiracDelta.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="DiracDelta.is_integer" class="variable">is_integer</dd>
-                <dd id="DiracDelta.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="DiracDelta.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="DiracDelta.is_irrational" class="variable">is_irrational</dd>
+                <dd id="DiracDelta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="DiracDelta.is_positive" class="variable">is_positive</dd>
+                <dd id="DiracDelta.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="DiracDelta.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="DiracDelta.is_negative" class="variable">is_negative</dd>
+                <dd id="DiracDelta.is_composite" class="variable">is_composite</dd>
                 <dd id="DiracDelta.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="DiracDelta.is_zero" class="variable">is_zero</dd>
                 <dd id="DiracDelta.is_even" class="variable">is_even</dd>
-                <dd id="DiracDelta.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="DiracDelta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="DiracDelta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="DiracDelta.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="DiracDelta.is_positive" class="variable">is_positive</dd>
+                <dd id="DiracDelta.is_odd" class="variable">is_odd</dd>
+                <dd id="DiracDelta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="DiracDelta.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="DiracDelta.is_irrational" class="variable">is_irrational</dd>
+                <dd id="DiracDelta.is_integer" class="variable">is_integer</dd>
                 <dd id="DiracDelta.is_rational" class="variable">is_rational</dd>
-                <dd id="DiracDelta.is_composite" class="variable">is_composite</dd>
-                <dd id="DiracDelta.is_negative" class="variable">is_negative</dd>
-                <dd id="DiracDelta.is_polar" class="variable">is_polar</dd>
-                <dd id="DiracDelta.is_odd" class="variable">is_odd</dd>
-                <dd id="DiracDelta.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="DiracDelta.is_nonzero" class="variable">is_nonzero</dd>
                 <dd id="DiracDelta.is_prime" class="variable">is_prime</dd>
+                <dd id="DiracDelta.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="DiracDelta.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="DiracDelta.is_polar" class="variable">is_polar</dd>
+                <dd id="DiracDelta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="DiracDelta.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="DiracDelta.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -35248,14 +36015,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Heaviside">
-                                <div class="attr class">
-        <a class="headerlink" href="#Heaviside">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Heaviside</span><wbr>(<span class="base">sympy.functions.special.delta_functions.Heaviside</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Heaviside</span><wbr>(<span class="base">sympy.functions.special.delta_functions.Heaviside</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Heaviside"></a>
     
             <div class="docstring"><p>Heaviside step function.</p>
 
@@ -35279,7 +36046,7 @@ <h1 id="explanation">Explanation</h1>
 can be given. Using <code>Heaviside(x, nan)</code> gives an expression that will
 evaluate to nan for x=0.</p>
 
-<p>.. versionchanged:: 1.9 <code>Heaviside(0)</code> now returns 1/2 (before: undefined)</p>
+<p><em>Changed in version 1.9 <code>Heaviside(0)</code> now returns 1/2 (before: undefined).</em></p>
 
 <h1 id="examples">Examples</h1>
 
@@ -35303,108 +36070,121 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://mathworld.wolfram.com/HeavisideStepFunction.html
-.. [2] http://dlmf.nist.gov/1.16#iv</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Heaviside.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Heaviside.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Heaviside</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Heaviside</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Heaviside.__init__"></a>
     
     
 
                             </div>
                             <div id="Heaviside.is_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.is_real">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.is_real"></a>
     
     
 
                             </div>
                             <div id="Heaviside.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;real&#39;: True, &#39;imaginary&#39;: False, &#39;commutative&#39;: True, &#39;infinite&#39;: False, &#39;finite&#39;: True, &#39;hermitian&#39;: True, &#39;extended_real&#39;: True, &#39;complex&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;real&#39;: True, &#39;hermitian&#39;: True, &#39;commutative&#39;: True, &#39;extended_real&#39;: True, &#39;complex&#39;: True, &#39;infinite&#39;: False, &#39;imaginary&#39;: False, &#39;finite&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.default_assumptions"></a>
     
     
 
                             </div>
-                            <div id="Heaviside.is_hermitian" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.is_hermitian">#&nbsp;&nbsp</a>
+                            <div id="Heaviside.is_imaginary" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_imaginary</span><span class="default_value"> = False</span>
 
-        <span class="name">is_hermitian</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.is_imaginary"></a>
     
     
 
                             </div>
                             <div id="Heaviside.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.is_commutative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.is_commutative"></a>
     
     
 
                             </div>
-                            <div id="Heaviside.is_extended_real" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.is_extended_real">#&nbsp;&nbsp</a>
+                            <div id="Heaviside.is_infinite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_infinite</span><span class="default_value"> = False</span>
 
-        <span class="name">is_extended_real</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.is_infinite"></a>
     
     
 
                             </div>
-                            <div id="Heaviside.is_complex" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.is_complex">#&nbsp;&nbsp</a>
+                            <div id="Heaviside.is_finite" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_finite</span><span class="default_value"> = True</span>
 
-        <span class="name">is_complex</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.is_finite"></a>
     
     
 
                             </div>
-                            <div id="Heaviside.is_infinite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.is_infinite">#&nbsp;&nbsp</a>
+                            <div id="Heaviside.is_hermitian" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_hermitian</span><span class="default_value"> = True</span>
 
-        <span class="name">is_infinite</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.is_hermitian"></a>
     
     
 
                             </div>
-                            <div id="Heaviside.is_imaginary" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.is_imaginary">#&nbsp;&nbsp</a>
+                            <div id="Heaviside.is_extended_real" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_extended_real</span><span class="default_value"> = True</span>
 
-        <span class="name">is_imaginary</span><span class="default_value"> = False</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.is_extended_real"></a>
     
     
 
                             </div>
-                            <div id="Heaviside.is_finite" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Heaviside.is_finite">#&nbsp;&nbsp</a>
+                            <div id="Heaviside.is_complex" class="classattr">
+                                <div class="attr variable">
+            <span class="name">is_complex</span><span class="default_value"> = True</span>
 
-        <span class="name">is_finite</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#Heaviside.is_complex"></a>
     
     
 
@@ -35560,29 +36340,29 @@ <h5>Inherited Members</h5>
                 <dd id="Heaviside.simplify" class="function">simplify</dd>
                 <dd id="Heaviside.refine" class="function">refine</dd>
                 <dd id="Heaviside.rewrite" class="function">rewrite</dd>
-                <dd id="Heaviside.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Heaviside.is_integer" class="variable">is_integer</dd>
-                <dd id="Heaviside.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Heaviside.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Heaviside.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Heaviside.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Heaviside.is_positive" class="variable">is_positive</dd>
+                <dd id="Heaviside.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Heaviside.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Heaviside.is_negative" class="variable">is_negative</dd>
+                <dd id="Heaviside.is_composite" class="variable">is_composite</dd>
                 <dd id="Heaviside.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Heaviside.is_zero" class="variable">is_zero</dd>
                 <dd id="Heaviside.is_even" class="variable">is_even</dd>
-                <dd id="Heaviside.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Heaviside.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Heaviside.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Heaviside.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Heaviside.is_positive" class="variable">is_positive</dd>
+                <dd id="Heaviside.is_odd" class="variable">is_odd</dd>
+                <dd id="Heaviside.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Heaviside.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Heaviside.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Heaviside.is_integer" class="variable">is_integer</dd>
                 <dd id="Heaviside.is_rational" class="variable">is_rational</dd>
-                <dd id="Heaviside.is_composite" class="variable">is_composite</dd>
-                <dd id="Heaviside.is_negative" class="variable">is_negative</dd>
-                <dd id="Heaviside.is_polar" class="variable">is_polar</dd>
-                <dd id="Heaviside.is_odd" class="variable">is_odd</dd>
-                <dd id="Heaviside.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Heaviside.is_nonzero" class="variable">is_nonzero</dd>
                 <dd id="Heaviside.is_prime" class="variable">is_prime</dd>
+                <dd id="Heaviside.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Heaviside.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Heaviside.is_polar" class="variable">is_polar</dd>
+                <dd id="Heaviside.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Heaviside.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Heaviside.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -35594,14 +36374,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="besselj">
-                                <div class="attr class">
-        <a class="headerlink" href="#besselj">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">besselj</span><wbr>(<span class="base">sympy.functions.special.bessel.besselj</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">besselj</span><wbr>(<span class="base">sympy.functions.special.bessel.besselj</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#besselj"></a>
     
             <div class="docstring"><p>Bessel function of the first kind.</p>
 
@@ -35660,33 +36440,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9",
-       Handbook of Mathematical Functions with Formulas, Graphs, and
-       Mathematical Tables
-.. [2] Luke, Y. L. (1969), The Special Functions and Their
-       Approximations, Volume 1
-.. [3] https://en.wikipedia.org/wiki/Bessel_function
-.. [4] http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="besselj.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#besselj.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">besselj</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">besselj</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#besselj.__init__"></a>
     
     
 
                             </div>
                             <div id="besselj.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#besselj.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#besselj.default_assumptions"></a>
     
     
 
@@ -35846,37 +36626,37 @@ <h5>Inherited Members</h5>
                 <dd id="besselj.simplify" class="function">simplify</dd>
                 <dd id="besselj.refine" class="function">refine</dd>
                 <dd id="besselj.rewrite" class="function">rewrite</dd>
-                <dd id="besselj.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="besselj.is_integer" class="variable">is_integer</dd>
-                <dd id="besselj.is_complex" class="variable">is_complex</dd>
-                <dd id="besselj.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="besselj.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="besselj.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="besselj.is_irrational" class="variable">is_irrational</dd>
+                <dd id="besselj.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="besselj.is_positive" class="variable">is_positive</dd>
+                <dd id="besselj.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="besselj.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="besselj.is_negative" class="variable">is_negative</dd>
+                <dd id="besselj.is_composite" class="variable">is_composite</dd>
+                <dd id="besselj.is_finite" class="variable">is_finite</dd>
                 <dd id="besselj.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="besselj.is_zero" class="variable">is_zero</dd>
                 <dd id="besselj.is_even" class="variable">is_even</dd>
-                <dd id="besselj.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="besselj.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="besselj.is_commutative" class="variable">is_commutative</dd>
-                <dd id="besselj.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="besselj.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="besselj.is_real" class="variable">is_real</dd>
-                <dd id="besselj.is_positive" class="variable">is_positive</dd>
-                <dd id="besselj.is_finite" class="variable">is_finite</dd>
-                <dd id="besselj.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="besselj.is_infinite" class="variable">is_infinite</dd>
-                <dd id="besselj.is_rational" class="variable">is_rational</dd>
-                <dd id="besselj.is_composite" class="variable">is_composite</dd>
-                <dd id="besselj.is_negative" class="variable">is_negative</dd>
-                <dd id="besselj.is_polar" class="variable">is_polar</dd>
+                <dd id="besselj.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="besselj.is_odd" class="variable">is_odd</dd>
-                <dd id="besselj.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="besselj.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="besselj.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="besselj.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="besselj.is_irrational" class="variable">is_irrational</dd>
+                <dd id="besselj.is_integer" class="variable">is_integer</dd>
+                <dd id="besselj.is_rational" class="variable">is_rational</dd>
+                <dd id="besselj.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="besselj.is_complex" class="variable">is_complex</dd>
                 <dd id="besselj.is_prime" class="variable">is_prime</dd>
+                <dd id="besselj.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="besselj.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="besselj.is_real" class="variable">is_real</dd>
+                <dd id="besselj.is_commutative" class="variable">is_commutative</dd>
                 <dd id="besselj.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="besselj.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="besselj.is_polar" class="variable">is_polar</dd>
+                <dd id="besselj.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="besselj.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="besselj.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -35888,14 +36668,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="bessely">
-                                <div class="attr class">
-        <a class="headerlink" href="#bessely">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">bessely</span><wbr>(<span class="base">sympy.functions.special.bessel.bessely</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">bessely</span><wbr>(<span class="base">sympy.functions.special.bessel.bessely</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#bessely"></a>
     
             <div class="docstring"><p>Bessel function of the second kind.</p>
 
@@ -35929,27 +36709,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="bessely.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#bessely.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">bessely</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">bessely</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#bessely.__init__"></a>
     
     
 
                             </div>
                             <div id="bessely.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#bessely.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#bessely.default_assumptions"></a>
     
     
 
@@ -36109,37 +36895,37 @@ <h5>Inherited Members</h5>
                 <dd id="bessely.simplify" class="function">simplify</dd>
                 <dd id="bessely.refine" class="function">refine</dd>
                 <dd id="bessely.rewrite" class="function">rewrite</dd>
-                <dd id="bessely.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="bessely.is_integer" class="variable">is_integer</dd>
-                <dd id="bessely.is_complex" class="variable">is_complex</dd>
-                <dd id="bessely.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="bessely.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="bessely.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="bessely.is_irrational" class="variable">is_irrational</dd>
+                <dd id="bessely.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="bessely.is_positive" class="variable">is_positive</dd>
+                <dd id="bessely.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="bessely.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="bessely.is_negative" class="variable">is_negative</dd>
+                <dd id="bessely.is_composite" class="variable">is_composite</dd>
+                <dd id="bessely.is_finite" class="variable">is_finite</dd>
                 <dd id="bessely.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="bessely.is_zero" class="variable">is_zero</dd>
                 <dd id="bessely.is_even" class="variable">is_even</dd>
-                <dd id="bessely.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="bessely.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="bessely.is_commutative" class="variable">is_commutative</dd>
-                <dd id="bessely.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="bessely.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="bessely.is_real" class="variable">is_real</dd>
-                <dd id="bessely.is_positive" class="variable">is_positive</dd>
-                <dd id="bessely.is_finite" class="variable">is_finite</dd>
-                <dd id="bessely.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="bessely.is_infinite" class="variable">is_infinite</dd>
-                <dd id="bessely.is_rational" class="variable">is_rational</dd>
-                <dd id="bessely.is_composite" class="variable">is_composite</dd>
-                <dd id="bessely.is_negative" class="variable">is_negative</dd>
-                <dd id="bessely.is_polar" class="variable">is_polar</dd>
+                <dd id="bessely.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="bessely.is_odd" class="variable">is_odd</dd>
-                <dd id="bessely.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="bessely.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="bessely.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="bessely.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="bessely.is_irrational" class="variable">is_irrational</dd>
+                <dd id="bessely.is_integer" class="variable">is_integer</dd>
+                <dd id="bessely.is_rational" class="variable">is_rational</dd>
+                <dd id="bessely.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="bessely.is_complex" class="variable">is_complex</dd>
                 <dd id="bessely.is_prime" class="variable">is_prime</dd>
+                <dd id="bessely.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="bessely.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="bessely.is_real" class="variable">is_real</dd>
+                <dd id="bessely.is_commutative" class="variable">is_commutative</dd>
                 <dd id="bessely.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="bessely.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="bessely.is_polar" class="variable">is_polar</dd>
+                <dd id="bessely.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="bessely.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="bessely.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -36151,14 +36937,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="besseli">
-                                <div class="attr class">
-        <a class="headerlink" href="#besseli">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">besseli</span><wbr>(<span class="base">sympy.functions.special.bessel.besseli</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">besseli</span><wbr>(<span class="base">sympy.functions.special.bessel.besseli</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#besseli"></a>
     
             <div class="docstring"><p>Modified Bessel function of the first kind.</p>
 
@@ -36191,27 +36977,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="besseli.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#besseli.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">besseli</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">besseli</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#besseli.__init__"></a>
     
     
 
                             </div>
                             <div id="besseli.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#besseli.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#besseli.default_assumptions"></a>
     
     
 
@@ -36371,37 +37163,37 @@ <h5>Inherited Members</h5>
                 <dd id="besseli.simplify" class="function">simplify</dd>
                 <dd id="besseli.refine" class="function">refine</dd>
                 <dd id="besseli.rewrite" class="function">rewrite</dd>
-                <dd id="besseli.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="besseli.is_integer" class="variable">is_integer</dd>
-                <dd id="besseli.is_complex" class="variable">is_complex</dd>
-                <dd id="besseli.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="besseli.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="besseli.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="besseli.is_irrational" class="variable">is_irrational</dd>
+                <dd id="besseli.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="besseli.is_positive" class="variable">is_positive</dd>
+                <dd id="besseli.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="besseli.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="besseli.is_negative" class="variable">is_negative</dd>
+                <dd id="besseli.is_composite" class="variable">is_composite</dd>
+                <dd id="besseli.is_finite" class="variable">is_finite</dd>
                 <dd id="besseli.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="besseli.is_zero" class="variable">is_zero</dd>
                 <dd id="besseli.is_even" class="variable">is_even</dd>
-                <dd id="besseli.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="besseli.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="besseli.is_commutative" class="variable">is_commutative</dd>
-                <dd id="besseli.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="besseli.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="besseli.is_real" class="variable">is_real</dd>
-                <dd id="besseli.is_positive" class="variable">is_positive</dd>
-                <dd id="besseli.is_finite" class="variable">is_finite</dd>
-                <dd id="besseli.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="besseli.is_infinite" class="variable">is_infinite</dd>
-                <dd id="besseli.is_rational" class="variable">is_rational</dd>
-                <dd id="besseli.is_composite" class="variable">is_composite</dd>
-                <dd id="besseli.is_negative" class="variable">is_negative</dd>
-                <dd id="besseli.is_polar" class="variable">is_polar</dd>
+                <dd id="besseli.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="besseli.is_odd" class="variable">is_odd</dd>
-                <dd id="besseli.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="besseli.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="besseli.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="besseli.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="besseli.is_irrational" class="variable">is_irrational</dd>
+                <dd id="besseli.is_integer" class="variable">is_integer</dd>
+                <dd id="besseli.is_rational" class="variable">is_rational</dd>
+                <dd id="besseli.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="besseli.is_complex" class="variable">is_complex</dd>
                 <dd id="besseli.is_prime" class="variable">is_prime</dd>
+                <dd id="besseli.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="besseli.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="besseli.is_real" class="variable">is_real</dd>
+                <dd id="besseli.is_commutative" class="variable">is_commutative</dd>
                 <dd id="besseli.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="besseli.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="besseli.is_polar" class="variable">is_polar</dd>
+                <dd id="besseli.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="besseli.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="besseli.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -36413,14 +37205,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="besselk">
-                                <div class="attr class">
-        <a class="headerlink" href="#besselk">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">besselk</span><wbr>(<span class="base">sympy.functions.special.bessel.besselk</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">besselk</span><wbr>(<span class="base">sympy.functions.special.bessel.besselk</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#besselk"></a>
     
             <div class="docstring"><p>Modified Bessel function of the second kind.</p>
 
@@ -36451,27 +37243,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="besselk.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#besselk.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">besselk</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">besselk</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#besselk.__init__"></a>
     
     
 
                             </div>
                             <div id="besselk.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#besselk.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#besselk.default_assumptions"></a>
     
     
 
@@ -36631,37 +37429,37 @@ <h5>Inherited Members</h5>
                 <dd id="besselk.simplify" class="function">simplify</dd>
                 <dd id="besselk.refine" class="function">refine</dd>
                 <dd id="besselk.rewrite" class="function">rewrite</dd>
-                <dd id="besselk.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="besselk.is_integer" class="variable">is_integer</dd>
-                <dd id="besselk.is_complex" class="variable">is_complex</dd>
-                <dd id="besselk.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="besselk.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="besselk.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="besselk.is_irrational" class="variable">is_irrational</dd>
+                <dd id="besselk.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="besselk.is_positive" class="variable">is_positive</dd>
+                <dd id="besselk.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="besselk.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="besselk.is_negative" class="variable">is_negative</dd>
+                <dd id="besselk.is_composite" class="variable">is_composite</dd>
+                <dd id="besselk.is_finite" class="variable">is_finite</dd>
                 <dd id="besselk.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="besselk.is_zero" class="variable">is_zero</dd>
                 <dd id="besselk.is_even" class="variable">is_even</dd>
-                <dd id="besselk.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="besselk.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="besselk.is_commutative" class="variable">is_commutative</dd>
-                <dd id="besselk.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="besselk.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="besselk.is_real" class="variable">is_real</dd>
-                <dd id="besselk.is_positive" class="variable">is_positive</dd>
-                <dd id="besselk.is_finite" class="variable">is_finite</dd>
-                <dd id="besselk.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="besselk.is_infinite" class="variable">is_infinite</dd>
-                <dd id="besselk.is_rational" class="variable">is_rational</dd>
-                <dd id="besselk.is_composite" class="variable">is_composite</dd>
-                <dd id="besselk.is_negative" class="variable">is_negative</dd>
-                <dd id="besselk.is_polar" class="variable">is_polar</dd>
+                <dd id="besselk.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="besselk.is_odd" class="variable">is_odd</dd>
-                <dd id="besselk.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="besselk.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="besselk.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="besselk.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="besselk.is_irrational" class="variable">is_irrational</dd>
+                <dd id="besselk.is_integer" class="variable">is_integer</dd>
+                <dd id="besselk.is_rational" class="variable">is_rational</dd>
+                <dd id="besselk.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="besselk.is_complex" class="variable">is_complex</dd>
                 <dd id="besselk.is_prime" class="variable">is_prime</dd>
+                <dd id="besselk.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="besselk.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="besselk.is_real" class="variable">is_real</dd>
+                <dd id="besselk.is_commutative" class="variable">is_commutative</dd>
                 <dd id="besselk.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="besselk.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="besselk.is_polar" class="variable">is_polar</dd>
+                <dd id="besselk.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="besselk.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="besselk.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -36673,14 +37471,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="hankel1">
-                                <div class="attr class">
-        <a class="headerlink" href="#hankel1">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">hankel1</span><wbr>(<span class="base">sympy.functions.special.bessel.hankel1</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">hankel1</span><wbr>(<span class="base">sympy.functions.special.bessel.hankel1</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#hankel1"></a>
     
             <div class="docstring"><p>Hankel function of the first kind.</p>
 
@@ -36710,27 +37508,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="hankel1.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#hankel1.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">hankel1</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">hankel1</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#hankel1.__init__"></a>
     
     
 
                             </div>
                             <div id="hankel1.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#hankel1.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#hankel1.default_assumptions"></a>
     
     
 
@@ -36887,37 +37691,37 @@ <h5>Inherited Members</h5>
                 <dd id="hankel1.simplify" class="function">simplify</dd>
                 <dd id="hankel1.refine" class="function">refine</dd>
                 <dd id="hankel1.rewrite" class="function">rewrite</dd>
-                <dd id="hankel1.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="hankel1.is_integer" class="variable">is_integer</dd>
-                <dd id="hankel1.is_complex" class="variable">is_complex</dd>
-                <dd id="hankel1.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="hankel1.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="hankel1.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="hankel1.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hankel1.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hankel1.is_positive" class="variable">is_positive</dd>
+                <dd id="hankel1.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="hankel1.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="hankel1.is_negative" class="variable">is_negative</dd>
+                <dd id="hankel1.is_composite" class="variable">is_composite</dd>
+                <dd id="hankel1.is_finite" class="variable">is_finite</dd>
                 <dd id="hankel1.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="hankel1.is_zero" class="variable">is_zero</dd>
                 <dd id="hankel1.is_even" class="variable">is_even</dd>
-                <dd id="hankel1.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="hankel1.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="hankel1.is_commutative" class="variable">is_commutative</dd>
-                <dd id="hankel1.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="hankel1.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="hankel1.is_real" class="variable">is_real</dd>
-                <dd id="hankel1.is_positive" class="variable">is_positive</dd>
-                <dd id="hankel1.is_finite" class="variable">is_finite</dd>
-                <dd id="hankel1.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="hankel1.is_infinite" class="variable">is_infinite</dd>
-                <dd id="hankel1.is_rational" class="variable">is_rational</dd>
-                <dd id="hankel1.is_composite" class="variable">is_composite</dd>
-                <dd id="hankel1.is_negative" class="variable">is_negative</dd>
-                <dd id="hankel1.is_polar" class="variable">is_polar</dd>
+                <dd id="hankel1.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="hankel1.is_odd" class="variable">is_odd</dd>
-                <dd id="hankel1.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="hankel1.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hankel1.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="hankel1.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="hankel1.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hankel1.is_integer" class="variable">is_integer</dd>
+                <dd id="hankel1.is_rational" class="variable">is_rational</dd>
+                <dd id="hankel1.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="hankel1.is_complex" class="variable">is_complex</dd>
                 <dd id="hankel1.is_prime" class="variable">is_prime</dd>
+                <dd id="hankel1.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="hankel1.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="hankel1.is_real" class="variable">is_real</dd>
+                <dd id="hankel1.is_commutative" class="variable">is_commutative</dd>
                 <dd id="hankel1.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="hankel1.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="hankel1.is_polar" class="variable">is_polar</dd>
+                <dd id="hankel1.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="hankel1.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="hankel1.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -36929,14 +37733,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="hankel2">
-                                <div class="attr class">
-        <a class="headerlink" href="#hankel2">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">hankel2</span><wbr>(<span class="base">sympy.functions.special.bessel.hankel2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">hankel2</span><wbr>(<span class="base">sympy.functions.special.bessel.hankel2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#hankel2"></a>
     
             <div class="docstring"><p>Hankel function of the second kind.</p>
 
@@ -36967,27 +37771,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="hankel2.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#hankel2.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">hankel2</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">hankel2</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#hankel2.__init__"></a>
     
     
 
                             </div>
                             <div id="hankel2.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#hankel2.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#hankel2.default_assumptions"></a>
     
     
 
@@ -37144,37 +37954,37 @@ <h5>Inherited Members</h5>
                 <dd id="hankel2.simplify" class="function">simplify</dd>
                 <dd id="hankel2.refine" class="function">refine</dd>
                 <dd id="hankel2.rewrite" class="function">rewrite</dd>
-                <dd id="hankel2.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="hankel2.is_integer" class="variable">is_integer</dd>
-                <dd id="hankel2.is_complex" class="variable">is_complex</dd>
-                <dd id="hankel2.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="hankel2.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="hankel2.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="hankel2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hankel2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hankel2.is_positive" class="variable">is_positive</dd>
+                <dd id="hankel2.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="hankel2.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="hankel2.is_negative" class="variable">is_negative</dd>
+                <dd id="hankel2.is_composite" class="variable">is_composite</dd>
+                <dd id="hankel2.is_finite" class="variable">is_finite</dd>
                 <dd id="hankel2.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="hankel2.is_zero" class="variable">is_zero</dd>
                 <dd id="hankel2.is_even" class="variable">is_even</dd>
-                <dd id="hankel2.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="hankel2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="hankel2.is_commutative" class="variable">is_commutative</dd>
-                <dd id="hankel2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="hankel2.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="hankel2.is_real" class="variable">is_real</dd>
-                <dd id="hankel2.is_positive" class="variable">is_positive</dd>
-                <dd id="hankel2.is_finite" class="variable">is_finite</dd>
-                <dd id="hankel2.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="hankel2.is_infinite" class="variable">is_infinite</dd>
-                <dd id="hankel2.is_rational" class="variable">is_rational</dd>
-                <dd id="hankel2.is_composite" class="variable">is_composite</dd>
-                <dd id="hankel2.is_negative" class="variable">is_negative</dd>
-                <dd id="hankel2.is_polar" class="variable">is_polar</dd>
+                <dd id="hankel2.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="hankel2.is_odd" class="variable">is_odd</dd>
-                <dd id="hankel2.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="hankel2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hankel2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="hankel2.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="hankel2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hankel2.is_integer" class="variable">is_integer</dd>
+                <dd id="hankel2.is_rational" class="variable">is_rational</dd>
+                <dd id="hankel2.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="hankel2.is_complex" class="variable">is_complex</dd>
                 <dd id="hankel2.is_prime" class="variable">is_prime</dd>
+                <dd id="hankel2.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="hankel2.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="hankel2.is_real" class="variable">is_real</dd>
+                <dd id="hankel2.is_commutative" class="variable">is_commutative</dd>
                 <dd id="hankel2.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="hankel2.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="hankel2.is_polar" class="variable">is_polar</dd>
+                <dd id="hankel2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="hankel2.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="hankel2.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -37186,14 +37996,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="jn">
-                                <div class="attr class">
-        <a class="headerlink" href="#jn">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">jn</span><wbr>(<span class="base">sympy.functions.special.bessel.jn</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">jn</span><wbr>(<span class="base">sympy.functions.special.bessel.jn</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#jn"></a>
     
             <div class="docstring"><p>Spherical Bessel function of the first kind.</p>
 
@@ -37215,10 +38025,10 @@ <h1 id="explanation">Explanation</h1>
 <p>The spherical Bessel functions of integral order are
 calculated using the formula:</p>
 
-<p>.. math:: j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},</p>
+<p>$$j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},$$</p>
 
 <p>where the coefficients $f_n(z)$ are available as
-:func:<code>sympy.polys.orthopolys.spherical_bessel_fn</code>.</p>
+<code>sympy.polys.orthopolys.spherical_bessel_fn</code>.</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -37245,27 +38055,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://dlmf.nist.gov/10.47</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="jn.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#jn.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">jn</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">jn</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#jn.__init__"></a>
     
     
 
                             </div>
                             <div id="jn.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#jn.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#jn.default_assumptions"></a>
     
     
 
@@ -37428,37 +38244,37 @@ <h5>Inherited Members</h5>
                 <dd id="jn.simplify" class="function">simplify</dd>
                 <dd id="jn.refine" class="function">refine</dd>
                 <dd id="jn.rewrite" class="function">rewrite</dd>
-                <dd id="jn.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="jn.is_integer" class="variable">is_integer</dd>
-                <dd id="jn.is_complex" class="variable">is_complex</dd>
-                <dd id="jn.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="jn.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="jn.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="jn.is_irrational" class="variable">is_irrational</dd>
+                <dd id="jn.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="jn.is_positive" class="variable">is_positive</dd>
+                <dd id="jn.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="jn.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="jn.is_negative" class="variable">is_negative</dd>
+                <dd id="jn.is_composite" class="variable">is_composite</dd>
+                <dd id="jn.is_finite" class="variable">is_finite</dd>
                 <dd id="jn.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="jn.is_zero" class="variable">is_zero</dd>
                 <dd id="jn.is_even" class="variable">is_even</dd>
-                <dd id="jn.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="jn.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="jn.is_commutative" class="variable">is_commutative</dd>
-                <dd id="jn.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="jn.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="jn.is_real" class="variable">is_real</dd>
-                <dd id="jn.is_positive" class="variable">is_positive</dd>
-                <dd id="jn.is_finite" class="variable">is_finite</dd>
-                <dd id="jn.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="jn.is_infinite" class="variable">is_infinite</dd>
-                <dd id="jn.is_rational" class="variable">is_rational</dd>
-                <dd id="jn.is_composite" class="variable">is_composite</dd>
-                <dd id="jn.is_negative" class="variable">is_negative</dd>
-                <dd id="jn.is_polar" class="variable">is_polar</dd>
+                <dd id="jn.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="jn.is_odd" class="variable">is_odd</dd>
-                <dd id="jn.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="jn.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="jn.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="jn.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="jn.is_irrational" class="variable">is_irrational</dd>
+                <dd id="jn.is_integer" class="variable">is_integer</dd>
+                <dd id="jn.is_rational" class="variable">is_rational</dd>
+                <dd id="jn.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="jn.is_complex" class="variable">is_complex</dd>
                 <dd id="jn.is_prime" class="variable">is_prime</dd>
+                <dd id="jn.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="jn.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="jn.is_real" class="variable">is_real</dd>
+                <dd id="jn.is_commutative" class="variable">is_commutative</dd>
                 <dd id="jn.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="jn.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="jn.is_polar" class="variable">is_polar</dd>
+                <dd id="jn.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="jn.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="jn.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -37470,14 +38286,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="yn">
-                                <div class="attr class">
-        <a class="headerlink" href="#yn">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">yn</span><wbr>(<span class="base">sympy.functions.special.bessel.yn</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">yn</span><wbr>(<span class="base">sympy.functions.special.bessel.yn</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#yn"></a>
     
             <div class="docstring"><p>Spherical Bessel function of the second kind.</p>
 
@@ -37493,7 +38309,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>For integral orders $n$, $y_n$ is calculated using the formula:</p>
 
-<p>.. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z)</p>
+<p>$$y_n(z) = (-1)^{n+1} j_{-n-1}(z)$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -37518,27 +38334,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://dlmf.nist.gov/10.47</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="yn.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#yn.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">yn</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">yn</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#yn.__init__"></a>
     
     
 
                             </div>
                             <div id="yn.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#yn.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#yn.default_assumptions"></a>
     
     
 
@@ -37698,37 +38520,37 @@ <h5>Inherited Members</h5>
                 <dd id="yn.simplify" class="function">simplify</dd>
                 <dd id="yn.refine" class="function">refine</dd>
                 <dd id="yn.rewrite" class="function">rewrite</dd>
-                <dd id="yn.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="yn.is_integer" class="variable">is_integer</dd>
-                <dd id="yn.is_complex" class="variable">is_complex</dd>
-                <dd id="yn.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="yn.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="yn.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="yn.is_irrational" class="variable">is_irrational</dd>
+                <dd id="yn.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="yn.is_positive" class="variable">is_positive</dd>
+                <dd id="yn.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="yn.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="yn.is_negative" class="variable">is_negative</dd>
+                <dd id="yn.is_composite" class="variable">is_composite</dd>
+                <dd id="yn.is_finite" class="variable">is_finite</dd>
                 <dd id="yn.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="yn.is_zero" class="variable">is_zero</dd>
                 <dd id="yn.is_even" class="variable">is_even</dd>
-                <dd id="yn.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="yn.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="yn.is_commutative" class="variable">is_commutative</dd>
-                <dd id="yn.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="yn.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="yn.is_real" class="variable">is_real</dd>
-                <dd id="yn.is_positive" class="variable">is_positive</dd>
-                <dd id="yn.is_finite" class="variable">is_finite</dd>
-                <dd id="yn.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="yn.is_infinite" class="variable">is_infinite</dd>
-                <dd id="yn.is_rational" class="variable">is_rational</dd>
-                <dd id="yn.is_composite" class="variable">is_composite</dd>
-                <dd id="yn.is_negative" class="variable">is_negative</dd>
-                <dd id="yn.is_polar" class="variable">is_polar</dd>
+                <dd id="yn.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="yn.is_odd" class="variable">is_odd</dd>
-                <dd id="yn.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="yn.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="yn.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="yn.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="yn.is_irrational" class="variable">is_irrational</dd>
+                <dd id="yn.is_integer" class="variable">is_integer</dd>
+                <dd id="yn.is_rational" class="variable">is_rational</dd>
+                <dd id="yn.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="yn.is_complex" class="variable">is_complex</dd>
                 <dd id="yn.is_prime" class="variable">is_prime</dd>
+                <dd id="yn.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="yn.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="yn.is_real" class="variable">is_real</dd>
+                <dd id="yn.is_commutative" class="variable">is_commutative</dd>
                 <dd id="yn.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="yn.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="yn.is_polar" class="variable">is_polar</dd>
+                <dd id="yn.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="yn.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="yn.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -37740,14 +38562,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="hn1">
-                                <div class="attr class">
-        <a class="headerlink" href="#hn1">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">hn1</span><wbr>(<span class="base">sympy.functions.special.bessel.hn1</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">hn1</span><wbr>(<span class="base">sympy.functions.special.bessel.hn1</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#hn1"></a>
     
             <div class="docstring"><p>Spherical Hankel function of the first kind.</p>
 
@@ -37755,14 +38577,14 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined as</p>
 
-<p>.. math:: h_\nu^(1)(z) = j_\nu(z) + i y_\nu(z),</p>
+<p>$$h_\nu^(1)(z) = j_\nu(z) + i y_\nu(z),$$</p>
 
 <p>where $j_\nu(z)$ and $y_\nu(z)$ are the spherical
 Bessel function of the first and second kinds.</p>
 
 <p>For integral orders $n$, $h_n^(1)$ is calculated using the formula:</p>
 
-<p>.. math:: h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z)</p>
+<p>$$h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z)$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -37789,27 +38611,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://dlmf.nist.gov/10.47</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="hn1.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#hn1.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">hn1</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">hn1</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#hn1.__init__"></a>
     
     
 
                             </div>
                             <div id="hn1.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#hn1.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#hn1.default_assumptions"></a>
     
     
 
@@ -37969,37 +38797,37 @@ <h5>Inherited Members</h5>
                 <dd id="hn1.simplify" class="function">simplify</dd>
                 <dd id="hn1.refine" class="function">refine</dd>
                 <dd id="hn1.rewrite" class="function">rewrite</dd>
-                <dd id="hn1.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="hn1.is_integer" class="variable">is_integer</dd>
-                <dd id="hn1.is_complex" class="variable">is_complex</dd>
-                <dd id="hn1.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="hn1.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="hn1.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="hn1.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hn1.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hn1.is_positive" class="variable">is_positive</dd>
+                <dd id="hn1.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="hn1.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="hn1.is_negative" class="variable">is_negative</dd>
+                <dd id="hn1.is_composite" class="variable">is_composite</dd>
+                <dd id="hn1.is_finite" class="variable">is_finite</dd>
                 <dd id="hn1.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="hn1.is_zero" class="variable">is_zero</dd>
                 <dd id="hn1.is_even" class="variable">is_even</dd>
-                <dd id="hn1.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="hn1.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="hn1.is_commutative" class="variable">is_commutative</dd>
-                <dd id="hn1.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="hn1.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="hn1.is_real" class="variable">is_real</dd>
-                <dd id="hn1.is_positive" class="variable">is_positive</dd>
-                <dd id="hn1.is_finite" class="variable">is_finite</dd>
-                <dd id="hn1.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="hn1.is_infinite" class="variable">is_infinite</dd>
-                <dd id="hn1.is_rational" class="variable">is_rational</dd>
-                <dd id="hn1.is_composite" class="variable">is_composite</dd>
-                <dd id="hn1.is_negative" class="variable">is_negative</dd>
-                <dd id="hn1.is_polar" class="variable">is_polar</dd>
+                <dd id="hn1.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="hn1.is_odd" class="variable">is_odd</dd>
-                <dd id="hn1.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="hn1.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hn1.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="hn1.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="hn1.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hn1.is_integer" class="variable">is_integer</dd>
+                <dd id="hn1.is_rational" class="variable">is_rational</dd>
+                <dd id="hn1.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="hn1.is_complex" class="variable">is_complex</dd>
                 <dd id="hn1.is_prime" class="variable">is_prime</dd>
+                <dd id="hn1.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="hn1.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="hn1.is_real" class="variable">is_real</dd>
+                <dd id="hn1.is_commutative" class="variable">is_commutative</dd>
                 <dd id="hn1.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="hn1.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="hn1.is_polar" class="variable">is_polar</dd>
+                <dd id="hn1.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="hn1.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="hn1.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -38011,14 +38839,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="hn2">
-                                <div class="attr class">
-        <a class="headerlink" href="#hn2">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">hn2</span><wbr>(<span class="base">sympy.functions.special.bessel.hn2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">hn2</span><wbr>(<span class="base">sympy.functions.special.bessel.hn2</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#hn2"></a>
     
             <div class="docstring"><p>Spherical Hankel function of the second kind.</p>
 
@@ -38026,14 +38854,14 @@ <h1 id="explanation">Explanation</h1>
 
 <p>This function is defined as</p>
 
-<p>.. math:: h_\nu^(2)(z) = j_\nu(z) - i y_\nu(z),</p>
+<p>$$h_\nu^(2)(z) = j_\nu(z) - i y_\nu(z),$$</p>
 
 <p>where $j_\nu(z)$ and $y_\nu(z)$ are the spherical
 Bessel function of the first and second kinds.</p>
 
 <p>For integral orders $n$, $h_n^(2)$ is calculated using the formula:</p>
 
-<p>.. math:: h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z)</p>
+<p>$$h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z)$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -38060,27 +38888,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] http://dlmf.nist.gov/10.47</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="hn2.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#hn2.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">hn2</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">hn2</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#hn2.__init__"></a>
     
     
 
                             </div>
                             <div id="hn2.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#hn2.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#hn2.default_assumptions"></a>
     
     
 
@@ -38240,37 +39074,37 @@ <h5>Inherited Members</h5>
                 <dd id="hn2.simplify" class="function">simplify</dd>
                 <dd id="hn2.refine" class="function">refine</dd>
                 <dd id="hn2.rewrite" class="function">rewrite</dd>
-                <dd id="hn2.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="hn2.is_integer" class="variable">is_integer</dd>
-                <dd id="hn2.is_complex" class="variable">is_complex</dd>
-                <dd id="hn2.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="hn2.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="hn2.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="hn2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hn2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hn2.is_positive" class="variable">is_positive</dd>
+                <dd id="hn2.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="hn2.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="hn2.is_negative" class="variable">is_negative</dd>
+                <dd id="hn2.is_composite" class="variable">is_composite</dd>
+                <dd id="hn2.is_finite" class="variable">is_finite</dd>
                 <dd id="hn2.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="hn2.is_zero" class="variable">is_zero</dd>
                 <dd id="hn2.is_even" class="variable">is_even</dd>
-                <dd id="hn2.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="hn2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="hn2.is_commutative" class="variable">is_commutative</dd>
-                <dd id="hn2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="hn2.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="hn2.is_real" class="variable">is_real</dd>
-                <dd id="hn2.is_positive" class="variable">is_positive</dd>
-                <dd id="hn2.is_finite" class="variable">is_finite</dd>
-                <dd id="hn2.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="hn2.is_infinite" class="variable">is_infinite</dd>
-                <dd id="hn2.is_rational" class="variable">is_rational</dd>
-                <dd id="hn2.is_composite" class="variable">is_composite</dd>
-                <dd id="hn2.is_negative" class="variable">is_negative</dd>
-                <dd id="hn2.is_polar" class="variable">is_polar</dd>
+                <dd id="hn2.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="hn2.is_odd" class="variable">is_odd</dd>
-                <dd id="hn2.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="hn2.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hn2.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="hn2.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="hn2.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hn2.is_integer" class="variable">is_integer</dd>
+                <dd id="hn2.is_rational" class="variable">is_rational</dd>
+                <dd id="hn2.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="hn2.is_complex" class="variable">is_complex</dd>
                 <dd id="hn2.is_prime" class="variable">is_prime</dd>
+                <dd id="hn2.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="hn2.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="hn2.is_real" class="variable">is_real</dd>
+                <dd id="hn2.is_commutative" class="variable">is_commutative</dd>
                 <dd id="hn2.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="hn2.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="hn2.is_polar" class="variable">is_polar</dd>
+                <dd id="hn2.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="hn2.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="hn2.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -38282,14 +39116,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="airyai">
-                                <div class="attr class">
-        <a class="headerlink" href="#airyai">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">airyai</span><wbr>(<span class="base">sympy.functions.special.bessel.airyai</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">airyai</span><wbr>(<span class="base">sympy.functions.special.bessel.airyai</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#airyai"></a>
     
             <div class="docstring"><p>The Airy function $\operatorname{Ai}$ of the first kind.</p>
 
@@ -38298,14 +39132,12 @@ <h1 id="explanation">Explanation</h1>
 <p>The Airy function $\operatorname{Ai}(z)$ is defined to be the function
 satisfying Airy's differential equation</p>
 
-<p>.. math::
-    \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.</p>
+<p>$$\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.$$</p>
 
 <p>Equivalently, for real $z$</p>
 
-<p>.. math::
-    \operatorname{Ai}(z) := \frac{1}{\pi}
-    \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.</p>
+<p>$$\operatorname{Ai}(z) := \frac{1}{\pi}
+\int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -38375,30 +39207,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Airy_function
-.. [2] http://dlmf.nist.gov/9
-.. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions
-.. [4] http://mathworld.wolfram.com/AiryFunctions.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="airyai.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#airyai.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">airyai</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">airyai</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#airyai.__init__"></a>
     
     
 
                             </div>
                             <div id="airyai.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#airyai.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#airyai.default_assumptions"></a>
     
     
 
@@ -38558,37 +39393,37 @@ <h5>Inherited Members</h5>
                 <dd id="airyai.simplify" class="function">simplify</dd>
                 <dd id="airyai.refine" class="function">refine</dd>
                 <dd id="airyai.rewrite" class="function">rewrite</dd>
-                <dd id="airyai.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="airyai.is_integer" class="variable">is_integer</dd>
-                <dd id="airyai.is_complex" class="variable">is_complex</dd>
-                <dd id="airyai.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="airyai.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="airyai.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="airyai.is_irrational" class="variable">is_irrational</dd>
+                <dd id="airyai.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="airyai.is_positive" class="variable">is_positive</dd>
+                <dd id="airyai.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="airyai.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="airyai.is_negative" class="variable">is_negative</dd>
+                <dd id="airyai.is_composite" class="variable">is_composite</dd>
+                <dd id="airyai.is_finite" class="variable">is_finite</dd>
                 <dd id="airyai.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="airyai.is_zero" class="variable">is_zero</dd>
                 <dd id="airyai.is_even" class="variable">is_even</dd>
-                <dd id="airyai.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="airyai.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="airyai.is_commutative" class="variable">is_commutative</dd>
-                <dd id="airyai.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="airyai.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="airyai.is_real" class="variable">is_real</dd>
-                <dd id="airyai.is_positive" class="variable">is_positive</dd>
-                <dd id="airyai.is_finite" class="variable">is_finite</dd>
-                <dd id="airyai.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="airyai.is_infinite" class="variable">is_infinite</dd>
-                <dd id="airyai.is_rational" class="variable">is_rational</dd>
-                <dd id="airyai.is_composite" class="variable">is_composite</dd>
-                <dd id="airyai.is_negative" class="variable">is_negative</dd>
-                <dd id="airyai.is_polar" class="variable">is_polar</dd>
+                <dd id="airyai.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="airyai.is_odd" class="variable">is_odd</dd>
-                <dd id="airyai.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="airyai.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="airyai.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="airyai.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="airyai.is_irrational" class="variable">is_irrational</dd>
+                <dd id="airyai.is_integer" class="variable">is_integer</dd>
+                <dd id="airyai.is_rational" class="variable">is_rational</dd>
+                <dd id="airyai.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="airyai.is_complex" class="variable">is_complex</dd>
                 <dd id="airyai.is_prime" class="variable">is_prime</dd>
+                <dd id="airyai.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="airyai.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="airyai.is_real" class="variable">is_real</dd>
+                <dd id="airyai.is_commutative" class="variable">is_commutative</dd>
                 <dd id="airyai.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="airyai.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="airyai.is_polar" class="variable">is_polar</dd>
+                <dd id="airyai.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="airyai.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="airyai.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -38600,14 +39435,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="airybi">
-                                <div class="attr class">
-        <a class="headerlink" href="#airybi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">airybi</span><wbr>(<span class="base">sympy.functions.special.bessel.airybi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">airybi</span><wbr>(<span class="base">sympy.functions.special.bessel.airybi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#airybi"></a>
     
             <div class="docstring"><p>The Airy function $\operatorname{Bi}$ of the second kind.</p>
 
@@ -38616,16 +39451,14 @@ <h1 id="explanation">Explanation</h1>
 <p>The Airy function $\operatorname{Bi}(z)$ is defined to be the function
 satisfying Airy's differential equation</p>
 
-<p>.. math::
-    \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.</p>
+<p>$$\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.$$</p>
 
 <p>Equivalently, for real $z$</p>
 
-<p>.. math::
-    \operatorname{Bi}(z) := \frac{1}{\pi}
-             \int_0^\infty
-               \exp\left(-\frac{t^3}{3} + z t\right)
-               + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.</p>
+<p>$$\operatorname{Bi}(z) := \frac{1}{\pi}
+         \int_0^\infty
+           \exp\left(-\frac{t^3}{3} + z t\right)
+           + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -38695,30 +39528,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Airy_function
-.. [2] http://dlmf.nist.gov/9
-.. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions
-.. [4] http://mathworld.wolfram.com/AiryFunctions.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="airybi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#airybi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">airybi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">airybi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#airybi.__init__"></a>
     
     
 
                             </div>
                             <div id="airybi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#airybi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#airybi.default_assumptions"></a>
     
     
 
@@ -38878,37 +39714,37 @@ <h5>Inherited Members</h5>
                 <dd id="airybi.simplify" class="function">simplify</dd>
                 <dd id="airybi.refine" class="function">refine</dd>
                 <dd id="airybi.rewrite" class="function">rewrite</dd>
-                <dd id="airybi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="airybi.is_integer" class="variable">is_integer</dd>
-                <dd id="airybi.is_complex" class="variable">is_complex</dd>
-                <dd id="airybi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="airybi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="airybi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="airybi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="airybi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="airybi.is_positive" class="variable">is_positive</dd>
+                <dd id="airybi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="airybi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="airybi.is_negative" class="variable">is_negative</dd>
+                <dd id="airybi.is_composite" class="variable">is_composite</dd>
+                <dd id="airybi.is_finite" class="variable">is_finite</dd>
                 <dd id="airybi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="airybi.is_zero" class="variable">is_zero</dd>
                 <dd id="airybi.is_even" class="variable">is_even</dd>
-                <dd id="airybi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="airybi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="airybi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="airybi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="airybi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="airybi.is_real" class="variable">is_real</dd>
-                <dd id="airybi.is_positive" class="variable">is_positive</dd>
-                <dd id="airybi.is_finite" class="variable">is_finite</dd>
-                <dd id="airybi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="airybi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="airybi.is_rational" class="variable">is_rational</dd>
-                <dd id="airybi.is_composite" class="variable">is_composite</dd>
-                <dd id="airybi.is_negative" class="variable">is_negative</dd>
-                <dd id="airybi.is_polar" class="variable">is_polar</dd>
+                <dd id="airybi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="airybi.is_odd" class="variable">is_odd</dd>
-                <dd id="airybi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="airybi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="airybi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="airybi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="airybi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="airybi.is_integer" class="variable">is_integer</dd>
+                <dd id="airybi.is_rational" class="variable">is_rational</dd>
+                <dd id="airybi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="airybi.is_complex" class="variable">is_complex</dd>
                 <dd id="airybi.is_prime" class="variable">is_prime</dd>
+                <dd id="airybi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="airybi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="airybi.is_real" class="variable">is_real</dd>
+                <dd id="airybi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="airybi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="airybi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="airybi.is_polar" class="variable">is_polar</dd>
+                <dd id="airybi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="airybi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="airybi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -38920,14 +39756,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="airyaiprime">
-                                <div class="attr class">
-        <a class="headerlink" href="#airyaiprime">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">airyaiprime</span><wbr>(<span class="base">sympy.functions.special.bessel.airyaiprime</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">airyaiprime</span><wbr>(<span class="base">sympy.functions.special.bessel.airyaiprime</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#airyaiprime"></a>
     
             <div class="docstring"><p>The derivative $\operatorname{Ai}^\prime$ of the Airy function of the first
 kind.</p>
@@ -38937,8 +39773,7 @@ <h1 id="explanation">Explanation</h1>
 <p>The Airy function $\operatorname{Ai}^\prime(z)$ is defined to be the
 function</p>
 
-<p>.. math::
-    \operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}.</p>
+<p>$$\operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}.$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -39006,30 +39841,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Airy_function
-.. [2] http://dlmf.nist.gov/9
-.. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions
-.. [4] http://mathworld.wolfram.com/AiryFunctions.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="airyaiprime.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#airyaiprime.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">airyaiprime</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">airyaiprime</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#airyaiprime.__init__"></a>
     
     
 
                             </div>
                             <div id="airyaiprime.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#airyaiprime.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#airyaiprime.default_assumptions"></a>
     
     
 
@@ -39189,37 +40027,37 @@ <h5>Inherited Members</h5>
                 <dd id="airyaiprime.simplify" class="function">simplify</dd>
                 <dd id="airyaiprime.refine" class="function">refine</dd>
                 <dd id="airyaiprime.rewrite" class="function">rewrite</dd>
-                <dd id="airyaiprime.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="airyaiprime.is_integer" class="variable">is_integer</dd>
-                <dd id="airyaiprime.is_complex" class="variable">is_complex</dd>
-                <dd id="airyaiprime.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="airyaiprime.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="airyaiprime.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="airyaiprime.is_irrational" class="variable">is_irrational</dd>
+                <dd id="airyaiprime.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="airyaiprime.is_positive" class="variable">is_positive</dd>
+                <dd id="airyaiprime.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="airyaiprime.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="airyaiprime.is_negative" class="variable">is_negative</dd>
+                <dd id="airyaiprime.is_composite" class="variable">is_composite</dd>
+                <dd id="airyaiprime.is_finite" class="variable">is_finite</dd>
                 <dd id="airyaiprime.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="airyaiprime.is_zero" class="variable">is_zero</dd>
                 <dd id="airyaiprime.is_even" class="variable">is_even</dd>
-                <dd id="airyaiprime.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="airyaiprime.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="airyaiprime.is_commutative" class="variable">is_commutative</dd>
-                <dd id="airyaiprime.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="airyaiprime.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="airyaiprime.is_real" class="variable">is_real</dd>
-                <dd id="airyaiprime.is_positive" class="variable">is_positive</dd>
-                <dd id="airyaiprime.is_finite" class="variable">is_finite</dd>
-                <dd id="airyaiprime.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="airyaiprime.is_infinite" class="variable">is_infinite</dd>
-                <dd id="airyaiprime.is_rational" class="variable">is_rational</dd>
-                <dd id="airyaiprime.is_composite" class="variable">is_composite</dd>
-                <dd id="airyaiprime.is_negative" class="variable">is_negative</dd>
-                <dd id="airyaiprime.is_polar" class="variable">is_polar</dd>
+                <dd id="airyaiprime.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="airyaiprime.is_odd" class="variable">is_odd</dd>
-                <dd id="airyaiprime.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="airyaiprime.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="airyaiprime.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="airyaiprime.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="airyaiprime.is_irrational" class="variable">is_irrational</dd>
+                <dd id="airyaiprime.is_integer" class="variable">is_integer</dd>
+                <dd id="airyaiprime.is_rational" class="variable">is_rational</dd>
+                <dd id="airyaiprime.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="airyaiprime.is_complex" class="variable">is_complex</dd>
                 <dd id="airyaiprime.is_prime" class="variable">is_prime</dd>
+                <dd id="airyaiprime.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="airyaiprime.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="airyaiprime.is_real" class="variable">is_real</dd>
+                <dd id="airyaiprime.is_commutative" class="variable">is_commutative</dd>
                 <dd id="airyaiprime.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="airyaiprime.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="airyaiprime.is_polar" class="variable">is_polar</dd>
+                <dd id="airyaiprime.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="airyaiprime.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="airyaiprime.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -39231,14 +40069,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="airybiprime">
-                                <div class="attr class">
-        <a class="headerlink" href="#airybiprime">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">airybiprime</span><wbr>(<span class="base">sympy.functions.special.bessel.airybiprime</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">airybiprime</span><wbr>(<span class="base">sympy.functions.special.bessel.airybiprime</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#airybiprime"></a>
     
             <div class="docstring"><p>The derivative $\operatorname{Bi}^\prime$ of the Airy function of the first
 kind.</p>
@@ -39248,8 +40086,7 @@ <h1 id="explanation">Explanation</h1>
 <p>The Airy function $\operatorname{Bi}^\prime(z)$ is defined to be the
 function</p>
 
-<p>.. math::
-    \operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}.</p>
+<p>$$\operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}.$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -39319,30 +40156,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Airy_function
-.. [2] http://dlmf.nist.gov/9
-.. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions
-.. [4] http://mathworld.wolfram.com/AiryFunctions.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="airybiprime.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#airybiprime.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">airybiprime</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">airybiprime</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#airybiprime.__init__"></a>
     
     
 
                             </div>
                             <div id="airybiprime.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#airybiprime.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#airybiprime.default_assumptions"></a>
     
     
 
@@ -39502,37 +40342,37 @@ <h5>Inherited Members</h5>
                 <dd id="airybiprime.simplify" class="function">simplify</dd>
                 <dd id="airybiprime.refine" class="function">refine</dd>
                 <dd id="airybiprime.rewrite" class="function">rewrite</dd>
-                <dd id="airybiprime.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="airybiprime.is_integer" class="variable">is_integer</dd>
-                <dd id="airybiprime.is_complex" class="variable">is_complex</dd>
-                <dd id="airybiprime.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="airybiprime.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="airybiprime.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="airybiprime.is_irrational" class="variable">is_irrational</dd>
+                <dd id="airybiprime.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="airybiprime.is_positive" class="variable">is_positive</dd>
+                <dd id="airybiprime.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="airybiprime.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="airybiprime.is_negative" class="variable">is_negative</dd>
+                <dd id="airybiprime.is_composite" class="variable">is_composite</dd>
+                <dd id="airybiprime.is_finite" class="variable">is_finite</dd>
                 <dd id="airybiprime.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="airybiprime.is_zero" class="variable">is_zero</dd>
                 <dd id="airybiprime.is_even" class="variable">is_even</dd>
-                <dd id="airybiprime.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="airybiprime.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="airybiprime.is_commutative" class="variable">is_commutative</dd>
-                <dd id="airybiprime.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="airybiprime.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="airybiprime.is_real" class="variable">is_real</dd>
-                <dd id="airybiprime.is_positive" class="variable">is_positive</dd>
-                <dd id="airybiprime.is_finite" class="variable">is_finite</dd>
-                <dd id="airybiprime.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="airybiprime.is_infinite" class="variable">is_infinite</dd>
-                <dd id="airybiprime.is_rational" class="variable">is_rational</dd>
-                <dd id="airybiprime.is_composite" class="variable">is_composite</dd>
-                <dd id="airybiprime.is_negative" class="variable">is_negative</dd>
-                <dd id="airybiprime.is_polar" class="variable">is_polar</dd>
+                <dd id="airybiprime.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="airybiprime.is_odd" class="variable">is_odd</dd>
-                <dd id="airybiprime.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="airybiprime.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="airybiprime.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="airybiprime.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="airybiprime.is_irrational" class="variable">is_irrational</dd>
+                <dd id="airybiprime.is_integer" class="variable">is_integer</dd>
+                <dd id="airybiprime.is_rational" class="variable">is_rational</dd>
+                <dd id="airybiprime.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="airybiprime.is_complex" class="variable">is_complex</dd>
                 <dd id="airybiprime.is_prime" class="variable">is_prime</dd>
+                <dd id="airybiprime.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="airybiprime.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="airybiprime.is_real" class="variable">is_real</dd>
+                <dd id="airybiprime.is_commutative" class="variable">is_commutative</dd>
                 <dd id="airybiprime.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="airybiprime.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="airybiprime.is_polar" class="variable">is_polar</dd>
+                <dd id="airybiprime.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="airybiprime.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="airybiprime.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -39544,14 +40384,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="marcumq">
-                                <div class="attr class">
-        <a class="headerlink" href="#marcumq">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">marcumq</span><wbr>(<span class="base">sympy.functions.special.bessel.marcumq</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">marcumq</span><wbr>(<span class="base">sympy.functions.special.bessel.marcumq</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#marcumq"></a>
     
             <div class="docstring"><p>The Marcum Q-function.</p>
 
@@ -39559,8 +40399,7 @@ <h1 id="explanation">Explanation</h1>
 
 <p>The Marcum Q-function is defined by the meromorphic continuation of</p>
 
-<p>.. math::
-    Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx</p>
+<p>$$Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -39595,28 +40434,33 @@ <h1 id="examples">Examples</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Marcum_Q-function
-.. [2] http://mathworld.wolfram.com/MarcumQ-Function.html</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="marcumq.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#marcumq.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">marcumq</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">marcumq</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#marcumq.__init__"></a>
     
     
 
                             </div>
                             <div id="marcumq.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#marcumq.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#marcumq.default_assumptions"></a>
     
     
 
@@ -39771,37 +40615,37 @@ <h5>Inherited Members</h5>
                 <dd id="marcumq.simplify" class="function">simplify</dd>
                 <dd id="marcumq.refine" class="function">refine</dd>
                 <dd id="marcumq.rewrite" class="function">rewrite</dd>
-                <dd id="marcumq.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="marcumq.is_integer" class="variable">is_integer</dd>
-                <dd id="marcumq.is_complex" class="variable">is_complex</dd>
-                <dd id="marcumq.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="marcumq.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="marcumq.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="marcumq.is_irrational" class="variable">is_irrational</dd>
+                <dd id="marcumq.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="marcumq.is_positive" class="variable">is_positive</dd>
+                <dd id="marcumq.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="marcumq.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="marcumq.is_negative" class="variable">is_negative</dd>
+                <dd id="marcumq.is_composite" class="variable">is_composite</dd>
+                <dd id="marcumq.is_finite" class="variable">is_finite</dd>
                 <dd id="marcumq.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="marcumq.is_zero" class="variable">is_zero</dd>
                 <dd id="marcumq.is_even" class="variable">is_even</dd>
-                <dd id="marcumq.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="marcumq.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="marcumq.is_commutative" class="variable">is_commutative</dd>
-                <dd id="marcumq.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="marcumq.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="marcumq.is_real" class="variable">is_real</dd>
-                <dd id="marcumq.is_positive" class="variable">is_positive</dd>
-                <dd id="marcumq.is_finite" class="variable">is_finite</dd>
-                <dd id="marcumq.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="marcumq.is_infinite" class="variable">is_infinite</dd>
-                <dd id="marcumq.is_rational" class="variable">is_rational</dd>
-                <dd id="marcumq.is_composite" class="variable">is_composite</dd>
-                <dd id="marcumq.is_negative" class="variable">is_negative</dd>
-                <dd id="marcumq.is_polar" class="variable">is_polar</dd>
+                <dd id="marcumq.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="marcumq.is_odd" class="variable">is_odd</dd>
-                <dd id="marcumq.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="marcumq.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="marcumq.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="marcumq.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="marcumq.is_irrational" class="variable">is_irrational</dd>
+                <dd id="marcumq.is_integer" class="variable">is_integer</dd>
+                <dd id="marcumq.is_rational" class="variable">is_rational</dd>
+                <dd id="marcumq.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="marcumq.is_complex" class="variable">is_complex</dd>
                 <dd id="marcumq.is_prime" class="variable">is_prime</dd>
+                <dd id="marcumq.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="marcumq.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="marcumq.is_real" class="variable">is_real</dd>
+                <dd id="marcumq.is_commutative" class="variable">is_commutative</dd>
                 <dd id="marcumq.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="marcumq.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="marcumq.is_polar" class="variable">is_polar</dd>
+                <dd id="marcumq.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="marcumq.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="marcumq.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -39813,14 +40657,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="hyper">
-                                <div class="attr class">
-        <a class="headerlink" href="#hyper">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">hyper</span><wbr>(<span class="base">sympy.functions.special.hyper.hyper</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">hyper</span><wbr>(<span class="base">sympy.functions.special.hyper.hyper</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#hyper"></a>
     
             <div class="docstring"><p>The generalized hypergeometric function is defined by a series where
 the ratios of successive terms are a rational function of the summation
@@ -39934,39 +40778,44 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations,
-       Volume 1
-.. [2] https://en.wikipedia.org/wiki/Generalized_hypergeometric_function</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="hyper.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#hyper.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">hyper</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">hyper</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#hyper.__init__"></a>
     
     
 
                             </div>
                             <div id="hyper.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#hyper.is_commutative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#hyper.is_commutative"></a>
     
     
 
                             </div>
                             <div id="hyper.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#hyper.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;commutative&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;commutative&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#hyper.default_assumptions"></a>
     
     
 
@@ -40127,36 +40976,36 @@ <h5>Inherited Members</h5>
                 <dd id="hyper.simplify" class="function">simplify</dd>
                 <dd id="hyper.refine" class="function">refine</dd>
                 <dd id="hyper.rewrite" class="function">rewrite</dd>
-                <dd id="hyper.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="hyper.is_integer" class="variable">is_integer</dd>
-                <dd id="hyper.is_complex" class="variable">is_complex</dd>
-                <dd id="hyper.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="hyper.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="hyper.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="hyper.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hyper.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hyper.is_positive" class="variable">is_positive</dd>
+                <dd id="hyper.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="hyper.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="hyper.is_negative" class="variable">is_negative</dd>
+                <dd id="hyper.is_composite" class="variable">is_composite</dd>
+                <dd id="hyper.is_finite" class="variable">is_finite</dd>
                 <dd id="hyper.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="hyper.is_zero" class="variable">is_zero</dd>
                 <dd id="hyper.is_even" class="variable">is_even</dd>
-                <dd id="hyper.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="hyper.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="hyper.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="hyper.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="hyper.is_real" class="variable">is_real</dd>
-                <dd id="hyper.is_positive" class="variable">is_positive</dd>
-                <dd id="hyper.is_finite" class="variable">is_finite</dd>
-                <dd id="hyper.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="hyper.is_infinite" class="variable">is_infinite</dd>
-                <dd id="hyper.is_rational" class="variable">is_rational</dd>
-                <dd id="hyper.is_composite" class="variable">is_composite</dd>
-                <dd id="hyper.is_negative" class="variable">is_negative</dd>
-                <dd id="hyper.is_polar" class="variable">is_polar</dd>
+                <dd id="hyper.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="hyper.is_odd" class="variable">is_odd</dd>
-                <dd id="hyper.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="hyper.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hyper.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="hyper.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="hyper.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hyper.is_integer" class="variable">is_integer</dd>
+                <dd id="hyper.is_rational" class="variable">is_rational</dd>
+                <dd id="hyper.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="hyper.is_complex" class="variable">is_complex</dd>
                 <dd id="hyper.is_prime" class="variable">is_prime</dd>
+                <dd id="hyper.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="hyper.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="hyper.is_real" class="variable">is_real</dd>
                 <dd id="hyper.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="hyper.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="hyper.is_polar" class="variable">is_polar</dd>
+                <dd id="hyper.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="hyper.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="hyper.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -40168,14 +41017,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="meijerg">
-                                <div class="attr class">
-        <a class="headerlink" href="#meijerg">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">meijerg</span><wbr>(<span class="base">sympy.functions.special.hyper.meijerg</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">meijerg</span><wbr>(<span class="base">sympy.functions.special.hyper.meijerg</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#meijerg"></a>
     
             <div class="docstring"><p>The Meijer G-function is defined by a Mellin-Barnes type integral that
 resembles an inverse Mellin transform. It generalizes the hypergeometric
@@ -40300,39 +41149,44 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations,
-       Volume 1
-.. [2] https://en.wikipedia.org/wiki/Meijer_G-function</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="meijerg.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#meijerg.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">meijerg</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">meijerg</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#meijerg.__init__"></a>
     
     
 
                             </div>
                             <div id="meijerg.is_commutative" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#meijerg.is_commutative">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">is_commutative</span><span class="default_value"> = True</span>
 
-        <span class="name">is_commutative</span><span class="default_value"> = True</span>
+        
     </div>
-
+    <a class="headerlink" href="#meijerg.is_commutative"></a>
     
     
 
                             </div>
                             <div id="meijerg.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#meijerg.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {&#39;commutative&#39;: True}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {&#39;commutative&#39;: True}</span>
+        
     </div>
-
+    <a class="headerlink" href="#meijerg.default_assumptions"></a>
     
     
 
@@ -40498,36 +41352,36 @@ <h5>Inherited Members</h5>
                 <dd id="meijerg.simplify" class="function">simplify</dd>
                 <dd id="meijerg.refine" class="function">refine</dd>
                 <dd id="meijerg.rewrite" class="function">rewrite</dd>
-                <dd id="meijerg.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="meijerg.is_integer" class="variable">is_integer</dd>
-                <dd id="meijerg.is_complex" class="variable">is_complex</dd>
-                <dd id="meijerg.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="meijerg.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="meijerg.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="meijerg.is_irrational" class="variable">is_irrational</dd>
+                <dd id="meijerg.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="meijerg.is_positive" class="variable">is_positive</dd>
+                <dd id="meijerg.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="meijerg.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="meijerg.is_negative" class="variable">is_negative</dd>
+                <dd id="meijerg.is_composite" class="variable">is_composite</dd>
+                <dd id="meijerg.is_finite" class="variable">is_finite</dd>
                 <dd id="meijerg.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="meijerg.is_zero" class="variable">is_zero</dd>
                 <dd id="meijerg.is_even" class="variable">is_even</dd>
-                <dd id="meijerg.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="meijerg.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="meijerg.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="meijerg.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="meijerg.is_real" class="variable">is_real</dd>
-                <dd id="meijerg.is_positive" class="variable">is_positive</dd>
-                <dd id="meijerg.is_finite" class="variable">is_finite</dd>
-                <dd id="meijerg.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="meijerg.is_infinite" class="variable">is_infinite</dd>
-                <dd id="meijerg.is_rational" class="variable">is_rational</dd>
-                <dd id="meijerg.is_composite" class="variable">is_composite</dd>
-                <dd id="meijerg.is_negative" class="variable">is_negative</dd>
-                <dd id="meijerg.is_polar" class="variable">is_polar</dd>
+                <dd id="meijerg.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="meijerg.is_odd" class="variable">is_odd</dd>
-                <dd id="meijerg.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="meijerg.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="meijerg.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="meijerg.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="meijerg.is_irrational" class="variable">is_irrational</dd>
+                <dd id="meijerg.is_integer" class="variable">is_integer</dd>
+                <dd id="meijerg.is_rational" class="variable">is_rational</dd>
+                <dd id="meijerg.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="meijerg.is_complex" class="variable">is_complex</dd>
                 <dd id="meijerg.is_prime" class="variable">is_prime</dd>
+                <dd id="meijerg.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="meijerg.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="meijerg.is_real" class="variable">is_real</dd>
                 <dd id="meijerg.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="meijerg.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="meijerg.is_polar" class="variable">is_polar</dd>
+                <dd id="meijerg.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="meijerg.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="meijerg.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -40539,14 +41393,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="appellf1">
-                                <div class="attr class">
-        <a class="headerlink" href="#appellf1">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">appellf1</span><wbr>(<span class="base">sympy.functions.special.hyper.appellf1</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">appellf1</span><wbr>(<span class="base">sympy.functions.special.hyper.appellf1</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#appellf1"></a>
     
             <div class="docstring"><p>This is the Appell hypergeometric function of two variables as:</p>
 
@@ -40575,28 +41429,33 @@ <h1 id="examples">Examples</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Appell_series
-.. [2] http://functions.wolfram.com/HypergeometricFunctions/AppellF1/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="appellf1.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#appellf1.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">appellf1</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">appellf1</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#appellf1.__init__"></a>
     
     
 
                             </div>
                             <div id="appellf1.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#appellf1.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#appellf1.default_assumptions"></a>
     
     
 
@@ -40751,37 +41610,37 @@ <h5>Inherited Members</h5>
                 <dd id="appellf1.simplify" class="function">simplify</dd>
                 <dd id="appellf1.refine" class="function">refine</dd>
                 <dd id="appellf1.rewrite" class="function">rewrite</dd>
-                <dd id="appellf1.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="appellf1.is_integer" class="variable">is_integer</dd>
-                <dd id="appellf1.is_complex" class="variable">is_complex</dd>
-                <dd id="appellf1.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="appellf1.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="appellf1.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="appellf1.is_irrational" class="variable">is_irrational</dd>
+                <dd id="appellf1.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="appellf1.is_positive" class="variable">is_positive</dd>
+                <dd id="appellf1.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="appellf1.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="appellf1.is_negative" class="variable">is_negative</dd>
+                <dd id="appellf1.is_composite" class="variable">is_composite</dd>
+                <dd id="appellf1.is_finite" class="variable">is_finite</dd>
                 <dd id="appellf1.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="appellf1.is_zero" class="variable">is_zero</dd>
                 <dd id="appellf1.is_even" class="variable">is_even</dd>
-                <dd id="appellf1.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="appellf1.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="appellf1.is_commutative" class="variable">is_commutative</dd>
-                <dd id="appellf1.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="appellf1.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="appellf1.is_real" class="variable">is_real</dd>
-                <dd id="appellf1.is_positive" class="variable">is_positive</dd>
-                <dd id="appellf1.is_finite" class="variable">is_finite</dd>
-                <dd id="appellf1.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="appellf1.is_infinite" class="variable">is_infinite</dd>
-                <dd id="appellf1.is_rational" class="variable">is_rational</dd>
-                <dd id="appellf1.is_composite" class="variable">is_composite</dd>
-                <dd id="appellf1.is_negative" class="variable">is_negative</dd>
-                <dd id="appellf1.is_polar" class="variable">is_polar</dd>
+                <dd id="appellf1.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="appellf1.is_odd" class="variable">is_odd</dd>
-                <dd id="appellf1.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="appellf1.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="appellf1.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="appellf1.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="appellf1.is_irrational" class="variable">is_irrational</dd>
+                <dd id="appellf1.is_integer" class="variable">is_integer</dd>
+                <dd id="appellf1.is_rational" class="variable">is_rational</dd>
+                <dd id="appellf1.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="appellf1.is_complex" class="variable">is_complex</dd>
                 <dd id="appellf1.is_prime" class="variable">is_prime</dd>
+                <dd id="appellf1.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="appellf1.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="appellf1.is_real" class="variable">is_real</dd>
+                <dd id="appellf1.is_commutative" class="variable">is_commutative</dd>
                 <dd id="appellf1.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="appellf1.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="appellf1.is_polar" class="variable">is_polar</dd>
+                <dd id="appellf1.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="appellf1.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="appellf1.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -40793,14 +41652,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="legendre">
-                                <div class="attr class">
-        <a class="headerlink" href="#legendre">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">legendre</span><wbr>(<span class="base">sympy.functions.special.polynomials.legendre</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">legendre</span><wbr>(<span class="base">sympy.functions.special.polynomials.legendre</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#legendre"></a>
     
             <div class="docstring"><p><code>legendre(n, x)</code> gives the nth Legendre polynomial of x, $P_n(x)$</p>
 
@@ -40843,30 +41702,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Legendre_polynomial
-.. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
-.. [3] http://functions.wolfram.com/Polynomials/LegendreP/
-.. [4] http://functions.wolfram.com/Polynomials/LegendreP2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="legendre.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#legendre.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">legendre</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">legendre</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#legendre.__init__"></a>
     
     
 
                             </div>
                             <div id="legendre.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#legendre.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#legendre.default_assumptions"></a>
     
     
 
@@ -41021,37 +41883,37 @@ <h5>Inherited Members</h5>
                 <dd id="legendre.simplify" class="function">simplify</dd>
                 <dd id="legendre.refine" class="function">refine</dd>
                 <dd id="legendre.rewrite" class="function">rewrite</dd>
-                <dd id="legendre.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="legendre.is_integer" class="variable">is_integer</dd>
-                <dd id="legendre.is_complex" class="variable">is_complex</dd>
-                <dd id="legendre.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="legendre.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="legendre.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="legendre.is_irrational" class="variable">is_irrational</dd>
+                <dd id="legendre.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="legendre.is_positive" class="variable">is_positive</dd>
+                <dd id="legendre.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="legendre.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="legendre.is_negative" class="variable">is_negative</dd>
+                <dd id="legendre.is_composite" class="variable">is_composite</dd>
+                <dd id="legendre.is_finite" class="variable">is_finite</dd>
                 <dd id="legendre.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="legendre.is_zero" class="variable">is_zero</dd>
                 <dd id="legendre.is_even" class="variable">is_even</dd>
-                <dd id="legendre.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="legendre.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="legendre.is_commutative" class="variable">is_commutative</dd>
-                <dd id="legendre.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="legendre.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="legendre.is_real" class="variable">is_real</dd>
-                <dd id="legendre.is_positive" class="variable">is_positive</dd>
-                <dd id="legendre.is_finite" class="variable">is_finite</dd>
-                <dd id="legendre.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="legendre.is_infinite" class="variable">is_infinite</dd>
-                <dd id="legendre.is_rational" class="variable">is_rational</dd>
-                <dd id="legendre.is_composite" class="variable">is_composite</dd>
-                <dd id="legendre.is_negative" class="variable">is_negative</dd>
-                <dd id="legendre.is_polar" class="variable">is_polar</dd>
+                <dd id="legendre.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="legendre.is_odd" class="variable">is_odd</dd>
-                <dd id="legendre.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="legendre.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="legendre.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="legendre.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="legendre.is_irrational" class="variable">is_irrational</dd>
+                <dd id="legendre.is_integer" class="variable">is_integer</dd>
+                <dd id="legendre.is_rational" class="variable">is_rational</dd>
+                <dd id="legendre.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="legendre.is_complex" class="variable">is_complex</dd>
                 <dd id="legendre.is_prime" class="variable">is_prime</dd>
+                <dd id="legendre.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="legendre.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="legendre.is_real" class="variable">is_real</dd>
+                <dd id="legendre.is_commutative" class="variable">is_commutative</dd>
                 <dd id="legendre.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="legendre.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="legendre.is_polar" class="variable">is_polar</dd>
+                <dd id="legendre.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="legendre.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="legendre.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -41063,22 +41925,21 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="assoc_legendre">
-                                <div class="attr class">
-        <a class="headerlink" href="#assoc_legendre">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">assoc_legendre</span><wbr>(<span class="base">sympy.functions.special.polynomials.assoc_legendre</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">assoc_legendre</span><wbr>(<span class="base">sympy.functions.special.polynomials.assoc_legendre</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#assoc_legendre"></a>
     
             <div class="docstring"><p><code>assoc_legendre(n, m, x)</code> gives $P_n^m(x)$, where n and m are
 the degree and order or an expression which is related to the nth
 order Legendre polynomial, $P_n(x)$ in the following manner:</p>
 
-<p>.. math::
-    P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}
-               \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}</p>
+<p>$$P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}
+           \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}$$</p>
 
 <h1 id="explanation">Explanation</h1>
 
@@ -41120,30 +41981,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
-.. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
-.. [3] http://functions.wolfram.com/Polynomials/LegendreP/
-.. [4] http://functions.wolfram.com/Polynomials/LegendreP2/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="assoc_legendre.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#assoc_legendre.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">assoc_legendre</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">assoc_legendre</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#assoc_legendre.__init__"></a>
     
     
 
                             </div>
                             <div id="assoc_legendre.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#assoc_legendre.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#assoc_legendre.default_assumptions"></a>
     
     
 
@@ -41298,37 +42162,37 @@ <h5>Inherited Members</h5>
                 <dd id="assoc_legendre.simplify" class="function">simplify</dd>
                 <dd id="assoc_legendre.refine" class="function">refine</dd>
                 <dd id="assoc_legendre.rewrite" class="function">rewrite</dd>
-                <dd id="assoc_legendre.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="assoc_legendre.is_integer" class="variable">is_integer</dd>
-                <dd id="assoc_legendre.is_complex" class="variable">is_complex</dd>
-                <dd id="assoc_legendre.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="assoc_legendre.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="assoc_legendre.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="assoc_legendre.is_irrational" class="variable">is_irrational</dd>
+                <dd id="assoc_legendre.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="assoc_legendre.is_positive" class="variable">is_positive</dd>
+                <dd id="assoc_legendre.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="assoc_legendre.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="assoc_legendre.is_negative" class="variable">is_negative</dd>
+                <dd id="assoc_legendre.is_composite" class="variable">is_composite</dd>
+                <dd id="assoc_legendre.is_finite" class="variable">is_finite</dd>
                 <dd id="assoc_legendre.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="assoc_legendre.is_zero" class="variable">is_zero</dd>
                 <dd id="assoc_legendre.is_even" class="variable">is_even</dd>
-                <dd id="assoc_legendre.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="assoc_legendre.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="assoc_legendre.is_commutative" class="variable">is_commutative</dd>
-                <dd id="assoc_legendre.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="assoc_legendre.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="assoc_legendre.is_real" class="variable">is_real</dd>
-                <dd id="assoc_legendre.is_positive" class="variable">is_positive</dd>
-                <dd id="assoc_legendre.is_finite" class="variable">is_finite</dd>
-                <dd id="assoc_legendre.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="assoc_legendre.is_infinite" class="variable">is_infinite</dd>
-                <dd id="assoc_legendre.is_rational" class="variable">is_rational</dd>
-                <dd id="assoc_legendre.is_composite" class="variable">is_composite</dd>
-                <dd id="assoc_legendre.is_negative" class="variable">is_negative</dd>
-                <dd id="assoc_legendre.is_polar" class="variable">is_polar</dd>
+                <dd id="assoc_legendre.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="assoc_legendre.is_odd" class="variable">is_odd</dd>
-                <dd id="assoc_legendre.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="assoc_legendre.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="assoc_legendre.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="assoc_legendre.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="assoc_legendre.is_irrational" class="variable">is_irrational</dd>
+                <dd id="assoc_legendre.is_integer" class="variable">is_integer</dd>
+                <dd id="assoc_legendre.is_rational" class="variable">is_rational</dd>
+                <dd id="assoc_legendre.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="assoc_legendre.is_complex" class="variable">is_complex</dd>
                 <dd id="assoc_legendre.is_prime" class="variable">is_prime</dd>
+                <dd id="assoc_legendre.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="assoc_legendre.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="assoc_legendre.is_real" class="variable">is_real</dd>
+                <dd id="assoc_legendre.is_commutative" class="variable">is_commutative</dd>
                 <dd id="assoc_legendre.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="assoc_legendre.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="assoc_legendre.is_polar" class="variable">is_polar</dd>
+                <dd id="assoc_legendre.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="assoc_legendre.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="assoc_legendre.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -41340,14 +42204,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="hermite">
-                                <div class="attr class">
-        <a class="headerlink" href="#hermite">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">hermite</span><wbr>(<span class="base">sympy.functions.special.polynomials.hermite</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">hermite</span><wbr>(<span class="base">sympy.functions.special.polynomials.hermite</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#hermite"></a>
     
             <div class="docstring"><p><code>hermite(n, x)</code> gives the nth Hermite polynomial in x, $H_n(x)$</p>
 
@@ -41390,29 +42254,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
-.. [2] http://mathworld.wolfram.com/HermitePolynomial.html
-.. [3] http://functions.wolfram.com/Polynomials/HermiteH/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="hermite.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#hermite.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">hermite</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">hermite</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#hermite.__init__"></a>
     
     
 
                             </div>
                             <div id="hermite.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#hermite.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#hermite.default_assumptions"></a>
     
     
 
@@ -41567,37 +42435,37 @@ <h5>Inherited Members</h5>
                 <dd id="hermite.simplify" class="function">simplify</dd>
                 <dd id="hermite.refine" class="function">refine</dd>
                 <dd id="hermite.rewrite" class="function">rewrite</dd>
-                <dd id="hermite.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="hermite.is_integer" class="variable">is_integer</dd>
-                <dd id="hermite.is_complex" class="variable">is_complex</dd>
-                <dd id="hermite.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="hermite.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="hermite.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="hermite.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hermite.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hermite.is_positive" class="variable">is_positive</dd>
+                <dd id="hermite.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="hermite.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="hermite.is_negative" class="variable">is_negative</dd>
+                <dd id="hermite.is_composite" class="variable">is_composite</dd>
+                <dd id="hermite.is_finite" class="variable">is_finite</dd>
                 <dd id="hermite.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="hermite.is_zero" class="variable">is_zero</dd>
                 <dd id="hermite.is_even" class="variable">is_even</dd>
-                <dd id="hermite.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="hermite.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="hermite.is_commutative" class="variable">is_commutative</dd>
-                <dd id="hermite.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="hermite.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="hermite.is_real" class="variable">is_real</dd>
-                <dd id="hermite.is_positive" class="variable">is_positive</dd>
-                <dd id="hermite.is_finite" class="variable">is_finite</dd>
-                <dd id="hermite.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="hermite.is_infinite" class="variable">is_infinite</dd>
-                <dd id="hermite.is_rational" class="variable">is_rational</dd>
-                <dd id="hermite.is_composite" class="variable">is_composite</dd>
-                <dd id="hermite.is_negative" class="variable">is_negative</dd>
-                <dd id="hermite.is_polar" class="variable">is_polar</dd>
+                <dd id="hermite.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="hermite.is_odd" class="variable">is_odd</dd>
-                <dd id="hermite.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="hermite.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="hermite.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="hermite.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="hermite.is_irrational" class="variable">is_irrational</dd>
+                <dd id="hermite.is_integer" class="variable">is_integer</dd>
+                <dd id="hermite.is_rational" class="variable">is_rational</dd>
+                <dd id="hermite.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="hermite.is_complex" class="variable">is_complex</dd>
                 <dd id="hermite.is_prime" class="variable">is_prime</dd>
+                <dd id="hermite.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="hermite.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="hermite.is_real" class="variable">is_real</dd>
+                <dd id="hermite.is_commutative" class="variable">is_commutative</dd>
                 <dd id="hermite.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="hermite.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="hermite.is_polar" class="variable">is_polar</dd>
+                <dd id="hermite.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="hermite.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="hermite.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -41609,14 +42477,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="chebyshevt">
-                                <div class="attr class">
-        <a class="headerlink" href="#chebyshevt">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">chebyshevt</span><wbr>(<span class="base">sympy.functions.special.polynomials.chebyshevt</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">chebyshevt</span><wbr>(<span class="base">sympy.functions.special.polynomials.chebyshevt</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#chebyshevt"></a>
     
             <div class="docstring"><p>Chebyshev polynomial of the first kind, $T_n(x)$.</p>
 
@@ -41675,31 +42543,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
-.. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
-.. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
-.. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
-.. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="chebyshevt.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#chebyshevt.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">chebyshevt</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">chebyshevt</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#chebyshevt.__init__"></a>
     
     
 
                             </div>
                             <div id="chebyshevt.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#chebyshevt.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#chebyshevt.default_assumptions"></a>
     
     
 
@@ -41854,37 +42724,37 @@ <h5>Inherited Members</h5>
                 <dd id="chebyshevt.simplify" class="function">simplify</dd>
                 <dd id="chebyshevt.refine" class="function">refine</dd>
                 <dd id="chebyshevt.rewrite" class="function">rewrite</dd>
-                <dd id="chebyshevt.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="chebyshevt.is_integer" class="variable">is_integer</dd>
-                <dd id="chebyshevt.is_complex" class="variable">is_complex</dd>
-                <dd id="chebyshevt.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="chebyshevt.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="chebyshevt.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="chebyshevt.is_irrational" class="variable">is_irrational</dd>
+                <dd id="chebyshevt.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="chebyshevt.is_positive" class="variable">is_positive</dd>
+                <dd id="chebyshevt.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="chebyshevt.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="chebyshevt.is_negative" class="variable">is_negative</dd>
+                <dd id="chebyshevt.is_composite" class="variable">is_composite</dd>
+                <dd id="chebyshevt.is_finite" class="variable">is_finite</dd>
                 <dd id="chebyshevt.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="chebyshevt.is_zero" class="variable">is_zero</dd>
                 <dd id="chebyshevt.is_even" class="variable">is_even</dd>
-                <dd id="chebyshevt.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="chebyshevt.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="chebyshevt.is_commutative" class="variable">is_commutative</dd>
-                <dd id="chebyshevt.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="chebyshevt.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="chebyshevt.is_real" class="variable">is_real</dd>
-                <dd id="chebyshevt.is_positive" class="variable">is_positive</dd>
-                <dd id="chebyshevt.is_finite" class="variable">is_finite</dd>
-                <dd id="chebyshevt.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="chebyshevt.is_infinite" class="variable">is_infinite</dd>
-                <dd id="chebyshevt.is_rational" class="variable">is_rational</dd>
-                <dd id="chebyshevt.is_composite" class="variable">is_composite</dd>
-                <dd id="chebyshevt.is_negative" class="variable">is_negative</dd>
-                <dd id="chebyshevt.is_polar" class="variable">is_polar</dd>
+                <dd id="chebyshevt.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="chebyshevt.is_odd" class="variable">is_odd</dd>
-                <dd id="chebyshevt.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="chebyshevt.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="chebyshevt.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="chebyshevt.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="chebyshevt.is_irrational" class="variable">is_irrational</dd>
+                <dd id="chebyshevt.is_integer" class="variable">is_integer</dd>
+                <dd id="chebyshevt.is_rational" class="variable">is_rational</dd>
+                <dd id="chebyshevt.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="chebyshevt.is_complex" class="variable">is_complex</dd>
                 <dd id="chebyshevt.is_prime" class="variable">is_prime</dd>
+                <dd id="chebyshevt.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="chebyshevt.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="chebyshevt.is_real" class="variable">is_real</dd>
+                <dd id="chebyshevt.is_commutative" class="variable">is_commutative</dd>
                 <dd id="chebyshevt.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="chebyshevt.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="chebyshevt.is_polar" class="variable">is_polar</dd>
+                <dd id="chebyshevt.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="chebyshevt.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="chebyshevt.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -41896,14 +42766,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="chebyshevu">
-                                <div class="attr class">
-        <a class="headerlink" href="#chebyshevu">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">chebyshevu</span><wbr>(<span class="base">sympy.functions.special.polynomials.chebyshevu</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">chebyshevu</span><wbr>(<span class="base">sympy.functions.special.polynomials.chebyshevu</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#chebyshevu"></a>
     
             <div class="docstring"><p>Chebyshev polynomial of the second kind, $U_n(x)$.</p>
 
@@ -41962,31 +42832,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
-.. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
-.. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
-.. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
-.. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="chebyshevu.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#chebyshevu.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">chebyshevu</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">chebyshevu</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#chebyshevu.__init__"></a>
     
     
 
                             </div>
                             <div id="chebyshevu.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#chebyshevu.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#chebyshevu.default_assumptions"></a>
     
     
 
@@ -42141,37 +43013,37 @@ <h5>Inherited Members</h5>
                 <dd id="chebyshevu.simplify" class="function">simplify</dd>
                 <dd id="chebyshevu.refine" class="function">refine</dd>
                 <dd id="chebyshevu.rewrite" class="function">rewrite</dd>
-                <dd id="chebyshevu.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="chebyshevu.is_integer" class="variable">is_integer</dd>
-                <dd id="chebyshevu.is_complex" class="variable">is_complex</dd>
-                <dd id="chebyshevu.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="chebyshevu.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="chebyshevu.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="chebyshevu.is_irrational" class="variable">is_irrational</dd>
+                <dd id="chebyshevu.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="chebyshevu.is_positive" class="variable">is_positive</dd>
+                <dd id="chebyshevu.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="chebyshevu.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="chebyshevu.is_negative" class="variable">is_negative</dd>
+                <dd id="chebyshevu.is_composite" class="variable">is_composite</dd>
+                <dd id="chebyshevu.is_finite" class="variable">is_finite</dd>
                 <dd id="chebyshevu.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="chebyshevu.is_zero" class="variable">is_zero</dd>
                 <dd id="chebyshevu.is_even" class="variable">is_even</dd>
-                <dd id="chebyshevu.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="chebyshevu.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="chebyshevu.is_commutative" class="variable">is_commutative</dd>
-                <dd id="chebyshevu.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="chebyshevu.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="chebyshevu.is_real" class="variable">is_real</dd>
-                <dd id="chebyshevu.is_positive" class="variable">is_positive</dd>
-                <dd id="chebyshevu.is_finite" class="variable">is_finite</dd>
-                <dd id="chebyshevu.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="chebyshevu.is_infinite" class="variable">is_infinite</dd>
-                <dd id="chebyshevu.is_rational" class="variable">is_rational</dd>
-                <dd id="chebyshevu.is_composite" class="variable">is_composite</dd>
-                <dd id="chebyshevu.is_negative" class="variable">is_negative</dd>
-                <dd id="chebyshevu.is_polar" class="variable">is_polar</dd>
+                <dd id="chebyshevu.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="chebyshevu.is_odd" class="variable">is_odd</dd>
-                <dd id="chebyshevu.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="chebyshevu.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="chebyshevu.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="chebyshevu.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="chebyshevu.is_irrational" class="variable">is_irrational</dd>
+                <dd id="chebyshevu.is_integer" class="variable">is_integer</dd>
+                <dd id="chebyshevu.is_rational" class="variable">is_rational</dd>
+                <dd id="chebyshevu.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="chebyshevu.is_complex" class="variable">is_complex</dd>
                 <dd id="chebyshevu.is_prime" class="variable">is_prime</dd>
+                <dd id="chebyshevu.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="chebyshevu.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="chebyshevu.is_real" class="variable">is_real</dd>
+                <dd id="chebyshevu.is_commutative" class="variable">is_commutative</dd>
                 <dd id="chebyshevu.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="chebyshevu.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="chebyshevu.is_polar" class="variable">is_polar</dd>
+                <dd id="chebyshevu.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="chebyshevu.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="chebyshevu.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -42183,14 +43055,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="chebyshevu_root">
-                                <div class="attr class">
-        <a class="headerlink" href="#chebyshevu_root">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">chebyshevu_root</span><wbr>(<span class="base">sympy.functions.special.polynomials.chebyshevu_root</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">chebyshevu_root</span><wbr>(<span class="base">sympy.functions.special.polynomials.chebyshevu_root</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#chebyshevu_root"></a>
     
             <div class="docstring"><p><code>chebyshevu_root(n, k)</code> returns the kth root (indexed from zero) of the
 nth Chebyshev polynomial of the second kind; that is, if 0 &lt;= k &lt; n,
@@ -42222,22 +43094,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="chebyshevu_root.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#chebyshevu_root.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">chebyshevu_root</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">chebyshevu_root</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#chebyshevu_root.__init__"></a>
     
     
 
                             </div>
                             <div id="chebyshevu_root.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#chebyshevu_root.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#chebyshevu_root.default_assumptions"></a>
     
     
 
@@ -42392,37 +43266,37 @@ <h5>Inherited Members</h5>
                 <dd id="chebyshevu_root.simplify" class="function">simplify</dd>
                 <dd id="chebyshevu_root.refine" class="function">refine</dd>
                 <dd id="chebyshevu_root.rewrite" class="function">rewrite</dd>
-                <dd id="chebyshevu_root.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="chebyshevu_root.is_integer" class="variable">is_integer</dd>
-                <dd id="chebyshevu_root.is_complex" class="variable">is_complex</dd>
-                <dd id="chebyshevu_root.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="chebyshevu_root.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="chebyshevu_root.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="chebyshevu_root.is_irrational" class="variable">is_irrational</dd>
+                <dd id="chebyshevu_root.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="chebyshevu_root.is_positive" class="variable">is_positive</dd>
+                <dd id="chebyshevu_root.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="chebyshevu_root.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="chebyshevu_root.is_negative" class="variable">is_negative</dd>
+                <dd id="chebyshevu_root.is_composite" class="variable">is_composite</dd>
+                <dd id="chebyshevu_root.is_finite" class="variable">is_finite</dd>
                 <dd id="chebyshevu_root.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="chebyshevu_root.is_zero" class="variable">is_zero</dd>
                 <dd id="chebyshevu_root.is_even" class="variable">is_even</dd>
-                <dd id="chebyshevu_root.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="chebyshevu_root.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="chebyshevu_root.is_commutative" class="variable">is_commutative</dd>
-                <dd id="chebyshevu_root.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="chebyshevu_root.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="chebyshevu_root.is_real" class="variable">is_real</dd>
-                <dd id="chebyshevu_root.is_positive" class="variable">is_positive</dd>
-                <dd id="chebyshevu_root.is_finite" class="variable">is_finite</dd>
-                <dd id="chebyshevu_root.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="chebyshevu_root.is_infinite" class="variable">is_infinite</dd>
-                <dd id="chebyshevu_root.is_rational" class="variable">is_rational</dd>
-                <dd id="chebyshevu_root.is_composite" class="variable">is_composite</dd>
-                <dd id="chebyshevu_root.is_negative" class="variable">is_negative</dd>
-                <dd id="chebyshevu_root.is_polar" class="variable">is_polar</dd>
+                <dd id="chebyshevu_root.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="chebyshevu_root.is_odd" class="variable">is_odd</dd>
-                <dd id="chebyshevu_root.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="chebyshevu_root.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="chebyshevu_root.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="chebyshevu_root.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="chebyshevu_root.is_irrational" class="variable">is_irrational</dd>
+                <dd id="chebyshevu_root.is_integer" class="variable">is_integer</dd>
+                <dd id="chebyshevu_root.is_rational" class="variable">is_rational</dd>
+                <dd id="chebyshevu_root.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="chebyshevu_root.is_complex" class="variable">is_complex</dd>
                 <dd id="chebyshevu_root.is_prime" class="variable">is_prime</dd>
+                <dd id="chebyshevu_root.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="chebyshevu_root.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="chebyshevu_root.is_real" class="variable">is_real</dd>
+                <dd id="chebyshevu_root.is_commutative" class="variable">is_commutative</dd>
                 <dd id="chebyshevu_root.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="chebyshevu_root.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="chebyshevu_root.is_polar" class="variable">is_polar</dd>
+                <dd id="chebyshevu_root.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="chebyshevu_root.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="chebyshevu_root.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -42434,14 +43308,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="chebyshevt_root">
-                                <div class="attr class">
-        <a class="headerlink" href="#chebyshevt_root">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">chebyshevt_root</span><wbr>(<span class="base">sympy.functions.special.polynomials.chebyshevt_root</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">chebyshevt_root</span><wbr>(<span class="base">sympy.functions.special.polynomials.chebyshevt_root</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#chebyshevt_root"></a>
     
             <div class="docstring"><p><code>chebyshev_root(n, k)</code> returns the kth root (indexed from zero) of
 the nth Chebyshev polynomial of the first kind; that is, if
@@ -42474,22 +43348,24 @@ <h1 id="see-also">See Also</h1>
 
 
                             <div id="chebyshevt_root.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#chebyshevt_root.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">chebyshevt_root</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">chebyshevt_root</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#chebyshevt_root.__init__"></a>
     
     
 
                             </div>
                             <div id="chebyshevt_root.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#chebyshevt_root.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#chebyshevt_root.default_assumptions"></a>
     
     
 
@@ -42644,37 +43520,37 @@ <h5>Inherited Members</h5>
                 <dd id="chebyshevt_root.simplify" class="function">simplify</dd>
                 <dd id="chebyshevt_root.refine" class="function">refine</dd>
                 <dd id="chebyshevt_root.rewrite" class="function">rewrite</dd>
-                <dd id="chebyshevt_root.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="chebyshevt_root.is_integer" class="variable">is_integer</dd>
-                <dd id="chebyshevt_root.is_complex" class="variable">is_complex</dd>
-                <dd id="chebyshevt_root.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="chebyshevt_root.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="chebyshevt_root.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="chebyshevt_root.is_irrational" class="variable">is_irrational</dd>
+                <dd id="chebyshevt_root.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="chebyshevt_root.is_positive" class="variable">is_positive</dd>
+                <dd id="chebyshevt_root.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="chebyshevt_root.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="chebyshevt_root.is_negative" class="variable">is_negative</dd>
+                <dd id="chebyshevt_root.is_composite" class="variable">is_composite</dd>
+                <dd id="chebyshevt_root.is_finite" class="variable">is_finite</dd>
                 <dd id="chebyshevt_root.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="chebyshevt_root.is_zero" class="variable">is_zero</dd>
                 <dd id="chebyshevt_root.is_even" class="variable">is_even</dd>
-                <dd id="chebyshevt_root.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="chebyshevt_root.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="chebyshevt_root.is_commutative" class="variable">is_commutative</dd>
-                <dd id="chebyshevt_root.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="chebyshevt_root.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="chebyshevt_root.is_real" class="variable">is_real</dd>
-                <dd id="chebyshevt_root.is_positive" class="variable">is_positive</dd>
-                <dd id="chebyshevt_root.is_finite" class="variable">is_finite</dd>
-                <dd id="chebyshevt_root.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="chebyshevt_root.is_infinite" class="variable">is_infinite</dd>
-                <dd id="chebyshevt_root.is_rational" class="variable">is_rational</dd>
-                <dd id="chebyshevt_root.is_composite" class="variable">is_composite</dd>
-                <dd id="chebyshevt_root.is_negative" class="variable">is_negative</dd>
-                <dd id="chebyshevt_root.is_polar" class="variable">is_polar</dd>
+                <dd id="chebyshevt_root.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="chebyshevt_root.is_odd" class="variable">is_odd</dd>
-                <dd id="chebyshevt_root.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="chebyshevt_root.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="chebyshevt_root.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="chebyshevt_root.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="chebyshevt_root.is_irrational" class="variable">is_irrational</dd>
+                <dd id="chebyshevt_root.is_integer" class="variable">is_integer</dd>
+                <dd id="chebyshevt_root.is_rational" class="variable">is_rational</dd>
+                <dd id="chebyshevt_root.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="chebyshevt_root.is_complex" class="variable">is_complex</dd>
                 <dd id="chebyshevt_root.is_prime" class="variable">is_prime</dd>
+                <dd id="chebyshevt_root.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="chebyshevt_root.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="chebyshevt_root.is_real" class="variable">is_real</dd>
+                <dd id="chebyshevt_root.is_commutative" class="variable">is_commutative</dd>
                 <dd id="chebyshevt_root.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="chebyshevt_root.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="chebyshevt_root.is_polar" class="variable">is_polar</dd>
+                <dd id="chebyshevt_root.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="chebyshevt_root.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="chebyshevt_root.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -42686,14 +43562,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="laguerre">
-                                <div class="attr class">
-        <a class="headerlink" href="#laguerre">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">laguerre</span><wbr>(<span class="base">sympy.functions.special.polynomials.laguerre</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">laguerre</span><wbr>(<span class="base">sympy.functions.special.polynomials.laguerre</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#laguerre"></a>
     
             <div class="docstring"><p>Returns the nth Laguerre polynomial in x, $L_n(x)$.</p>
 
@@ -42741,30 +43617,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial
-.. [2] http://mathworld.wolfram.com/LaguerrePolynomial.html
-.. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
-.. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="laguerre.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#laguerre.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">laguerre</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">laguerre</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#laguerre.__init__"></a>
     
     
 
                             </div>
                             <div id="laguerre.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#laguerre.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#laguerre.default_assumptions"></a>
     
     
 
@@ -42919,37 +43798,37 @@ <h5>Inherited Members</h5>
                 <dd id="laguerre.simplify" class="function">simplify</dd>
                 <dd id="laguerre.refine" class="function">refine</dd>
                 <dd id="laguerre.rewrite" class="function">rewrite</dd>
-                <dd id="laguerre.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="laguerre.is_integer" class="variable">is_integer</dd>
-                <dd id="laguerre.is_complex" class="variable">is_complex</dd>
-                <dd id="laguerre.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="laguerre.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="laguerre.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="laguerre.is_irrational" class="variable">is_irrational</dd>
+                <dd id="laguerre.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="laguerre.is_positive" class="variable">is_positive</dd>
+                <dd id="laguerre.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="laguerre.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="laguerre.is_negative" class="variable">is_negative</dd>
+                <dd id="laguerre.is_composite" class="variable">is_composite</dd>
+                <dd id="laguerre.is_finite" class="variable">is_finite</dd>
                 <dd id="laguerre.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="laguerre.is_zero" class="variable">is_zero</dd>
                 <dd id="laguerre.is_even" class="variable">is_even</dd>
-                <dd id="laguerre.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="laguerre.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="laguerre.is_commutative" class="variable">is_commutative</dd>
-                <dd id="laguerre.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="laguerre.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="laguerre.is_real" class="variable">is_real</dd>
-                <dd id="laguerre.is_positive" class="variable">is_positive</dd>
-                <dd id="laguerre.is_finite" class="variable">is_finite</dd>
-                <dd id="laguerre.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="laguerre.is_infinite" class="variable">is_infinite</dd>
-                <dd id="laguerre.is_rational" class="variable">is_rational</dd>
-                <dd id="laguerre.is_composite" class="variable">is_composite</dd>
-                <dd id="laguerre.is_negative" class="variable">is_negative</dd>
-                <dd id="laguerre.is_polar" class="variable">is_polar</dd>
+                <dd id="laguerre.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="laguerre.is_odd" class="variable">is_odd</dd>
-                <dd id="laguerre.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="laguerre.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="laguerre.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="laguerre.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="laguerre.is_irrational" class="variable">is_irrational</dd>
+                <dd id="laguerre.is_integer" class="variable">is_integer</dd>
+                <dd id="laguerre.is_rational" class="variable">is_rational</dd>
+                <dd id="laguerre.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="laguerre.is_complex" class="variable">is_complex</dd>
                 <dd id="laguerre.is_prime" class="variable">is_prime</dd>
+                <dd id="laguerre.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="laguerre.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="laguerre.is_real" class="variable">is_real</dd>
+                <dd id="laguerre.is_commutative" class="variable">is_commutative</dd>
                 <dd id="laguerre.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="laguerre.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="laguerre.is_polar" class="variable">is_polar</dd>
+                <dd id="laguerre.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="laguerre.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="laguerre.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -42961,14 +43840,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="assoc_laguerre">
-                                <div class="attr class">
-        <a class="headerlink" href="#assoc_laguerre">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">assoc_laguerre</span><wbr>(<span class="base">sympy.functions.special.polynomials.assoc_laguerre</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">assoc_laguerre</span><wbr>(<span class="base">sympy.functions.special.polynomials.assoc_laguerre</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#assoc_laguerre"></a>
     
             <div class="docstring"><p>Returns the nth generalized Laguerre polynomial in x, $L_n(x)$.</p>
 
@@ -43033,30 +43912,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials
-.. [2] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
-.. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
-.. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="assoc_laguerre.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#assoc_laguerre.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">assoc_laguerre</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">assoc_laguerre</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#assoc_laguerre.__init__"></a>
     
     
 
                             </div>
                             <div id="assoc_laguerre.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#assoc_laguerre.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#assoc_laguerre.default_assumptions"></a>
     
     
 
@@ -43211,37 +44093,37 @@ <h5>Inherited Members</h5>
                 <dd id="assoc_laguerre.simplify" class="function">simplify</dd>
                 <dd id="assoc_laguerre.refine" class="function">refine</dd>
                 <dd id="assoc_laguerre.rewrite" class="function">rewrite</dd>
-                <dd id="assoc_laguerre.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="assoc_laguerre.is_integer" class="variable">is_integer</dd>
-                <dd id="assoc_laguerre.is_complex" class="variable">is_complex</dd>
-                <dd id="assoc_laguerre.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="assoc_laguerre.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="assoc_laguerre.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="assoc_laguerre.is_irrational" class="variable">is_irrational</dd>
+                <dd id="assoc_laguerre.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="assoc_laguerre.is_positive" class="variable">is_positive</dd>
+                <dd id="assoc_laguerre.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="assoc_laguerre.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="assoc_laguerre.is_negative" class="variable">is_negative</dd>
+                <dd id="assoc_laguerre.is_composite" class="variable">is_composite</dd>
+                <dd id="assoc_laguerre.is_finite" class="variable">is_finite</dd>
                 <dd id="assoc_laguerre.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="assoc_laguerre.is_zero" class="variable">is_zero</dd>
                 <dd id="assoc_laguerre.is_even" class="variable">is_even</dd>
-                <dd id="assoc_laguerre.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="assoc_laguerre.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="assoc_laguerre.is_commutative" class="variable">is_commutative</dd>
-                <dd id="assoc_laguerre.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="assoc_laguerre.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="assoc_laguerre.is_real" class="variable">is_real</dd>
-                <dd id="assoc_laguerre.is_positive" class="variable">is_positive</dd>
-                <dd id="assoc_laguerre.is_finite" class="variable">is_finite</dd>
-                <dd id="assoc_laguerre.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="assoc_laguerre.is_infinite" class="variable">is_infinite</dd>
-                <dd id="assoc_laguerre.is_rational" class="variable">is_rational</dd>
-                <dd id="assoc_laguerre.is_composite" class="variable">is_composite</dd>
-                <dd id="assoc_laguerre.is_negative" class="variable">is_negative</dd>
-                <dd id="assoc_laguerre.is_polar" class="variable">is_polar</dd>
+                <dd id="assoc_laguerre.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="assoc_laguerre.is_odd" class="variable">is_odd</dd>
-                <dd id="assoc_laguerre.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="assoc_laguerre.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="assoc_laguerre.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="assoc_laguerre.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="assoc_laguerre.is_irrational" class="variable">is_irrational</dd>
+                <dd id="assoc_laguerre.is_integer" class="variable">is_integer</dd>
+                <dd id="assoc_laguerre.is_rational" class="variable">is_rational</dd>
+                <dd id="assoc_laguerre.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="assoc_laguerre.is_complex" class="variable">is_complex</dd>
                 <dd id="assoc_laguerre.is_prime" class="variable">is_prime</dd>
+                <dd id="assoc_laguerre.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="assoc_laguerre.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="assoc_laguerre.is_real" class="variable">is_real</dd>
+                <dd id="assoc_laguerre.is_commutative" class="variable">is_commutative</dd>
                 <dd id="assoc_laguerre.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="assoc_laguerre.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="assoc_laguerre.is_polar" class="variable">is_polar</dd>
+                <dd id="assoc_laguerre.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="assoc_laguerre.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="assoc_laguerre.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -43253,14 +44135,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="gegenbauer">
-                                <div class="attr class">
-        <a class="headerlink" href="#gegenbauer">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">gegenbauer</span><wbr>(<span class="base">sympy.functions.special.polynomials.gegenbauer</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">gegenbauer</span><wbr>(<span class="base">sympy.functions.special.polynomials.gegenbauer</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#gegenbauer"></a>
     
             <div class="docstring"><p>Gegenbauer polynomial $C_n^{\left(\alpha\right)}(x)$.</p>
 
@@ -43323,29 +44205,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials
-.. [2] http://mathworld.wolfram.com/GegenbauerPolynomial.html
-.. [3] http://functions.wolfram.com/Polynomials/GegenbauerC3/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="gegenbauer.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#gegenbauer.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">gegenbauer</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">gegenbauer</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#gegenbauer.__init__"></a>
     
     
 
                             </div>
                             <div id="gegenbauer.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#gegenbauer.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#gegenbauer.default_assumptions"></a>
     
     
 
@@ -43500,37 +44386,37 @@ <h5>Inherited Members</h5>
                 <dd id="gegenbauer.simplify" class="function">simplify</dd>
                 <dd id="gegenbauer.refine" class="function">refine</dd>
                 <dd id="gegenbauer.rewrite" class="function">rewrite</dd>
-                <dd id="gegenbauer.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="gegenbauer.is_integer" class="variable">is_integer</dd>
-                <dd id="gegenbauer.is_complex" class="variable">is_complex</dd>
-                <dd id="gegenbauer.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="gegenbauer.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="gegenbauer.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="gegenbauer.is_irrational" class="variable">is_irrational</dd>
+                <dd id="gegenbauer.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="gegenbauer.is_positive" class="variable">is_positive</dd>
+                <dd id="gegenbauer.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="gegenbauer.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="gegenbauer.is_negative" class="variable">is_negative</dd>
+                <dd id="gegenbauer.is_composite" class="variable">is_composite</dd>
+                <dd id="gegenbauer.is_finite" class="variable">is_finite</dd>
                 <dd id="gegenbauer.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="gegenbauer.is_zero" class="variable">is_zero</dd>
                 <dd id="gegenbauer.is_even" class="variable">is_even</dd>
-                <dd id="gegenbauer.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="gegenbauer.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="gegenbauer.is_commutative" class="variable">is_commutative</dd>
-                <dd id="gegenbauer.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="gegenbauer.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="gegenbauer.is_real" class="variable">is_real</dd>
-                <dd id="gegenbauer.is_positive" class="variable">is_positive</dd>
-                <dd id="gegenbauer.is_finite" class="variable">is_finite</dd>
-                <dd id="gegenbauer.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="gegenbauer.is_infinite" class="variable">is_infinite</dd>
-                <dd id="gegenbauer.is_rational" class="variable">is_rational</dd>
-                <dd id="gegenbauer.is_composite" class="variable">is_composite</dd>
-                <dd id="gegenbauer.is_negative" class="variable">is_negative</dd>
-                <dd id="gegenbauer.is_polar" class="variable">is_polar</dd>
+                <dd id="gegenbauer.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="gegenbauer.is_odd" class="variable">is_odd</dd>
-                <dd id="gegenbauer.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="gegenbauer.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="gegenbauer.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="gegenbauer.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="gegenbauer.is_irrational" class="variable">is_irrational</dd>
+                <dd id="gegenbauer.is_integer" class="variable">is_integer</dd>
+                <dd id="gegenbauer.is_rational" class="variable">is_rational</dd>
+                <dd id="gegenbauer.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="gegenbauer.is_complex" class="variable">is_complex</dd>
                 <dd id="gegenbauer.is_prime" class="variable">is_prime</dd>
+                <dd id="gegenbauer.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="gegenbauer.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="gegenbauer.is_real" class="variable">is_real</dd>
+                <dd id="gegenbauer.is_commutative" class="variable">is_commutative</dd>
                 <dd id="gegenbauer.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="gegenbauer.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="gegenbauer.is_polar" class="variable">is_polar</dd>
+                <dd id="gegenbauer.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="gegenbauer.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="gegenbauer.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -43542,14 +44428,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="jacobi">
-                                <div class="attr class">
-        <a class="headerlink" href="#jacobi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">jacobi</span><wbr>(<span class="base">sympy.functions.special.polynomials.jacobi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">jacobi</span><wbr>(<span class="base">sympy.functions.special.polynomials.jacobi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#jacobi"></a>
     
             <div class="docstring"><p>Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.</p>
 
@@ -43631,29 +44517,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
-.. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
-.. [3] http://functions.wolfram.com/Polynomials/JacobiP/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="jacobi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#jacobi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">jacobi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">jacobi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#jacobi.__init__"></a>
     
     
 
                             </div>
                             <div id="jacobi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#jacobi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#jacobi.default_assumptions"></a>
     
     
 
@@ -43808,37 +44698,37 @@ <h5>Inherited Members</h5>
                 <dd id="jacobi.simplify" class="function">simplify</dd>
                 <dd id="jacobi.refine" class="function">refine</dd>
                 <dd id="jacobi.rewrite" class="function">rewrite</dd>
-                <dd id="jacobi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="jacobi.is_integer" class="variable">is_integer</dd>
-                <dd id="jacobi.is_complex" class="variable">is_complex</dd>
-                <dd id="jacobi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="jacobi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="jacobi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="jacobi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="jacobi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="jacobi.is_positive" class="variable">is_positive</dd>
+                <dd id="jacobi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="jacobi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="jacobi.is_negative" class="variable">is_negative</dd>
+                <dd id="jacobi.is_composite" class="variable">is_composite</dd>
+                <dd id="jacobi.is_finite" class="variable">is_finite</dd>
                 <dd id="jacobi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="jacobi.is_zero" class="variable">is_zero</dd>
                 <dd id="jacobi.is_even" class="variable">is_even</dd>
-                <dd id="jacobi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="jacobi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="jacobi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="jacobi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="jacobi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="jacobi.is_real" class="variable">is_real</dd>
-                <dd id="jacobi.is_positive" class="variable">is_positive</dd>
-                <dd id="jacobi.is_finite" class="variable">is_finite</dd>
-                <dd id="jacobi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="jacobi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="jacobi.is_rational" class="variable">is_rational</dd>
-                <dd id="jacobi.is_composite" class="variable">is_composite</dd>
-                <dd id="jacobi.is_negative" class="variable">is_negative</dd>
-                <dd id="jacobi.is_polar" class="variable">is_polar</dd>
+                <dd id="jacobi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="jacobi.is_odd" class="variable">is_odd</dd>
-                <dd id="jacobi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="jacobi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="jacobi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="jacobi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="jacobi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="jacobi.is_integer" class="variable">is_integer</dd>
+                <dd id="jacobi.is_rational" class="variable">is_rational</dd>
+                <dd id="jacobi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="jacobi.is_complex" class="variable">is_complex</dd>
                 <dd id="jacobi.is_prime" class="variable">is_prime</dd>
+                <dd id="jacobi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="jacobi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="jacobi.is_real" class="variable">is_real</dd>
+                <dd id="jacobi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="jacobi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="jacobi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="jacobi.is_polar" class="variable">is_polar</dd>
+                <dd id="jacobi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="jacobi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="jacobi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -43850,21 +44740,20 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Ynm">
-                                <div class="attr class">
-        <a class="headerlink" href="#Ynm">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Ynm</span><wbr>(<span class="base">sympy.functions.special.spherical_harmonics.Ynm</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Ynm</span><wbr>(<span class="base">sympy.functions.special.spherical_harmonics.Ynm</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Ynm"></a>
     
             <div class="docstring"><p>Spherical harmonics defined as</p>
 
-<p>.. math::
-    Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}
-                              \exp(i m \varphi)
-                              \mathrm{P}_n^m\left(\cos(\theta)\right)</p>
+<p>$$Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}
+                          \exp(i m \varphi)
+                          \mathrm{P}_n^m\left(\cos(\theta)\right)$$</p>
 
 <h1 id="explanation">Explanation</h1>
 
@@ -43989,30 +44878,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
-.. [2] http://mathworld.wolfram.com/SphericalHarmonic.html
-.. [3] http://functions.wolfram.com/Polynomials/SphericalHarmonicY/
-.. [4] http://dlmf.nist.gov/14.30</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Ynm.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Ynm.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Ynm</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Ynm</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Ynm.__init__"></a>
     
     
 
                             </div>
                             <div id="Ynm.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Ynm.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Ynm.default_assumptions"></a>
     
     
 
@@ -44167,37 +45059,37 @@ <h5>Inherited Members</h5>
                 <dd id="Ynm.simplify" class="function">simplify</dd>
                 <dd id="Ynm.refine" class="function">refine</dd>
                 <dd id="Ynm.rewrite" class="function">rewrite</dd>
-                <dd id="Ynm.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Ynm.is_integer" class="variable">is_integer</dd>
-                <dd id="Ynm.is_complex" class="variable">is_complex</dd>
-                <dd id="Ynm.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Ynm.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Ynm.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Ynm.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Ynm.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Ynm.is_positive" class="variable">is_positive</dd>
+                <dd id="Ynm.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Ynm.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Ynm.is_negative" class="variable">is_negative</dd>
+                <dd id="Ynm.is_composite" class="variable">is_composite</dd>
+                <dd id="Ynm.is_finite" class="variable">is_finite</dd>
                 <dd id="Ynm.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Ynm.is_zero" class="variable">is_zero</dd>
                 <dd id="Ynm.is_even" class="variable">is_even</dd>
-                <dd id="Ynm.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Ynm.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Ynm.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Ynm.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Ynm.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Ynm.is_real" class="variable">is_real</dd>
-                <dd id="Ynm.is_positive" class="variable">is_positive</dd>
-                <dd id="Ynm.is_finite" class="variable">is_finite</dd>
-                <dd id="Ynm.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Ynm.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Ynm.is_rational" class="variable">is_rational</dd>
-                <dd id="Ynm.is_composite" class="variable">is_composite</dd>
-                <dd id="Ynm.is_negative" class="variable">is_negative</dd>
-                <dd id="Ynm.is_polar" class="variable">is_polar</dd>
+                <dd id="Ynm.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Ynm.is_odd" class="variable">is_odd</dd>
-                <dd id="Ynm.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Ynm.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Ynm.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Ynm.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Ynm.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Ynm.is_integer" class="variable">is_integer</dd>
+                <dd id="Ynm.is_rational" class="variable">is_rational</dd>
+                <dd id="Ynm.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Ynm.is_complex" class="variable">is_complex</dd>
                 <dd id="Ynm.is_prime" class="variable">is_prime</dd>
+                <dd id="Ynm.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Ynm.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Ynm.is_real" class="variable">is_real</dd>
+                <dd id="Ynm.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Ynm.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Ynm.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Ynm.is_polar" class="variable">is_polar</dd>
+                <dd id="Ynm.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Ynm.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Ynm.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -44209,38 +45101,32 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="Znm">
-                                <div class="attr class">
-        <a class="headerlink" href="#Znm">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">Znm</span><wbr>(<span class="base">sympy.functions.special.spherical_harmonics.Znm</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">Znm</span><wbr>(<span class="base">sympy.functions.special.spherical_harmonics.Znm</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#Znm"></a>
     
             <div class="docstring"><p>Real spherical harmonics defined as</p>
 
-<p>.. math::</p>
-
-<pre><code>Z_n^m(\theta, \varphi) :=
+<p>$$Z_n^m(\theta, \varphi) :=
 \begin{cases}
-  \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &amp;\quad m &gt; 0 \\
-  Y_n^m(\theta, \varphi) &amp;\quad m = 0 \\
-  \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &amp;\quad m &lt; 0 \\
-\end{cases}
-</code></pre>
+  \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &amp;\quad m &gt; 0 \
+  Y_n^m(\theta, \varphi) &amp;\quad m = 0 \
+  \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &amp;\quad m &lt; 0 \
+\end{cases}$$</p>
 
 <p>which gives in simplified form</p>
 
-<p>.. math::</p>
-
-<pre><code>Z_n^m(\theta, \varphi) =
+<p>$$Z_n^m(\theta, \varphi) =
 \begin{cases}
-  \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &amp;\quad m &gt; 0 \\
-  Y_n^m(\theta, \varphi) &amp;\quad m = 0 \\
-  \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &amp;\quad m &lt; 0 \\
-\end{cases}
-</code></pre>
+  \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &amp;\quad m &gt; 0 \
+  Y_n^m(\theta, \varphi) &amp;\quad m = 0 \
+  \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &amp;\quad m &lt; 0 \
+\end{cases}$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -44269,29 +45155,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
-.. [2] http://mathworld.wolfram.com/SphericalHarmonic.html
-.. [3] http://functions.wolfram.com/Polynomials/SphericalHarmonicY/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="Znm.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#Znm.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">Znm</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">Znm</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#Znm.__init__"></a>
     
     
 
                             </div>
                             <div id="Znm.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#Znm.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#Znm.default_assumptions"></a>
     
     
 
@@ -44446,37 +45336,37 @@ <h5>Inherited Members</h5>
                 <dd id="Znm.simplify" class="function">simplify</dd>
                 <dd id="Znm.refine" class="function">refine</dd>
                 <dd id="Znm.rewrite" class="function">rewrite</dd>
-                <dd id="Znm.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="Znm.is_integer" class="variable">is_integer</dd>
-                <dd id="Znm.is_complex" class="variable">is_complex</dd>
-                <dd id="Znm.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="Znm.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="Znm.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="Znm.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Znm.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Znm.is_positive" class="variable">is_positive</dd>
+                <dd id="Znm.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="Znm.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="Znm.is_negative" class="variable">is_negative</dd>
+                <dd id="Znm.is_composite" class="variable">is_composite</dd>
+                <dd id="Znm.is_finite" class="variable">is_finite</dd>
                 <dd id="Znm.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="Znm.is_zero" class="variable">is_zero</dd>
                 <dd id="Znm.is_even" class="variable">is_even</dd>
-                <dd id="Znm.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="Znm.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="Znm.is_commutative" class="variable">is_commutative</dd>
-                <dd id="Znm.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="Znm.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="Znm.is_real" class="variable">is_real</dd>
-                <dd id="Znm.is_positive" class="variable">is_positive</dd>
-                <dd id="Znm.is_finite" class="variable">is_finite</dd>
-                <dd id="Znm.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="Znm.is_infinite" class="variable">is_infinite</dd>
-                <dd id="Znm.is_rational" class="variable">is_rational</dd>
-                <dd id="Znm.is_composite" class="variable">is_composite</dd>
-                <dd id="Znm.is_negative" class="variable">is_negative</dd>
-                <dd id="Znm.is_polar" class="variable">is_polar</dd>
+                <dd id="Znm.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="Znm.is_odd" class="variable">is_odd</dd>
-                <dd id="Znm.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="Znm.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="Znm.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="Znm.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="Znm.is_irrational" class="variable">is_irrational</dd>
+                <dd id="Znm.is_integer" class="variable">is_integer</dd>
+                <dd id="Znm.is_rational" class="variable">is_rational</dd>
+                <dd id="Znm.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="Znm.is_complex" class="variable">is_complex</dd>
                 <dd id="Znm.is_prime" class="variable">is_prime</dd>
+                <dd id="Znm.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="Znm.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="Znm.is_real" class="variable">is_real</dd>
+                <dd id="Znm.is_commutative" class="variable">is_commutative</dd>
                 <dd id="Znm.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="Znm.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="Znm.is_polar" class="variable">is_polar</dd>
+                <dd id="Znm.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="Znm.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="Znm.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -44488,18 +45378,18 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="elliptic_k">
-                                <div class="attr class">
-        <a class="headerlink" href="#elliptic_k">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">elliptic_k</span><wbr>(<span class="base">sympy.functions.special.elliptic_integrals.elliptic_k</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">elliptic_k</span><wbr>(<span class="base">sympy.functions.special.elliptic_integrals.elliptic_k</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#elliptic_k"></a>
     
             <div class="docstring"><p>The complete elliptic integral of the first kind, defined by</p>
 
-<p>.. math:: K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)</p>
+<p>$$K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)$$</p>
 
 <p>where $F\left(z\middle| m\right)$ is the Legendre incomplete
 elliptic integral of the first kind.</p>
@@ -44532,28 +45422,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
-.. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticK</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="elliptic_k.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#elliptic_k.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">elliptic_k</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">elliptic_k</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#elliptic_k.__init__"></a>
     
     
 
                             </div>
                             <div id="elliptic_k.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#elliptic_k.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#elliptic_k.default_assumptions"></a>
     
     
 
@@ -44708,37 +45603,37 @@ <h5>Inherited Members</h5>
                 <dd id="elliptic_k.simplify" class="function">simplify</dd>
                 <dd id="elliptic_k.refine" class="function">refine</dd>
                 <dd id="elliptic_k.rewrite" class="function">rewrite</dd>
-                <dd id="elliptic_k.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="elliptic_k.is_integer" class="variable">is_integer</dd>
-                <dd id="elliptic_k.is_complex" class="variable">is_complex</dd>
-                <dd id="elliptic_k.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="elliptic_k.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="elliptic_k.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="elliptic_k.is_irrational" class="variable">is_irrational</dd>
+                <dd id="elliptic_k.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="elliptic_k.is_positive" class="variable">is_positive</dd>
+                <dd id="elliptic_k.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="elliptic_k.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="elliptic_k.is_negative" class="variable">is_negative</dd>
+                <dd id="elliptic_k.is_composite" class="variable">is_composite</dd>
+                <dd id="elliptic_k.is_finite" class="variable">is_finite</dd>
                 <dd id="elliptic_k.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="elliptic_k.is_zero" class="variable">is_zero</dd>
                 <dd id="elliptic_k.is_even" class="variable">is_even</dd>
-                <dd id="elliptic_k.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="elliptic_k.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="elliptic_k.is_commutative" class="variable">is_commutative</dd>
-                <dd id="elliptic_k.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="elliptic_k.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="elliptic_k.is_real" class="variable">is_real</dd>
-                <dd id="elliptic_k.is_positive" class="variable">is_positive</dd>
-                <dd id="elliptic_k.is_finite" class="variable">is_finite</dd>
-                <dd id="elliptic_k.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="elliptic_k.is_infinite" class="variable">is_infinite</dd>
-                <dd id="elliptic_k.is_rational" class="variable">is_rational</dd>
-                <dd id="elliptic_k.is_composite" class="variable">is_composite</dd>
-                <dd id="elliptic_k.is_negative" class="variable">is_negative</dd>
-                <dd id="elliptic_k.is_polar" class="variable">is_polar</dd>
+                <dd id="elliptic_k.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="elliptic_k.is_odd" class="variable">is_odd</dd>
-                <dd id="elliptic_k.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="elliptic_k.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="elliptic_k.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="elliptic_k.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="elliptic_k.is_irrational" class="variable">is_irrational</dd>
+                <dd id="elliptic_k.is_integer" class="variable">is_integer</dd>
+                <dd id="elliptic_k.is_rational" class="variable">is_rational</dd>
+                <dd id="elliptic_k.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="elliptic_k.is_complex" class="variable">is_complex</dd>
                 <dd id="elliptic_k.is_prime" class="variable">is_prime</dd>
+                <dd id="elliptic_k.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="elliptic_k.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="elliptic_k.is_real" class="variable">is_real</dd>
+                <dd id="elliptic_k.is_commutative" class="variable">is_commutative</dd>
                 <dd id="elliptic_k.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="elliptic_k.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="elliptic_k.is_polar" class="variable">is_polar</dd>
+                <dd id="elliptic_k.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="elliptic_k.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="elliptic_k.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -44750,20 +45645,19 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="elliptic_f">
-                                <div class="attr class">
-        <a class="headerlink" href="#elliptic_f">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">elliptic_f</span><wbr>(<span class="base">sympy.functions.special.elliptic_integrals.elliptic_f</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">elliptic_f</span><wbr>(<span class="base">sympy.functions.special.elliptic_integrals.elliptic_f</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#elliptic_f"></a>
     
             <div class="docstring"><p>The Legendre incomplete elliptic integral of the first
 kind, defined by</p>
 
-<p>.. math:: F\left(z\middle| m\right) =
-          \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}</p>
+<p>$$F\left(z\middle| m\right) =\int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}$$</p>
 
 <h1 id="explanation">Explanation</h1>
 
@@ -44791,28 +45685,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
-.. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticF</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="elliptic_f.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#elliptic_f.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">elliptic_f</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">elliptic_f</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#elliptic_f.__init__"></a>
     
     
 
                             </div>
                             <div id="elliptic_f.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#elliptic_f.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#elliptic_f.default_assumptions"></a>
     
     
 
@@ -44967,37 +45866,37 @@ <h5>Inherited Members</h5>
                 <dd id="elliptic_f.simplify" class="function">simplify</dd>
                 <dd id="elliptic_f.refine" class="function">refine</dd>
                 <dd id="elliptic_f.rewrite" class="function">rewrite</dd>
-                <dd id="elliptic_f.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="elliptic_f.is_integer" class="variable">is_integer</dd>
-                <dd id="elliptic_f.is_complex" class="variable">is_complex</dd>
-                <dd id="elliptic_f.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="elliptic_f.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="elliptic_f.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="elliptic_f.is_irrational" class="variable">is_irrational</dd>
+                <dd id="elliptic_f.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="elliptic_f.is_positive" class="variable">is_positive</dd>
+                <dd id="elliptic_f.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="elliptic_f.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="elliptic_f.is_negative" class="variable">is_negative</dd>
+                <dd id="elliptic_f.is_composite" class="variable">is_composite</dd>
+                <dd id="elliptic_f.is_finite" class="variable">is_finite</dd>
                 <dd id="elliptic_f.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="elliptic_f.is_zero" class="variable">is_zero</dd>
                 <dd id="elliptic_f.is_even" class="variable">is_even</dd>
-                <dd id="elliptic_f.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="elliptic_f.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="elliptic_f.is_commutative" class="variable">is_commutative</dd>
-                <dd id="elliptic_f.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="elliptic_f.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="elliptic_f.is_real" class="variable">is_real</dd>
-                <dd id="elliptic_f.is_positive" class="variable">is_positive</dd>
-                <dd id="elliptic_f.is_finite" class="variable">is_finite</dd>
-                <dd id="elliptic_f.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="elliptic_f.is_infinite" class="variable">is_infinite</dd>
-                <dd id="elliptic_f.is_rational" class="variable">is_rational</dd>
-                <dd id="elliptic_f.is_composite" class="variable">is_composite</dd>
-                <dd id="elliptic_f.is_negative" class="variable">is_negative</dd>
-                <dd id="elliptic_f.is_polar" class="variable">is_polar</dd>
+                <dd id="elliptic_f.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="elliptic_f.is_odd" class="variable">is_odd</dd>
-                <dd id="elliptic_f.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="elliptic_f.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="elliptic_f.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="elliptic_f.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="elliptic_f.is_irrational" class="variable">is_irrational</dd>
+                <dd id="elliptic_f.is_integer" class="variable">is_integer</dd>
+                <dd id="elliptic_f.is_rational" class="variable">is_rational</dd>
+                <dd id="elliptic_f.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="elliptic_f.is_complex" class="variable">is_complex</dd>
                 <dd id="elliptic_f.is_prime" class="variable">is_prime</dd>
+                <dd id="elliptic_f.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="elliptic_f.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="elliptic_f.is_real" class="variable">is_real</dd>
+                <dd id="elliptic_f.is_commutative" class="variable">is_commutative</dd>
                 <dd id="elliptic_f.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="elliptic_f.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="elliptic_f.is_polar" class="variable">is_polar</dd>
+                <dd id="elliptic_f.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="elliptic_f.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="elliptic_f.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -45009,24 +45908,24 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="elliptic_e">
-                                <div class="attr class">
-        <a class="headerlink" href="#elliptic_e">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">elliptic_e</span><wbr>(<span class="base">sympy.functions.special.elliptic_integrals.elliptic_e</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">elliptic_e</span><wbr>(<span class="base">sympy.functions.special.elliptic_integrals.elliptic_e</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#elliptic_e"></a>
     
             <div class="docstring"><p>Called with two arguments $z$ and $m$, evaluates the
 incomplete elliptic integral of the second kind, defined by</p>
 
-<p>.. math:: E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt</p>
+<p>$$E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt$$</p>
 
 <p>Called with a single argument $m$, evaluates the Legendre complete
 elliptic integral of the second kind</p>
 
-<p>.. math:: E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)</p>
+<p>$$E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)$$</p>
 
 <h1 id="explanation">Explanation</h1>
 
@@ -45056,29 +45955,33 @@ <h1 id="examples">Examples</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
-.. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticE2
-.. [3] http://functions.wolfram.com/EllipticIntegrals/EllipticE</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="elliptic_e.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#elliptic_e.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">elliptic_e</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">elliptic_e</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#elliptic_e.__init__"></a>
     
     
 
                             </div>
                             <div id="elliptic_e.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#elliptic_e.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#elliptic_e.default_assumptions"></a>
     
     
 
@@ -45233,37 +46136,37 @@ <h5>Inherited Members</h5>
                 <dd id="elliptic_e.simplify" class="function">simplify</dd>
                 <dd id="elliptic_e.refine" class="function">refine</dd>
                 <dd id="elliptic_e.rewrite" class="function">rewrite</dd>
-                <dd id="elliptic_e.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="elliptic_e.is_integer" class="variable">is_integer</dd>
-                <dd id="elliptic_e.is_complex" class="variable">is_complex</dd>
-                <dd id="elliptic_e.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="elliptic_e.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="elliptic_e.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="elliptic_e.is_irrational" class="variable">is_irrational</dd>
+                <dd id="elliptic_e.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="elliptic_e.is_positive" class="variable">is_positive</dd>
+                <dd id="elliptic_e.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="elliptic_e.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="elliptic_e.is_negative" class="variable">is_negative</dd>
+                <dd id="elliptic_e.is_composite" class="variable">is_composite</dd>
+                <dd id="elliptic_e.is_finite" class="variable">is_finite</dd>
                 <dd id="elliptic_e.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="elliptic_e.is_zero" class="variable">is_zero</dd>
                 <dd id="elliptic_e.is_even" class="variable">is_even</dd>
-                <dd id="elliptic_e.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="elliptic_e.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="elliptic_e.is_commutative" class="variable">is_commutative</dd>
-                <dd id="elliptic_e.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="elliptic_e.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="elliptic_e.is_real" class="variable">is_real</dd>
-                <dd id="elliptic_e.is_positive" class="variable">is_positive</dd>
-                <dd id="elliptic_e.is_finite" class="variable">is_finite</dd>
-                <dd id="elliptic_e.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="elliptic_e.is_infinite" class="variable">is_infinite</dd>
-                <dd id="elliptic_e.is_rational" class="variable">is_rational</dd>
-                <dd id="elliptic_e.is_composite" class="variable">is_composite</dd>
-                <dd id="elliptic_e.is_negative" class="variable">is_negative</dd>
-                <dd id="elliptic_e.is_polar" class="variable">is_polar</dd>
+                <dd id="elliptic_e.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="elliptic_e.is_odd" class="variable">is_odd</dd>
-                <dd id="elliptic_e.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="elliptic_e.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="elliptic_e.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="elliptic_e.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="elliptic_e.is_irrational" class="variable">is_irrational</dd>
+                <dd id="elliptic_e.is_integer" class="variable">is_integer</dd>
+                <dd id="elliptic_e.is_rational" class="variable">is_rational</dd>
+                <dd id="elliptic_e.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="elliptic_e.is_complex" class="variable">is_complex</dd>
                 <dd id="elliptic_e.is_prime" class="variable">is_prime</dd>
+                <dd id="elliptic_e.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="elliptic_e.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="elliptic_e.is_real" class="variable">is_real</dd>
+                <dd id="elliptic_e.is_commutative" class="variable">is_commutative</dd>
                 <dd id="elliptic_e.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="elliptic_e.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="elliptic_e.is_polar" class="variable">is_polar</dd>
+                <dd id="elliptic_e.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="elliptic_e.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="elliptic_e.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -45275,26 +46178,24 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="elliptic_pi">
-                                <div class="attr class">
-        <a class="headerlink" href="#elliptic_pi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">elliptic_pi</span><wbr>(<span class="base">sympy.functions.special.elliptic_integrals.elliptic_pi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">elliptic_pi</span><wbr>(<span class="base">sympy.functions.special.elliptic_integrals.elliptic_pi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#elliptic_pi"></a>
     
             <div class="docstring"><p>Called with three arguments $n$, $z$ and $m$, evaluates the
 Legendre incomplete elliptic integral of the third kind, defined by</p>
 
-<p>.. math:: \Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt}
-          {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}</p>
+<p>$$\Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt}{\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}$$</p>
 
 <p>Called with two arguments $n$ and $m$, evaluates the complete
 elliptic integral of the third kind:</p>
 
-<p>.. math:: \Pi\left(n\middle| m\right) =
-          \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)</p>
+<p>$$\Pi\left(n\middle| m\right) =\Pi\left(n; \tfrac{\pi}{2}\middle| m\right)$$</p>
 
 <h1 id="explanation">Explanation</h1>
 
@@ -45319,29 +46220,33 @@ <h1 id="examples">Examples</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
-.. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticPi3
-.. [3] http://functions.wolfram.com/EllipticIntegrals/EllipticPi</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="elliptic_pi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#elliptic_pi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">elliptic_pi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">elliptic_pi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#elliptic_pi.__init__"></a>
     
     
 
                             </div>
                             <div id="elliptic_pi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#elliptic_pi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#elliptic_pi.default_assumptions"></a>
     
     
 
@@ -45496,37 +46401,37 @@ <h5>Inherited Members</h5>
                 <dd id="elliptic_pi.simplify" class="function">simplify</dd>
                 <dd id="elliptic_pi.refine" class="function">refine</dd>
                 <dd id="elliptic_pi.rewrite" class="function">rewrite</dd>
-                <dd id="elliptic_pi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="elliptic_pi.is_integer" class="variable">is_integer</dd>
-                <dd id="elliptic_pi.is_complex" class="variable">is_complex</dd>
-                <dd id="elliptic_pi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="elliptic_pi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="elliptic_pi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="elliptic_pi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="elliptic_pi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="elliptic_pi.is_positive" class="variable">is_positive</dd>
+                <dd id="elliptic_pi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="elliptic_pi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="elliptic_pi.is_negative" class="variable">is_negative</dd>
+                <dd id="elliptic_pi.is_composite" class="variable">is_composite</dd>
+                <dd id="elliptic_pi.is_finite" class="variable">is_finite</dd>
                 <dd id="elliptic_pi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="elliptic_pi.is_zero" class="variable">is_zero</dd>
                 <dd id="elliptic_pi.is_even" class="variable">is_even</dd>
-                <dd id="elliptic_pi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="elliptic_pi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="elliptic_pi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="elliptic_pi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="elliptic_pi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="elliptic_pi.is_real" class="variable">is_real</dd>
-                <dd id="elliptic_pi.is_positive" class="variable">is_positive</dd>
-                <dd id="elliptic_pi.is_finite" class="variable">is_finite</dd>
-                <dd id="elliptic_pi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="elliptic_pi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="elliptic_pi.is_rational" class="variable">is_rational</dd>
-                <dd id="elliptic_pi.is_composite" class="variable">is_composite</dd>
-                <dd id="elliptic_pi.is_negative" class="variable">is_negative</dd>
-                <dd id="elliptic_pi.is_polar" class="variable">is_polar</dd>
+                <dd id="elliptic_pi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="elliptic_pi.is_odd" class="variable">is_odd</dd>
-                <dd id="elliptic_pi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="elliptic_pi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="elliptic_pi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="elliptic_pi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="elliptic_pi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="elliptic_pi.is_integer" class="variable">is_integer</dd>
+                <dd id="elliptic_pi.is_rational" class="variable">is_rational</dd>
+                <dd id="elliptic_pi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="elliptic_pi.is_complex" class="variable">is_complex</dd>
                 <dd id="elliptic_pi.is_prime" class="variable">is_prime</dd>
+                <dd id="elliptic_pi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="elliptic_pi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="elliptic_pi.is_real" class="variable">is_real</dd>
+                <dd id="elliptic_pi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="elliptic_pi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="elliptic_pi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="elliptic_pi.is_polar" class="variable">is_polar</dd>
+                <dd id="elliptic_pi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="elliptic_pi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="elliptic_pi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -45538,20 +46443,19 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="beta">
-                                <div class="attr class">
-        <a class="headerlink" href="#beta">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">beta</span><wbr>(<span class="base">sympy.functions.special.beta_functions.beta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">beta</span><wbr>(<span class="base">sympy.functions.special.beta_functions.beta</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#beta"></a>
     
             <div class="docstring"><p>The beta integral is called the Eulerian integral of the first kind by
 Legendre:</p>
 
-<p>.. math::
-    \mathrm{B}(x,y)  \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t.</p>
+<p>$$\mathrm{B}(x,y)  \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t.$$</p>
 
 <h1 id="explanation">Explanation</h1>
 
@@ -45559,24 +46463,21 @@ <h1 id="explanation">Explanation</h1>
 with the gamma function. The Beta function is often used in probability
 theory and mathematical statistics. It satisfies properties like:</p>
 
-<p>.. math::
-    \mathrm{B}(a,1) = \frac{1}{a} \
-    \mathrm{B}(a,b) = \mathrm{B}(b,a)  \
-    \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}</p>
+<p>$$\mathrm{B}(a,1) = \frac{1}{a} \
+\mathrm{B}(a,b) = \mathrm{B}(b,a)  \
+\mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}$$</p>
 
 <p>Therefore for integral values of $a$ and $b$:</p>
 
-<p>.. math::
-    \mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!}</p>
+<p>$$\mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!}$$</p>
 
 <p>A special case of the Beta function when <code>x = y</code> is the
 Central Beta function. It satisfies properties like:</p>
 
-<p>.. math::
-    \mathrm{B}(x) = 2^{1 - 2x}\mathrm{B}(x, \frac{1}{2})
-    \mathrm{B}(x) = 2^{1 - 2x} cos(\pi x) \mathrm{B}(\frac{1}{2} - x, x)
-    \mathrm{B}(x) = \int_{0}^{1} \frac{t^x}{(1 + t)^{2x}} dt
-    \mathrm{B}(x) = \frac{2}{x} \prod_{n = 1}^{\infty} \frac{n(n + 2x)}{(n + x)^2}</p>
+<p>$$\mathrm{B}(x) = 2^{1 - 2x}\mathrm{B}(x, \frac{1}{2})
+\mathrm{B}(x) = 2^{1 - 2x} cos(\pi x) \mathrm{B}(\frac{1}{2} - x, x)
+\mathrm{B}(x) = \int_{0}^{1} \frac{t^x}{(1 + t)^{2x}} dt
+\mathrm{B}(x) = \frac{2}{x} \prod_{n = 1}^{\infty} \frac{n(n + 2x)}{(n + x)^2}$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -45630,29 +46531,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Beta_function
-.. [2] http://mathworld.wolfram.com/BetaFunction.html
-.. [3] http://dlmf.nist.gov/5.12</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="beta.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#beta.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">beta</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">beta</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#beta.__init__"></a>
     
     
 
                             </div>
                             <div id="beta.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#beta.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#beta.default_assumptions"></a>
     
     
 
@@ -45808,37 +46713,37 @@ <h5>Inherited Members</h5>
                 <dd id="beta.simplify" class="function">simplify</dd>
                 <dd id="beta.refine" class="function">refine</dd>
                 <dd id="beta.rewrite" class="function">rewrite</dd>
-                <dd id="beta.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="beta.is_integer" class="variable">is_integer</dd>
-                <dd id="beta.is_complex" class="variable">is_complex</dd>
-                <dd id="beta.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="beta.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="beta.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="beta.is_irrational" class="variable">is_irrational</dd>
+                <dd id="beta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="beta.is_positive" class="variable">is_positive</dd>
+                <dd id="beta.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="beta.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="beta.is_negative" class="variable">is_negative</dd>
+                <dd id="beta.is_composite" class="variable">is_composite</dd>
+                <dd id="beta.is_finite" class="variable">is_finite</dd>
                 <dd id="beta.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="beta.is_zero" class="variable">is_zero</dd>
                 <dd id="beta.is_even" class="variable">is_even</dd>
-                <dd id="beta.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="beta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="beta.is_commutative" class="variable">is_commutative</dd>
-                <dd id="beta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="beta.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="beta.is_real" class="variable">is_real</dd>
-                <dd id="beta.is_positive" class="variable">is_positive</dd>
-                <dd id="beta.is_finite" class="variable">is_finite</dd>
-                <dd id="beta.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="beta.is_infinite" class="variable">is_infinite</dd>
-                <dd id="beta.is_rational" class="variable">is_rational</dd>
-                <dd id="beta.is_composite" class="variable">is_composite</dd>
-                <dd id="beta.is_negative" class="variable">is_negative</dd>
-                <dd id="beta.is_polar" class="variable">is_polar</dd>
+                <dd id="beta.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="beta.is_odd" class="variable">is_odd</dd>
-                <dd id="beta.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="beta.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="beta.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="beta.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="beta.is_irrational" class="variable">is_irrational</dd>
+                <dd id="beta.is_integer" class="variable">is_integer</dd>
+                <dd id="beta.is_rational" class="variable">is_rational</dd>
+                <dd id="beta.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="beta.is_complex" class="variable">is_complex</dd>
                 <dd id="beta.is_prime" class="variable">is_prime</dd>
+                <dd id="beta.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="beta.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="beta.is_real" class="variable">is_real</dd>
+                <dd id="beta.is_commutative" class="variable">is_commutative</dd>
                 <dd id="beta.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="beta.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="beta.is_polar" class="variable">is_polar</dd>
+                <dd id="beta.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="beta.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="beta.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -45850,14 +46755,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="mathieus">
-                                <div class="attr class">
-        <a class="headerlink" href="#mathieus">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">mathieus</span><wbr>(<span class="base">sympy.functions.special.mathieu_functions.mathieus</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">mathieus</span><wbr>(<span class="base">sympy.functions.special.mathieu_functions.mathieus</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#mathieus"></a>
     
             <div class="docstring"><p>The Mathieu Sine function $S(a,q,z)$.</p>
 
@@ -45896,30 +46801,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Mathieu_function
-.. [2] http://dlmf.nist.gov/28
-.. [3] http://mathworld.wolfram.com/MathieuBase.html
-.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="mathieus.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#mathieus.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">mathieus</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">mathieus</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#mathieus.__init__"></a>
     
     
 
                             </div>
                             <div id="mathieus.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#mathieus.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#mathieus.default_assumptions"></a>
     
     
 
@@ -46078,37 +46986,37 @@ <h5>Inherited Members</h5>
                 <dd id="mathieus.simplify" class="function">simplify</dd>
                 <dd id="mathieus.refine" class="function">refine</dd>
                 <dd id="mathieus.rewrite" class="function">rewrite</dd>
-                <dd id="mathieus.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="mathieus.is_integer" class="variable">is_integer</dd>
-                <dd id="mathieus.is_complex" class="variable">is_complex</dd>
-                <dd id="mathieus.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="mathieus.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="mathieus.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="mathieus.is_irrational" class="variable">is_irrational</dd>
+                <dd id="mathieus.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="mathieus.is_positive" class="variable">is_positive</dd>
+                <dd id="mathieus.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="mathieus.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="mathieus.is_negative" class="variable">is_negative</dd>
+                <dd id="mathieus.is_composite" class="variable">is_composite</dd>
+                <dd id="mathieus.is_finite" class="variable">is_finite</dd>
                 <dd id="mathieus.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="mathieus.is_zero" class="variable">is_zero</dd>
                 <dd id="mathieus.is_even" class="variable">is_even</dd>
-                <dd id="mathieus.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="mathieus.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="mathieus.is_commutative" class="variable">is_commutative</dd>
-                <dd id="mathieus.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="mathieus.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="mathieus.is_real" class="variable">is_real</dd>
-                <dd id="mathieus.is_positive" class="variable">is_positive</dd>
-                <dd id="mathieus.is_finite" class="variable">is_finite</dd>
-                <dd id="mathieus.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="mathieus.is_infinite" class="variable">is_infinite</dd>
-                <dd id="mathieus.is_rational" class="variable">is_rational</dd>
-                <dd id="mathieus.is_composite" class="variable">is_composite</dd>
-                <dd id="mathieus.is_negative" class="variable">is_negative</dd>
-                <dd id="mathieus.is_polar" class="variable">is_polar</dd>
+                <dd id="mathieus.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="mathieus.is_odd" class="variable">is_odd</dd>
-                <dd id="mathieus.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="mathieus.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="mathieus.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="mathieus.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="mathieus.is_irrational" class="variable">is_irrational</dd>
+                <dd id="mathieus.is_integer" class="variable">is_integer</dd>
+                <dd id="mathieus.is_rational" class="variable">is_rational</dd>
+                <dd id="mathieus.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="mathieus.is_complex" class="variable">is_complex</dd>
                 <dd id="mathieus.is_prime" class="variable">is_prime</dd>
+                <dd id="mathieus.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="mathieus.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="mathieus.is_real" class="variable">is_real</dd>
+                <dd id="mathieus.is_commutative" class="variable">is_commutative</dd>
                 <dd id="mathieus.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="mathieus.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="mathieus.is_polar" class="variable">is_polar</dd>
+                <dd id="mathieus.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="mathieus.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="mathieus.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -46120,14 +47028,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="mathieuc">
-                                <div class="attr class">
-        <a class="headerlink" href="#mathieuc">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">mathieuc</span><wbr>(<span class="base">sympy.functions.special.mathieu_functions.mathieuc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">mathieuc</span><wbr>(<span class="base">sympy.functions.special.mathieu_functions.mathieuc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#mathieuc"></a>
     
             <div class="docstring"><p>The Mathieu Cosine function $C(a,q,z)$.</p>
 
@@ -46166,30 +47074,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Mathieu_function
-.. [2] http://dlmf.nist.gov/28
-.. [3] http://mathworld.wolfram.com/MathieuBase.html
-.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="mathieuc.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#mathieuc.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">mathieuc</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">mathieuc</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#mathieuc.__init__"></a>
     
     
 
                             </div>
                             <div id="mathieuc.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#mathieuc.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#mathieuc.default_assumptions"></a>
     
     
 
@@ -46348,37 +47259,37 @@ <h5>Inherited Members</h5>
                 <dd id="mathieuc.simplify" class="function">simplify</dd>
                 <dd id="mathieuc.refine" class="function">refine</dd>
                 <dd id="mathieuc.rewrite" class="function">rewrite</dd>
-                <dd id="mathieuc.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="mathieuc.is_integer" class="variable">is_integer</dd>
-                <dd id="mathieuc.is_complex" class="variable">is_complex</dd>
-                <dd id="mathieuc.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="mathieuc.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="mathieuc.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="mathieuc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="mathieuc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="mathieuc.is_positive" class="variable">is_positive</dd>
+                <dd id="mathieuc.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="mathieuc.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="mathieuc.is_negative" class="variable">is_negative</dd>
+                <dd id="mathieuc.is_composite" class="variable">is_composite</dd>
+                <dd id="mathieuc.is_finite" class="variable">is_finite</dd>
                 <dd id="mathieuc.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="mathieuc.is_zero" class="variable">is_zero</dd>
                 <dd id="mathieuc.is_even" class="variable">is_even</dd>
-                <dd id="mathieuc.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="mathieuc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="mathieuc.is_commutative" class="variable">is_commutative</dd>
-                <dd id="mathieuc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="mathieuc.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="mathieuc.is_real" class="variable">is_real</dd>
-                <dd id="mathieuc.is_positive" class="variable">is_positive</dd>
-                <dd id="mathieuc.is_finite" class="variable">is_finite</dd>
-                <dd id="mathieuc.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="mathieuc.is_infinite" class="variable">is_infinite</dd>
-                <dd id="mathieuc.is_rational" class="variable">is_rational</dd>
-                <dd id="mathieuc.is_composite" class="variable">is_composite</dd>
-                <dd id="mathieuc.is_negative" class="variable">is_negative</dd>
-                <dd id="mathieuc.is_polar" class="variable">is_polar</dd>
+                <dd id="mathieuc.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="mathieuc.is_odd" class="variable">is_odd</dd>
-                <dd id="mathieuc.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="mathieuc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="mathieuc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="mathieuc.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="mathieuc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="mathieuc.is_integer" class="variable">is_integer</dd>
+                <dd id="mathieuc.is_rational" class="variable">is_rational</dd>
+                <dd id="mathieuc.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="mathieuc.is_complex" class="variable">is_complex</dd>
                 <dd id="mathieuc.is_prime" class="variable">is_prime</dd>
+                <dd id="mathieuc.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="mathieuc.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="mathieuc.is_real" class="variable">is_real</dd>
+                <dd id="mathieuc.is_commutative" class="variable">is_commutative</dd>
                 <dd id="mathieuc.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="mathieuc.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="mathieuc.is_polar" class="variable">is_polar</dd>
+                <dd id="mathieuc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="mathieuc.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="mathieuc.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -46390,14 +47301,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="mathieusprime">
-                                <div class="attr class">
-        <a class="headerlink" href="#mathieusprime">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">mathieusprime</span><wbr>(<span class="base">sympy.functions.special.mathieu_functions.mathieusprime</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">mathieusprime</span><wbr>(<span class="base">sympy.functions.special.mathieu_functions.mathieusprime</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#mathieusprime"></a>
     
             <div class="docstring"><p>The derivative $S^{\prime}(a,q,z)$ of the Mathieu Sine function.</p>
 
@@ -46436,30 +47347,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Mathieu_function
-.. [2] http://dlmf.nist.gov/28
-.. [3] http://mathworld.wolfram.com/MathieuBase.html
-.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="mathieusprime.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#mathieusprime.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">mathieusprime</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">mathieusprime</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#mathieusprime.__init__"></a>
     
     
 
                             </div>
                             <div id="mathieusprime.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#mathieusprime.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#mathieusprime.default_assumptions"></a>
     
     
 
@@ -46618,37 +47532,37 @@ <h5>Inherited Members</h5>
                 <dd id="mathieusprime.simplify" class="function">simplify</dd>
                 <dd id="mathieusprime.refine" class="function">refine</dd>
                 <dd id="mathieusprime.rewrite" class="function">rewrite</dd>
-                <dd id="mathieusprime.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="mathieusprime.is_integer" class="variable">is_integer</dd>
-                <dd id="mathieusprime.is_complex" class="variable">is_complex</dd>
-                <dd id="mathieusprime.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="mathieusprime.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="mathieusprime.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="mathieusprime.is_irrational" class="variable">is_irrational</dd>
+                <dd id="mathieusprime.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="mathieusprime.is_positive" class="variable">is_positive</dd>
+                <dd id="mathieusprime.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="mathieusprime.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="mathieusprime.is_negative" class="variable">is_negative</dd>
+                <dd id="mathieusprime.is_composite" class="variable">is_composite</dd>
+                <dd id="mathieusprime.is_finite" class="variable">is_finite</dd>
                 <dd id="mathieusprime.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="mathieusprime.is_zero" class="variable">is_zero</dd>
                 <dd id="mathieusprime.is_even" class="variable">is_even</dd>
-                <dd id="mathieusprime.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="mathieusprime.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="mathieusprime.is_commutative" class="variable">is_commutative</dd>
-                <dd id="mathieusprime.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="mathieusprime.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="mathieusprime.is_real" class="variable">is_real</dd>
-                <dd id="mathieusprime.is_positive" class="variable">is_positive</dd>
-                <dd id="mathieusprime.is_finite" class="variable">is_finite</dd>
-                <dd id="mathieusprime.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="mathieusprime.is_infinite" class="variable">is_infinite</dd>
-                <dd id="mathieusprime.is_rational" class="variable">is_rational</dd>
-                <dd id="mathieusprime.is_composite" class="variable">is_composite</dd>
-                <dd id="mathieusprime.is_negative" class="variable">is_negative</dd>
-                <dd id="mathieusprime.is_polar" class="variable">is_polar</dd>
+                <dd id="mathieusprime.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="mathieusprime.is_odd" class="variable">is_odd</dd>
-                <dd id="mathieusprime.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="mathieusprime.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="mathieusprime.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="mathieusprime.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="mathieusprime.is_irrational" class="variable">is_irrational</dd>
+                <dd id="mathieusprime.is_integer" class="variable">is_integer</dd>
+                <dd id="mathieusprime.is_rational" class="variable">is_rational</dd>
+                <dd id="mathieusprime.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="mathieusprime.is_complex" class="variable">is_complex</dd>
                 <dd id="mathieusprime.is_prime" class="variable">is_prime</dd>
+                <dd id="mathieusprime.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="mathieusprime.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="mathieusprime.is_real" class="variable">is_real</dd>
+                <dd id="mathieusprime.is_commutative" class="variable">is_commutative</dd>
                 <dd id="mathieusprime.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="mathieusprime.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="mathieusprime.is_polar" class="variable">is_polar</dd>
+                <dd id="mathieusprime.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="mathieusprime.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="mathieusprime.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -46660,14 +47574,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="mathieucprime">
-                                <div class="attr class">
-        <a class="headerlink" href="#mathieucprime">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">mathieucprime</span><wbr>(<span class="base">sympy.functions.special.mathieu_functions.mathieucprime</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">mathieucprime</span><wbr>(<span class="base">sympy.functions.special.mathieu_functions.mathieucprime</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#mathieucprime"></a>
     
             <div class="docstring"><p>The derivative $C^{\prime}(a,q,z)$ of the Mathieu Cosine function.</p>
 
@@ -46706,30 +47620,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Mathieu_function
-.. [2] http://dlmf.nist.gov/28
-.. [3] http://mathworld.wolfram.com/MathieuBase.html
-.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="mathieucprime.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#mathieucprime.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">mathieucprime</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">mathieucprime</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#mathieucprime.__init__"></a>
     
     
 
                             </div>
                             <div id="mathieucprime.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#mathieucprime.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#mathieucprime.default_assumptions"></a>
     
     
 
@@ -46888,37 +47805,37 @@ <h5>Inherited Members</h5>
                 <dd id="mathieucprime.simplify" class="function">simplify</dd>
                 <dd id="mathieucprime.refine" class="function">refine</dd>
                 <dd id="mathieucprime.rewrite" class="function">rewrite</dd>
-                <dd id="mathieucprime.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="mathieucprime.is_integer" class="variable">is_integer</dd>
-                <dd id="mathieucprime.is_complex" class="variable">is_complex</dd>
-                <dd id="mathieucprime.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="mathieucprime.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="mathieucprime.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="mathieucprime.is_irrational" class="variable">is_irrational</dd>
+                <dd id="mathieucprime.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="mathieucprime.is_positive" class="variable">is_positive</dd>
+                <dd id="mathieucprime.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="mathieucprime.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="mathieucprime.is_negative" class="variable">is_negative</dd>
+                <dd id="mathieucprime.is_composite" class="variable">is_composite</dd>
+                <dd id="mathieucprime.is_finite" class="variable">is_finite</dd>
                 <dd id="mathieucprime.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="mathieucprime.is_zero" class="variable">is_zero</dd>
                 <dd id="mathieucprime.is_even" class="variable">is_even</dd>
-                <dd id="mathieucprime.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="mathieucprime.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="mathieucprime.is_commutative" class="variable">is_commutative</dd>
-                <dd id="mathieucprime.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="mathieucprime.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="mathieucprime.is_real" class="variable">is_real</dd>
-                <dd id="mathieucprime.is_positive" class="variable">is_positive</dd>
-                <dd id="mathieucprime.is_finite" class="variable">is_finite</dd>
-                <dd id="mathieucprime.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="mathieucprime.is_infinite" class="variable">is_infinite</dd>
-                <dd id="mathieucprime.is_rational" class="variable">is_rational</dd>
-                <dd id="mathieucprime.is_composite" class="variable">is_composite</dd>
-                <dd id="mathieucprime.is_negative" class="variable">is_negative</dd>
-                <dd id="mathieucprime.is_polar" class="variable">is_polar</dd>
+                <dd id="mathieucprime.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="mathieucprime.is_odd" class="variable">is_odd</dd>
-                <dd id="mathieucprime.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="mathieucprime.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="mathieucprime.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="mathieucprime.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="mathieucprime.is_irrational" class="variable">is_irrational</dd>
+                <dd id="mathieucprime.is_integer" class="variable">is_integer</dd>
+                <dd id="mathieucprime.is_rational" class="variable">is_rational</dd>
+                <dd id="mathieucprime.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="mathieucprime.is_complex" class="variable">is_complex</dd>
                 <dd id="mathieucprime.is_prime" class="variable">is_prime</dd>
+                <dd id="mathieucprime.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="mathieucprime.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="mathieucprime.is_real" class="variable">is_real</dd>
+                <dd id="mathieucprime.is_commutative" class="variable">is_commutative</dd>
                 <dd id="mathieucprime.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="mathieucprime.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="mathieucprime.is_polar" class="variable">is_polar</dd>
+                <dd id="mathieucprime.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="mathieucprime.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="mathieucprime.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -46930,14 +47847,14 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="riemann_xi">
-                                <div class="attr class">
-        <a class="headerlink" href="#riemann_xi">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">riemann_xi</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.riemann_xi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">riemann_xi</span><wbr>(<span class="base">sympy.functions.special.zeta_functions.riemann_xi</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#riemann_xi"></a>
     
             <div class="docstring"><p>Riemann Xi function.</p>
 
@@ -46955,27 +47872,33 @@ <h1 id="examples">Examples</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Riemann_Xi_function</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="riemann_xi.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#riemann_xi.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">riemann_xi</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">riemann_xi</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#riemann_xi.__init__"></a>
     
     
 
                             </div>
                             <div id="riemann_xi.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#riemann_xi.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#riemann_xi.default_assumptions"></a>
     
     
 
@@ -47130,37 +48053,37 @@ <h5>Inherited Members</h5>
                 <dd id="riemann_xi.simplify" class="function">simplify</dd>
                 <dd id="riemann_xi.refine" class="function">refine</dd>
                 <dd id="riemann_xi.rewrite" class="function">rewrite</dd>
-                <dd id="riemann_xi.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="riemann_xi.is_integer" class="variable">is_integer</dd>
-                <dd id="riemann_xi.is_complex" class="variable">is_complex</dd>
-                <dd id="riemann_xi.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="riemann_xi.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="riemann_xi.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="riemann_xi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="riemann_xi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="riemann_xi.is_positive" class="variable">is_positive</dd>
+                <dd id="riemann_xi.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="riemann_xi.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="riemann_xi.is_negative" class="variable">is_negative</dd>
+                <dd id="riemann_xi.is_composite" class="variable">is_composite</dd>
+                <dd id="riemann_xi.is_finite" class="variable">is_finite</dd>
                 <dd id="riemann_xi.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="riemann_xi.is_zero" class="variable">is_zero</dd>
                 <dd id="riemann_xi.is_even" class="variable">is_even</dd>
-                <dd id="riemann_xi.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="riemann_xi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="riemann_xi.is_commutative" class="variable">is_commutative</dd>
-                <dd id="riemann_xi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="riemann_xi.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="riemann_xi.is_real" class="variable">is_real</dd>
-                <dd id="riemann_xi.is_positive" class="variable">is_positive</dd>
-                <dd id="riemann_xi.is_finite" class="variable">is_finite</dd>
-                <dd id="riemann_xi.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="riemann_xi.is_infinite" class="variable">is_infinite</dd>
-                <dd id="riemann_xi.is_rational" class="variable">is_rational</dd>
-                <dd id="riemann_xi.is_composite" class="variable">is_composite</dd>
-                <dd id="riemann_xi.is_negative" class="variable">is_negative</dd>
-                <dd id="riemann_xi.is_polar" class="variable">is_polar</dd>
+                <dd id="riemann_xi.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="riemann_xi.is_odd" class="variable">is_odd</dd>
-                <dd id="riemann_xi.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="riemann_xi.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="riemann_xi.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="riemann_xi.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="riemann_xi.is_irrational" class="variable">is_irrational</dd>
+                <dd id="riemann_xi.is_integer" class="variable">is_integer</dd>
+                <dd id="riemann_xi.is_rational" class="variable">is_rational</dd>
+                <dd id="riemann_xi.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="riemann_xi.is_complex" class="variable">is_complex</dd>
                 <dd id="riemann_xi.is_prime" class="variable">is_prime</dd>
+                <dd id="riemann_xi.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="riemann_xi.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="riemann_xi.is_real" class="variable">is_real</dd>
+                <dd id="riemann_xi.is_commutative" class="variable">is_commutative</dd>
                 <dd id="riemann_xi.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="riemann_xi.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="riemann_xi.is_polar" class="variable">is_polar</dd>
+                <dd id="riemann_xi.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="riemann_xi.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="riemann_xi.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -47172,32 +48095,31 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="betainc">
-                                <div class="attr class">
-        <a class="headerlink" href="#betainc">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">betainc</span><wbr>(<span class="base">sympy.functions.special.beta_functions.betainc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">betainc</span><wbr>(<span class="base">sympy.functions.special.beta_functions.betainc</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#betainc"></a>
     
             <div class="docstring"><p>The Generalized Incomplete Beta function is defined as</p>
 
-<p>.. math::
-    \mathrm{B}_{(x_1, x_2)}(a, b) = \int_{x_1}^{x_2} t^{a - 1} (1 - t)^{b - 1} dt</p>
+<p>$$\mathrm{B}_{(x_1, x_2)}(a, b) = \int_{x_1}^{x_2} t^{a - 1} (1 - t)^{b - 1} dt$$</p>
 
 <p>The Incomplete Beta function is a special case
 of the Generalized Incomplete Beta function :</p>
 
-<p>.. math:: \mathrm{B}_z (a, b) = \mathrm{B}_{(0, z)}(a, b)</p>
+<p>$$\mathrm{B}_z (a, b) = \mathrm{B}_{(0, z)}(a, b)$$</p>
 
 <p>The Incomplete Beta function satisfies :</p>
 
-<p>.. math:: \mathrm{B}_z (a, b) = (-1)^a \mathrm{B}_{\frac{z}{z - 1}} (a, 1 - a - b)</p>
+<p>$$\mathrm{B}_z (a, b) = (-1)^a \mathrm{B}_{\frac{z}{z - 1}} (a, 1 - a - b)$$</p>
 
 <p>The Beta function is a special case of the Incomplete Beta function :</p>
 
-<p>.. math:: \mathrm{B}(a, b) = \mathrm{B}_{1}(a, b)</p>
+<p>$$\mathrm{B}(a, b) = \mathrm{B}_{1}(a, b)$$</p>
 
 <h1 id="examples">Examples</h1>
 
@@ -47248,30 +48170,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function
-.. [2] https://dlmf.nist.gov/8.17
-.. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/
-.. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="betainc.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#betainc.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">betainc</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">betainc</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#betainc.__init__"></a>
     
     
 
                             </div>
                             <div id="betainc.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#betainc.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#betainc.default_assumptions"></a>
     
     
 
@@ -47428,37 +48353,37 @@ <h5>Inherited Members</h5>
                 <dd id="betainc.simplify" class="function">simplify</dd>
                 <dd id="betainc.refine" class="function">refine</dd>
                 <dd id="betainc.rewrite" class="function">rewrite</dd>
-                <dd id="betainc.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="betainc.is_integer" class="variable">is_integer</dd>
-                <dd id="betainc.is_complex" class="variable">is_complex</dd>
-                <dd id="betainc.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="betainc.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="betainc.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="betainc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="betainc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="betainc.is_positive" class="variable">is_positive</dd>
+                <dd id="betainc.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="betainc.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="betainc.is_negative" class="variable">is_negative</dd>
+                <dd id="betainc.is_composite" class="variable">is_composite</dd>
+                <dd id="betainc.is_finite" class="variable">is_finite</dd>
                 <dd id="betainc.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="betainc.is_zero" class="variable">is_zero</dd>
                 <dd id="betainc.is_even" class="variable">is_even</dd>
-                <dd id="betainc.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="betainc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="betainc.is_commutative" class="variable">is_commutative</dd>
-                <dd id="betainc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="betainc.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="betainc.is_real" class="variable">is_real</dd>
-                <dd id="betainc.is_positive" class="variable">is_positive</dd>
-                <dd id="betainc.is_finite" class="variable">is_finite</dd>
-                <dd id="betainc.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="betainc.is_infinite" class="variable">is_infinite</dd>
-                <dd id="betainc.is_rational" class="variable">is_rational</dd>
-                <dd id="betainc.is_composite" class="variable">is_composite</dd>
-                <dd id="betainc.is_negative" class="variable">is_negative</dd>
-                <dd id="betainc.is_polar" class="variable">is_polar</dd>
+                <dd id="betainc.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="betainc.is_odd" class="variable">is_odd</dd>
-                <dd id="betainc.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="betainc.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="betainc.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="betainc.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="betainc.is_irrational" class="variable">is_irrational</dd>
+                <dd id="betainc.is_integer" class="variable">is_integer</dd>
+                <dd id="betainc.is_rational" class="variable">is_rational</dd>
+                <dd id="betainc.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="betainc.is_complex" class="variable">is_complex</dd>
                 <dd id="betainc.is_prime" class="variable">is_prime</dd>
+                <dd id="betainc.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="betainc.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="betainc.is_real" class="variable">is_real</dd>
+                <dd id="betainc.is_commutative" class="variable">is_commutative</dd>
                 <dd id="betainc.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="betainc.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="betainc.is_polar" class="variable">is_polar</dd>
+                <dd id="betainc.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="betainc.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="betainc.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
@@ -47470,24 +48395,23 @@ <h5>Inherited Members</h5>
                             </div>
                 </section>
                 <section id="betainc_regularized">
-                                <div class="attr class">
-        <a class="headerlink" href="#betainc_regularized">#&nbsp;&nbsp</a>
+                    <div class="attr class">
+            
+    <span class="def">class</span>
+    <span class="name">betainc_regularized</span><wbr>(<span class="base">sympy.functions.special.beta_functions.betainc_regularized</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
 
         
-        <span class="def">class</span>
-        <span class="name">betainc_regularized</span><wbr>(<span class="base">sympy.functions.special.beta_functions.betainc_regularized</span>, <span class="base"><a href="#EqnFunction">EqnFunction</a></span>):
     </div>
-
+    <a class="headerlink" href="#betainc_regularized"></a>
     
             <div class="docstring"><p>The Generalized Regularized Incomplete Beta function is given by</p>
 
-<p>.. math::
-    \mathrm{I}_{(x_1, x_2)}(a, b) = \frac{\mathrm{B}_{(x_1, x_2)}(a, b)}{\mathrm{B}(a, b)}</p>
+<p>$$\mathrm{I}_{(x_1, x_2)}(a, b) = \frac{\mathrm{B}_{(x_1, x_2)}(a, b)}{\mathrm{B}(a, b)}$$</p>
 
 <p>The Regularized Incomplete Beta function is a special case
 of the Generalized Regularized Incomplete Beta function :</p>
 
-<p>.. math:: \mathrm{I}_z (a, b) = \mathrm{I}_{(0, z)}(a, b)</p>
+<p>$$\mathrm{I}_z (a, b) = \mathrm{I}_{(0, z)}(a, b)$$</p>
 
 <p>The Regularized Incomplete Beta function is the cumulative distribution
 function of the beta distribution.</p>
@@ -47544,30 +48468,33 @@ <h1 id="see-also">See Also</h1>
 
 <h1 id="references">References</h1>
 
-<p>.. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function
-.. [2] https://dlmf.nist.gov/8.17
-.. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/
-.. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/</p>
+<div class="footnotes">
+<hr />
+<ol>
+</ol>
+</div>
 </div>
 
 
                             <div id="betainc_regularized.__init__" class="classattr">
-                                        <div class="attr function"><a class="headerlink" href="#betainc_regularized.__init__">#&nbsp;&nbsp</a>
+                                <div class="attr function">
+            
+        <span class="name">betainc_regularized</span><span class="signature pdoc-code condensed">()</span>
 
         
-            <span class="name">betainc_regularized</span><span class="signature">()</span>
     </div>
-
+    <a class="headerlink" href="#betainc_regularized.__init__"></a>
     
     
 
                             </div>
                             <div id="betainc_regularized.default_assumptions" class="classattr">
-                                            <div class="attr variable"><a class="headerlink" href="#betainc_regularized.default_assumptions">#&nbsp;&nbsp</a>
+                                <div class="attr variable">
+            <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
 
-        <span class="name">default_assumptions</span><span class="default_value"> = {}</span>
+        
     </div>
-
+    <a class="headerlink" href="#betainc_regularized.default_assumptions"></a>
     
     
 
@@ -47724,37 +48651,37 @@ <h5>Inherited Members</h5>
                 <dd id="betainc_regularized.simplify" class="function">simplify</dd>
                 <dd id="betainc_regularized.refine" class="function">refine</dd>
                 <dd id="betainc_regularized.rewrite" class="function">rewrite</dd>
-                <dd id="betainc_regularized.is_algebraic" class="variable">is_algebraic</dd>
-                <dd id="betainc_regularized.is_integer" class="variable">is_integer</dd>
-                <dd id="betainc_regularized.is_complex" class="variable">is_complex</dd>
-                <dd id="betainc_regularized.is_nonpositive" class="variable">is_nonpositive</dd>
                 <dd id="betainc_regularized.is_extended_positive" class="variable">is_extended_positive</dd>
-                <dd id="betainc_regularized.is_extended_real" class="variable">is_extended_real</dd>
-                <dd id="betainc_regularized.is_irrational" class="variable">is_irrational</dd>
+                <dd id="betainc_regularized.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="betainc_regularized.is_positive" class="variable">is_positive</dd>
+                <dd id="betainc_regularized.is_algebraic" class="variable">is_algebraic</dd>
+                <dd id="betainc_regularized.is_antihermitian" class="variable">is_antihermitian</dd>
+                <dd id="betainc_regularized.is_negative" class="variable">is_negative</dd>
+                <dd id="betainc_regularized.is_composite" class="variable">is_composite</dd>
+                <dd id="betainc_regularized.is_finite" class="variable">is_finite</dd>
                 <dd id="betainc_regularized.is_extended_nonpositive" class="variable">is_extended_nonpositive</dd>
-                <dd id="betainc_regularized.is_zero" class="variable">is_zero</dd>
                 <dd id="betainc_regularized.is_even" class="variable">is_even</dd>
-                <dd id="betainc_regularized.is_noninteger" class="variable">is_noninteger</dd>
-                <dd id="betainc_regularized.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
-                <dd id="betainc_regularized.is_commutative" class="variable">is_commutative</dd>
-                <dd id="betainc_regularized.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
                 <dd id="betainc_regularized.is_nonnegative" class="variable">is_nonnegative</dd>
-                <dd id="betainc_regularized.is_real" class="variable">is_real</dd>
-                <dd id="betainc_regularized.is_positive" class="variable">is_positive</dd>
-                <dd id="betainc_regularized.is_finite" class="variable">is_finite</dd>
-                <dd id="betainc_regularized.is_extended_negative" class="variable">is_extended_negative</dd>
                 <dd id="betainc_regularized.is_infinite" class="variable">is_infinite</dd>
-                <dd id="betainc_regularized.is_rational" class="variable">is_rational</dd>
-                <dd id="betainc_regularized.is_composite" class="variable">is_composite</dd>
-                <dd id="betainc_regularized.is_negative" class="variable">is_negative</dd>
-                <dd id="betainc_regularized.is_polar" class="variable">is_polar</dd>
+                <dd id="betainc_regularized.is_imaginary" class="variable">is_imaginary</dd>
                 <dd id="betainc_regularized.is_odd" class="variable">is_odd</dd>
-                <dd id="betainc_regularized.is_antihermitian" class="variable">is_antihermitian</dd>
-                <dd id="betainc_regularized.is_nonzero" class="variable">is_nonzero</dd>
+                <dd id="betainc_regularized.is_extended_nonzero" class="variable">is_extended_nonzero</dd>
+                <dd id="betainc_regularized.is_extended_negative" class="variable">is_extended_negative</dd>
+                <dd id="betainc_regularized.is_irrational" class="variable">is_irrational</dd>
+                <dd id="betainc_regularized.is_integer" class="variable">is_integer</dd>
+                <dd id="betainc_regularized.is_rational" class="variable">is_rational</dd>
+                <dd id="betainc_regularized.is_extended_real" class="variable">is_extended_real</dd>
+                <dd id="betainc_regularized.is_complex" class="variable">is_complex</dd>
                 <dd id="betainc_regularized.is_prime" class="variable">is_prime</dd>
+                <dd id="betainc_regularized.is_noninteger" class="variable">is_noninteger</dd>
                 <dd id="betainc_regularized.is_transcendental" class="variable">is_transcendental</dd>
+                <dd id="betainc_regularized.is_real" class="variable">is_real</dd>
+                <dd id="betainc_regularized.is_commutative" class="variable">is_commutative</dd>
                 <dd id="betainc_regularized.is_hermitian" class="variable">is_hermitian</dd>
-                <dd id="betainc_regularized.is_imaginary" class="variable">is_imaginary</dd>
+                <dd id="betainc_regularized.is_polar" class="variable">is_polar</dd>
+                <dd id="betainc_regularized.is_extended_nonnegative" class="variable">is_extended_nonnegative</dd>
+                <dd id="betainc_regularized.is_nonpositive" class="variable">is_nonpositive</dd>
+                <dd id="betainc_regularized.is_zero" class="variable">is_zero</dd>
 
             </div>
             <div><dt>sympy.core.evalf.EvalfMixin</dt>
diff --git a/docs/intro.md b/docs/intro.md
index 8dae213..2607ac8 100644
--- a/docs/intro.md
+++ b/docs/intro.md
@@ -31,13 +31,26 @@ over missed details such as a negative sign.
 
 A simple example as it would appear in a [Jupyter](https://jupyter.org) 
 notebook is shown immediately below:
-![screenshot of simple example](./resources/simple_example.png)
+![screenshot of simple example](https://gutow.github.io/Algebra_with_Sympy/resources/simple_example.png)
 
 Once the algebra is complete it is possible to substitute numbers with 
 units into the solved equation to calculate a numerical solution with 
 proper units.
 
-[More examples here](Demonstration%20of%20equation%20class.html).
+In IPython environments (IPython and Jupyter) there is also a shorthand 
+syntax for entering equations provided through the IPython preparser. An 
+equation can be specified as `eq1 =@ a/b = c/d`. If no Python name is 
+specified for the equation (`eq1` left out in the previous example), the 
+equation will still 
+be defined, but will not be easily accessible for further computation. The 
+`=@` symbol combination was chosen to avoid conflicts with reserved python 
+symbols while minimizing impacts on syntax highlighting and autoformatting. A 
+screenshot of an example in Jupyter is shown immediately below:
+
+![screenshot of short syntax](https://gutow.github.io/Algebra_with_Sympy/resources/short_syntax.png)
+
+[More examples of the capabilities of Algebra with Sympy are 
+here](https://gutow.github.io/Algebra_with_Sympy/Demonstration%20of%20equation%20class.html).
 
 Many math packages such as [SageMath](https://www.sagemath.org/) 
 and [Maxima](http://maxima.sourceforge.net/) have similar capabilities, 
@@ -91,6 +104,10 @@ github](https://github.com/gutow/Algebra_with_Sympy/issues).
 
 #### Change Log
 
+* 0.9.2
+  * `=@` shorthand syntax for defining equations in IPython compatible 
+    environments.
+  * Fixed bug where `root()` override called `sqrt()` on bare expressions.
 * 0.9.1
   * Equations labeled with their python name, if they have one.
   * Added flags to adjust human readable output and equation labeling.
diff --git a/docs/search.js b/docs/search.js
index 4c07e0f..306fa3a 100644
--- a/docs/search.js
+++ b/docs/search.js
@@ -1,6 +1,6 @@
 window.pdocSearch = (function(){
 /** elasticlunr - http://weixsong.github.io * Copyright (C) 2017 Oliver Nightingale * Copyright (C) 2017 Wei Song * MIT Licensed */!function(){function e(e){if(null===e||"object"!=typeof e)return e;var t=e.constructor();for(var n in e)e.hasOwnProperty(n)&&(t[n]=e[n]);return t}var t=function(e){var n=new t.Index;return n.pipeline.add(t.trimmer,t.stopWordFilter,t.stemmer),e&&e.call(n,n),n};t.version="0.9.5",lunr=t,t.utils={},t.utils.warn=function(e){return function(t){e.console&&console.warn&&console.warn(t)}}(this),t.utils.toString=function(e){return void 0===e||null===e?"":e.toString()},t.EventEmitter=function(){this.events={}},t.EventEmitter.prototype.addListener=function(){var e=Array.prototype.slice.call(arguments),t=e.pop(),n=e;if("function"!=typeof t)throw new TypeError("last argument must be a function");n.forEach(function(e){this.hasHandler(e)||(this.events[e]=[]),this.events[e].push(t)},this)},t.EventEmitter.prototype.removeListener=function(e,t){if(this.hasHandler(e)){var n=this.events[e].indexOf(t);-1!==n&&(this.events[e].splice(n,1),0==this.events[e].length&&delete this.events[e])}},t.EventEmitter.prototype.emit=function(e){if(this.hasHandler(e)){var t=Array.prototype.slice.call(arguments,1);this.events[e].forEach(function(e){e.apply(void 0,t)},this)}},t.EventEmitter.prototype.hasHandler=function(e){return e in this.events},t.tokenizer=function(e){if(!arguments.length||null===e||void 0===e)return[];if(Array.isArray(e)){var n=e.filter(function(e){return null===e||void 0===e?!1:!0});n=n.map(function(e){return t.utils.toString(e).toLowerCase()});var i=[];return n.forEach(function(e){var n=e.split(t.tokenizer.seperator);i=i.concat(n)},this),i}return e.toString().trim().toLowerCase().split(t.tokenizer.seperator)},t.tokenizer.defaultSeperator=/[\s\-]+/,t.tokenizer.seperator=t.tokenizer.defaultSeperator,t.tokenizer.setSeperator=function(e){null!==e&&void 0!==e&&"object"==typeof e&&(t.tokenizer.seperator=e)},t.tokenizer.resetSeperator=function(){t.tokenizer.seperator=t.tokenizer.defaultSeperator},t.tokenizer.getSeperator=function(){return t.tokenizer.seperator},t.Pipeline=function(){this._queue=[]},t.Pipeline.registeredFunctions={},t.Pipeline.registerFunction=function(e,n){n in t.Pipeline.registeredFunctions&&t.utils.warn("Overwriting existing registered function: "+n),e.label=n,t.Pipeline.registeredFunctions[n]=e},t.Pipeline.getRegisteredFunction=function(e){return e in t.Pipeline.registeredFunctions!=!0?null:t.Pipeline.registeredFunctions[e]},t.Pipeline.warnIfFunctionNotRegistered=function(e){var n=e.label&&e.label in this.registeredFunctions;n||t.utils.warn("Function is not registered with pipeline. This may cause problems when serialising the index.\n",e)},t.Pipeline.load=function(e){var n=new t.Pipeline;return e.forEach(function(e){var i=t.Pipeline.getRegisteredFunction(e);if(!i)throw new Error("Cannot load un-registered function: "+e);n.add(i)}),n},t.Pipeline.prototype.add=function(){var e=Array.prototype.slice.call(arguments);e.forEach(function(e){t.Pipeline.warnIfFunctionNotRegistered(e),this._queue.push(e)},this)},t.Pipeline.prototype.after=function(e,n){t.Pipeline.warnIfFunctionNotRegistered(n);var i=this._queue.indexOf(e);if(-1===i)throw new Error("Cannot find existingFn");this._queue.splice(i+1,0,n)},t.Pipeline.prototype.before=function(e,n){t.Pipeline.warnIfFunctionNotRegistered(n);var i=this._queue.indexOf(e);if(-1===i)throw new Error("Cannot find existingFn");this._queue.splice(i,0,n)},t.Pipeline.prototype.remove=function(e){var t=this._queue.indexOf(e);-1!==t&&this._queue.splice(t,1)},t.Pipeline.prototype.run=function(e){for(var t=[],n=e.length,i=this._queue.length,o=0;n>o;o++){for(var r=e[o],s=0;i>s&&(r=this._queue[s](r,o,e),void 0!==r&&null!==r);s++);void 0!==r&&null!==r&&t.push(r)}return t},t.Pipeline.prototype.reset=function(){this._queue=[]},t.Pipeline.prototype.get=function(){return this._queue},t.Pipeline.prototype.toJSON=function(){return this._queue.map(function(e){return t.Pipeline.warnIfFunctionNotRegistered(e),e.label})},t.Index=function(){this._fields=[],this._ref="id",this.pipeline=new t.Pipeline,this.documentStore=new t.DocumentStore,this.index={},this.eventEmitter=new t.EventEmitter,this._idfCache={},this.on("add","remove","update",function(){this._idfCache={}}.bind(this))},t.Index.prototype.on=function(){var e=Array.prototype.slice.call(arguments);return this.eventEmitter.addListener.apply(this.eventEmitter,e)},t.Index.prototype.off=function(e,t){return this.eventEmitter.removeListener(e,t)},t.Index.load=function(e){e.version!==t.version&&t.utils.warn("version mismatch: current "+t.version+" importing "+e.version);var n=new this;n._fields=e.fields,n._ref=e.ref,n.documentStore=t.DocumentStore.load(e.documentStore),n.pipeline=t.Pipeline.load(e.pipeline),n.index={};for(var i in e.index)n.index[i]=t.InvertedIndex.load(e.index[i]);return n},t.Index.prototype.addField=function(e){return this._fields.push(e),this.index[e]=new t.InvertedIndex,this},t.Index.prototype.setRef=function(e){return this._ref=e,this},t.Index.prototype.saveDocument=function(e){return this.documentStore=new t.DocumentStore(e),this},t.Index.prototype.addDoc=function(e,n){if(e){var n=void 0===n?!0:n,i=e[this._ref];this.documentStore.addDoc(i,e),this._fields.forEach(function(n){var o=this.pipeline.run(t.tokenizer(e[n]));this.documentStore.addFieldLength(i,n,o.length);var r={};o.forEach(function(e){e in r?r[e]+=1:r[e]=1},this);for(var s in r){var u=r[s];u=Math.sqrt(u),this.index[n].addToken(s,{ref:i,tf:u})}},this),n&&this.eventEmitter.emit("add",e,this)}},t.Index.prototype.removeDocByRef=function(e){if(e&&this.documentStore.isDocStored()!==!1&&this.documentStore.hasDoc(e)){var t=this.documentStore.getDoc(e);this.removeDoc(t,!1)}},t.Index.prototype.removeDoc=function(e,n){if(e){var n=void 0===n?!0:n,i=e[this._ref];this.documentStore.hasDoc(i)&&(this.documentStore.removeDoc(i),this._fields.forEach(function(n){var o=this.pipeline.run(t.tokenizer(e[n]));o.forEach(function(e){this.index[n].removeToken(e,i)},this)},this),n&&this.eventEmitter.emit("remove",e,this))}},t.Index.prototype.updateDoc=function(e,t){var t=void 0===t?!0:t;this.removeDocByRef(e[this._ref],!1),this.addDoc(e,!1),t&&this.eventEmitter.emit("update",e,this)},t.Index.prototype.idf=function(e,t){var n="@"+t+"/"+e;if(Object.prototype.hasOwnProperty.call(this._idfCache,n))return this._idfCache[n];var i=this.index[t].getDocFreq(e),o=1+Math.log(this.documentStore.length/(i+1));return this._idfCache[n]=o,o},t.Index.prototype.getFields=function(){return this._fields.slice()},t.Index.prototype.search=function(e,n){if(!e)return[];e="string"==typeof e?{any:e}:JSON.parse(JSON.stringify(e));var i=null;null!=n&&(i=JSON.stringify(n));for(var o=new t.Configuration(i,this.getFields()).get(),r={},s=Object.keys(e),u=0;u<s.length;u++){var a=s[u];r[a]=this.pipeline.run(t.tokenizer(e[a]))}var l={};for(var c in o){var d=r[c]||r.any;if(d){var f=this.fieldSearch(d,c,o),h=o[c].boost;for(var p in f)f[p]=f[p]*h;for(var p in f)p in l?l[p]+=f[p]:l[p]=f[p]}}var v,g=[];for(var p in l)v={ref:p,score:l[p]},this.documentStore.hasDoc(p)&&(v.doc=this.documentStore.getDoc(p)),g.push(v);return g.sort(function(e,t){return t.score-e.score}),g},t.Index.prototype.fieldSearch=function(e,t,n){var i=n[t].bool,o=n[t].expand,r=n[t].boost,s=null,u={};return 0!==r?(e.forEach(function(e){var n=[e];1==o&&(n=this.index[t].expandToken(e));var r={};n.forEach(function(n){var o=this.index[t].getDocs(n),a=this.idf(n,t);if(s&&"AND"==i){var l={};for(var c in s)c in o&&(l[c]=o[c]);o=l}n==e&&this.fieldSearchStats(u,n,o);for(var c in o){var d=this.index[t].getTermFrequency(n,c),f=this.documentStore.getFieldLength(c,t),h=1;0!=f&&(h=1/Math.sqrt(f));var p=1;n!=e&&(p=.15*(1-(n.length-e.length)/n.length));var v=d*a*h*p;c in r?r[c]+=v:r[c]=v}},this),s=this.mergeScores(s,r,i)},this),s=this.coordNorm(s,u,e.length)):void 0},t.Index.prototype.mergeScores=function(e,t,n){if(!e)return t;if("AND"==n){var i={};for(var o in t)o in e&&(i[o]=e[o]+t[o]);return i}for(var o in t)o in e?e[o]+=t[o]:e[o]=t[o];return e},t.Index.prototype.fieldSearchStats=function(e,t,n){for(var i in n)i in e?e[i].push(t):e[i]=[t]},t.Index.prototype.coordNorm=function(e,t,n){for(var i in e)if(i in t){var o=t[i].length;e[i]=e[i]*o/n}return e},t.Index.prototype.toJSON=function(){var e={};return this._fields.forEach(function(t){e[t]=this.index[t].toJSON()},this),{version:t.version,fields:this._fields,ref:this._ref,documentStore:this.documentStore.toJSON(),index:e,pipeline:this.pipeline.toJSON()}},t.Index.prototype.use=function(e){var t=Array.prototype.slice.call(arguments,1);t.unshift(this),e.apply(this,t)},t.DocumentStore=function(e){this._save=null===e||void 0===e?!0:e,this.docs={},this.docInfo={},this.length=0},t.DocumentStore.load=function(e){var t=new this;return t.length=e.length,t.docs=e.docs,t.docInfo=e.docInfo,t._save=e.save,t},t.DocumentStore.prototype.isDocStored=function(){return this._save},t.DocumentStore.prototype.addDoc=function(t,n){this.hasDoc(t)||this.length++,this.docs[t]=this._save===!0?e(n):null},t.DocumentStore.prototype.getDoc=function(e){return this.hasDoc(e)===!1?null:this.docs[e]},t.DocumentStore.prototype.hasDoc=function(e){return e in this.docs},t.DocumentStore.prototype.removeDoc=function(e){this.hasDoc(e)&&(delete this.docs[e],delete this.docInfo[e],this.length--)},t.DocumentStore.prototype.addFieldLength=function(e,t,n){null!==e&&void 0!==e&&0!=this.hasDoc(e)&&(this.docInfo[e]||(this.docInfo[e]={}),this.docInfo[e][t]=n)},t.DocumentStore.prototype.updateFieldLength=function(e,t,n){null!==e&&void 0!==e&&0!=this.hasDoc(e)&&this.addFieldLength(e,t,n)},t.DocumentStore.prototype.getFieldLength=function(e,t){return null===e||void 0===e?0:e in this.docs&&t in this.docInfo[e]?this.docInfo[e][t]:0},t.DocumentStore.prototype.toJSON=function(){return{docs:this.docs,docInfo:this.docInfo,length:this.length,save:this._save}},t.stemmer=function(){var e={ational:"ate",tional:"tion",enci:"ence",anci:"ance",izer:"ize",bli:"ble",alli:"al",entli:"ent",eli:"e",ousli:"ous",ization:"ize",ation:"ate",ator:"ate",alism:"al",iveness:"ive",fulness:"ful",ousness:"ous",aliti:"al",iviti:"ive",biliti:"ble",logi:"log"},t={icate:"ic",ative:"",alize:"al",iciti:"ic",ical:"ic",ful:"",ness:""},n="[^aeiou]",i="[aeiouy]",o=n+"[^aeiouy]*",r=i+"[aeiou]*",s="^("+o+")?"+r+o,u="^("+o+")?"+r+o+"("+r+")?$",a="^("+o+")?"+r+o+r+o,l="^("+o+")?"+i,c=new RegExp(s),d=new RegExp(a),f=new RegExp(u),h=new RegExp(l),p=/^(.+?)(ss|i)es$/,v=/^(.+?)([^s])s$/,g=/^(.+?)eed$/,m=/^(.+?)(ed|ing)$/,y=/.$/,S=/(at|bl|iz)$/,x=new RegExp("([^aeiouylsz])\\1$"),w=new RegExp("^"+o+i+"[^aeiouwxy]$"),I=/^(.+?[^aeiou])y$/,b=/^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/,E=/^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/,D=/^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/,F=/^(.+?)(s|t)(ion)$/,_=/^(.+?)e$/,P=/ll$/,k=new RegExp("^"+o+i+"[^aeiouwxy]$"),z=function(n){var i,o,r,s,u,a,l;if(n.length<3)return n;if(r=n.substr(0,1),"y"==r&&(n=r.toUpperCase()+n.substr(1)),s=p,u=v,s.test(n)?n=n.replace(s,"$1$2"):u.test(n)&&(n=n.replace(u,"$1$2")),s=g,u=m,s.test(n)){var z=s.exec(n);s=c,s.test(z[1])&&(s=y,n=n.replace(s,""))}else if(u.test(n)){var z=u.exec(n);i=z[1],u=h,u.test(i)&&(n=i,u=S,a=x,l=w,u.test(n)?n+="e":a.test(n)?(s=y,n=n.replace(s,"")):l.test(n)&&(n+="e"))}if(s=I,s.test(n)){var z=s.exec(n);i=z[1],n=i+"i"}if(s=b,s.test(n)){var z=s.exec(n);i=z[1],o=z[2],s=c,s.test(i)&&(n=i+e[o])}if(s=E,s.test(n)){var z=s.exec(n);i=z[1],o=z[2],s=c,s.test(i)&&(n=i+t[o])}if(s=D,u=F,s.test(n)){var z=s.exec(n);i=z[1],s=d,s.test(i)&&(n=i)}else if(u.test(n)){var z=u.exec(n);i=z[1]+z[2],u=d,u.test(i)&&(n=i)}if(s=_,s.test(n)){var z=s.exec(n);i=z[1],s=d,u=f,a=k,(s.test(i)||u.test(i)&&!a.test(i))&&(n=i)}return s=P,u=d,s.test(n)&&u.test(n)&&(s=y,n=n.replace(s,"")),"y"==r&&(n=r.toLowerCase()+n.substr(1)),n};return z}(),t.Pipeline.registerFunction(t.stemmer,"stemmer"),t.stopWordFilter=function(e){return e&&t.stopWordFilter.stopWords[e]!==!0?e:void 0},t.clearStopWords=function(){t.stopWordFilter.stopWords={}},t.addStopWords=function(e){null!=e&&Array.isArray(e)!==!1&&e.forEach(function(e){t.stopWordFilter.stopWords[e]=!0},this)},t.resetStopWords=function(){t.stopWordFilter.stopWords=t.defaultStopWords},t.defaultStopWords={"":!0,a:!0,able:!0,about:!0,across:!0,after:!0,all:!0,almost:!0,also:!0,am:!0,among:!0,an:!0,and:!0,any:!0,are:!0,as:!0,at:!0,be:!0,because:!0,been:!0,but:!0,by:!0,can:!0,cannot:!0,could:!0,dear:!0,did:!0,"do":!0,does:!0,either:!0,"else":!0,ever:!0,every:!0,"for":!0,from:!0,get:!0,got:!0,had:!0,has:!0,have:!0,he:!0,her:!0,hers:!0,him:!0,his:!0,how:!0,however:!0,i:!0,"if":!0,"in":!0,into:!0,is:!0,it:!0,its:!0,just:!0,least:!0,let:!0,like:!0,likely:!0,may:!0,me:!0,might:!0,most:!0,must:!0,my:!0,neither:!0,no:!0,nor:!0,not:!0,of:!0,off:!0,often:!0,on:!0,only:!0,or:!0,other:!0,our:!0,own:!0,rather:!0,said:!0,say:!0,says:!0,she:!0,should:!0,since:!0,so:!0,some:!0,than:!0,that:!0,the:!0,their:!0,them:!0,then:!0,there:!0,these:!0,they:!0,"this":!0,tis:!0,to:!0,too:!0,twas:!0,us:!0,wants:!0,was:!0,we:!0,were:!0,what:!0,when:!0,where:!0,which:!0,"while":!0,who:!0,whom:!0,why:!0,will:!0,"with":!0,would:!0,yet:!0,you:!0,your:!0},t.stopWordFilter.stopWords=t.defaultStopWords,t.Pipeline.registerFunction(t.stopWordFilter,"stopWordFilter"),t.trimmer=function(e){if(null===e||void 0===e)throw new Error("token should not be undefined");return e.replace(/^\W+/,"").replace(/\W+$/,"")},t.Pipeline.registerFunction(t.trimmer,"trimmer"),t.InvertedIndex=function(){this.root={docs:{},df:0}},t.InvertedIndex.load=function(e){var t=new this;return t.root=e.root,t},t.InvertedIndex.prototype.addToken=function(e,t,n){for(var n=n||this.root,i=0;i<=e.length-1;){var o=e[i];o in n||(n[o]={docs:{},df:0}),i+=1,n=n[o]}var r=t.ref;n.docs[r]?n.docs[r]={tf:t.tf}:(n.docs[r]={tf:t.tf},n.df+=1)},t.InvertedIndex.prototype.hasToken=function(e){if(!e)return!1;for(var t=this.root,n=0;n<e.length;n++){if(!t[e[n]])return!1;t=t[e[n]]}return!0},t.InvertedIndex.prototype.getNode=function(e){if(!e)return null;for(var t=this.root,n=0;n<e.length;n++){if(!t[e[n]])return null;t=t[e[n]]}return t},t.InvertedIndex.prototype.getDocs=function(e){var t=this.getNode(e);return null==t?{}:t.docs},t.InvertedIndex.prototype.getTermFrequency=function(e,t){var n=this.getNode(e);return null==n?0:t in n.docs?n.docs[t].tf:0},t.InvertedIndex.prototype.getDocFreq=function(e){var t=this.getNode(e);return null==t?0:t.df},t.InvertedIndex.prototype.removeToken=function(e,t){if(e){var n=this.getNode(e);null!=n&&t in n.docs&&(delete n.docs[t],n.df-=1)}},t.InvertedIndex.prototype.expandToken=function(e,t,n){if(null==e||""==e)return[];var t=t||[];if(void 0==n&&(n=this.getNode(e),null==n))return t;n.df>0&&t.push(e);for(var i in n)"docs"!==i&&"df"!==i&&this.expandToken(e+i,t,n[i]);return t},t.InvertedIndex.prototype.toJSON=function(){return{root:this.root}},t.Configuration=function(e,n){var e=e||"";if(void 0==n||null==n)throw new Error("fields should not be null");this.config={};var i;try{i=JSON.parse(e),this.buildUserConfig(i,n)}catch(o){t.utils.warn("user configuration parse failed, will use default configuration"),this.buildDefaultConfig(n)}},t.Configuration.prototype.buildDefaultConfig=function(e){this.reset(),e.forEach(function(e){this.config[e]={boost:1,bool:"OR",expand:!1}},this)},t.Configuration.prototype.buildUserConfig=function(e,n){var i="OR",o=!1;if(this.reset(),"bool"in e&&(i=e.bool||i),"expand"in e&&(o=e.expand||o),"fields"in e)for(var r in e.fields)if(n.indexOf(r)>-1){var s=e.fields[r],u=o;void 0!=s.expand&&(u=s.expand),this.config[r]={boost:s.boost||0===s.boost?s.boost:1,bool:s.bool||i,expand:u}}else t.utils.warn("field name in user configuration not found in index instance fields");else this.addAllFields2UserConfig(i,o,n)},t.Configuration.prototype.addAllFields2UserConfig=function(e,t,n){n.forEach(function(n){this.config[n]={boost:1,bool:e,expand:t}},this)},t.Configuration.prototype.get=function(){return 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-    /** pdoc search index */const docs = [{"fullname": "algebra_with_sympy", "modulename": "algebra_with_sympy", "type": "module", "doc": "<h2 id=\"algebraic-equations-with-sympy\">Algebraic Equations with SymPy</h2>\n\n<p><a href=\"#introduction\">Introduction</a> | <a href=\"#controlling-the-format-of-interactive-outputs\">Output Formatting</a>\n| <a href=\"#setupinstallation\">Installation</a> |\n<a href=\"#try-in-binder\">Try Live</a> | <a href=\"#issues-or-comments\">Issues or Comments</a> |\n<a href=\"#change-log\">Change Log</a> |\n<a href=\"#this-software-is-distributed-under-the-gnu-v3-licensehttpsgnuorglicenses\">License</a>\n| <a href=\"https://github.com/gutow/Algebra_with_Sympy\">GIT Repository</a>\n| <a href=\"https://pypi.org/project/Algebra-with-SymPy/\">PyPi Link</a></p>\n\n<h4 id=\"websitedocumentation-including-apihttpsgutowgithubioalgebra_with_sympy\"><a href=\"https://gutow.github.io/Algebra_with_Sympy/\">Website/Documentation (including API)</a></h4>\n\n<h4 id=\"introduction\">Introduction</h4>\n\n<p>This tool defines relations that all high school and college students would\nrecognize as mathematical equations. \nThey consist of a left hand side (lhs) and a right hand side (rhs) connected by\nthe relation operator \"=\".</p>\n\n<p>This tool applies operations to both sides of the equation simultaneously, just\nas students are taught to do when \nattempting to isolate (solve for) a variable. Thus the statement <code>Equation/b</code>\nyields a new equation <code>Equation.lhs/b = Equation.rhs/b</code></p>\n\n<p>The intent is to allow using the mathematical tools in SymPy to rearrange\nequations and perform algebra\nin a stepwise fashion using as close to standard mathematical notation as \npossible. In this way more people can successfully perform \nalgebraic rearrangements without stumbling\nover missed details such as a negative sign.</p>\n\n<p>A simple example as it would appear in a <a href=\"https://jupyter.org\">Jupyter</a> \nnotebook is shown immediately below:\n<img src=\"./resources/simple_example.png\" alt=\"screenshot of simple example\" /></p>\n\n<p>Once the algebra is complete it is possible to substitute numbers with \nunits into the solved equation to calculate a numerical solution with \nproper units.</p>\n\n<p><a href=\"Demonstration%20of%20equation%20class.html\">More examples here</a>.</p>\n\n<p>Many math packages such as <a href=\"https://www.sagemath.org/\">SageMath</a> \nand <a href=\"http://maxima.sourceforge.net/\">Maxima</a> have similar capabilities, \nbut require more knowledge of command syntax, plus they cannot easily be \ninstalled in a generic python environment.</p>\n\n<h4 id=\"controlling-the-format-of-interactive-outputs\">Controlling the Format of Interactive Outputs</h4>\n\n<ul>\n<li><p><strong>In graphical environments (Jupyter)</strong> you will get rendered Latex such as \n$\\frac{a}{b} = \\frac{c}{d}$. To also see the code representation (what can \nbe copy and pasted for \nadditional computation) set <code>algwsym_config.output.show_code = True</code>. \nThis will print the code version (e.g. <code>Equation(a,b/c)</code>) of the equation as \nwell. This code version can be accessed directly by calling <code>repr()</code> on the \nequation.</p></li>\n<li><p><strong>In interactive text environments (ipython and command line)</strong> the \nrepresentation (code version) is returned by default. Calling <code>print()</code> \nor <code>str()</code> on an equation will return the human readable version with an \nequals sign. To have the human readable version returned by default set \n<code>algwsym_config.output.human_text = True</code>. If combined with \n<code>algwsym_config.output.show_code = True</code>, both code and human readable \nversions will be shown.</p></li>\n<li><p><strong>The equation label</strong> can be turned off by setting\n<code>algwsym_config.output.label = False</code>.</p></li>\n</ul>\n\n<h4 id=\"setupinstallation\">Setup/Installation</h4>\n\n<ol>\n<li>Use pip to install in your python environment: \n<code>pip install -U Algebra-with-SymPy</code></li>\n<li>To use in a running python session issue\nthe following command : <code>from algebra_with_sympy import *</code>. \nThis will also import the SymPy tools. </li>\n<li>If you want to isolate this tool from the global namespace you are \nworking with change the import statement \nto <code>import algebra_with_sympy as spa</code>, where \n<code>spa</code> stands for \"SymPy Algebra\". Then all calls would be made to <code>\nspa.funcname()</code>.</li>\n</ol>\n\n<h4 id=\"try-in-binder\">Try in binder</h4>\n\n<p><a href=\"https://mybinder.org/v2/gh/gutow/Algebra_with_Sympy.git/master/?urlpath=/tree/Demonstration%20of%20equation%20class.ipynb\"><img src=\"https://mybinder.org/badge_logo.svg\" alt=\"Binder\" /></a></p>\n\n<h4 id=\"issues-or-comments\">Issues or Comments</h4>\n\n<ul>\n<li>Issues and bug reports should be <a href=\"https://github.com/gutow/Algebra_with_Sympy/issues\">filed on \ngithub</a>.</li>\n<li>Comments, questions, show and tell, etc. should go in the <a href=\"https://github.com/gutow/Algebra_with_Sympy/discussions\">project \ndiscussions</a>.</li>\n</ul>\n\n<h4 id=\"change-log\">Change Log</h4>\n\n<ul>\n<li>0.9.1\n<ul>\n<li>Equations labeled with their python name, if they have one.</li>\n<li>Added flags to adjust human readable output and equation labeling.</li>\n<li>Accept equation as function argument in any position.</li>\n<li>First pass at <code>solve()</code> accepting equations.</li>\n<li>Added override of <code>root()</code> to avoid warning messages.</li>\n<li>More unit tests.</li>\n<li>First pass at documentation.</li>\n</ul></li>\n<li>0.9.0 functionality equivalent to extension of SymPy in\n<a href=\"https://github.com/sympy/sympy/pull/21333\">PR#21333</a>.</li>\n</ul>\n\n<h5 id=\"this-software-is-distributed-under-the-gnu-v3-licensehttpsgnuorglicenses\"><a href=\"https://gnu.org/licenses\">This software is distributed under the GNU V3 license</a></h5>\n\n<p>This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU General Public License as published by\n    the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n    GNU General Public License for more details.</p>\n\n<p>Copyright - Jonathan Gutow 2021, 2022</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation", "modulename": "algebra_with_sympy.algebraic_equation", "type": "module", "doc": "<h1 id=\"algebraic-equations-with-sympy\">Algebraic Equations with SymPy</h1>\n\n<p>These tools define relations that all high school and college students would\nrecognize as mathematical equations. They consist of a left hand side (lhs)\nand a right hand side (rhs) connected by a relation operator such as \"=\". At\npresent the \"=\" relation operator is the only option. The relation operator may\nnot be set.</p>\n\n<p>This class should not be confused with the Boolean class <code>Equality</code>\n(abbreviated <code>Eq</code>) which specifies that the equality of two objects is\n<code>True</code>.</p>\n\n<p>This tool applies operations to both sides of the equation simultaneously, just\nas students are taught to do when attempting to isolate (solve for) a\nvariable. Thus the statement <code>Equation/b</code> yields a new equation\n<code>Equation.lhs/b = Equation.rhs/b</code></p>\n\n<p>The intent is to allow using the mathematical tools in SymPy to rearrange\nequations and perform algebra in a stepwise fashion. In this way more people\ncan successfully perform algebraic rearrangements without stumbling over\nmissed details such as a negative sign. This mimics the capabilities available\nin <a href=\"https://www.sagemath.org/\">SageMath</a> and\n<a href=\"http://maxima.sourceforge.net/\">Maxima</a>.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config", "type": "class", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.__init__", "type": "function", "doc": "<p>This is a class to hold parameters that control behavior of\nthe algebra_with_sympy package.</p>\n\n<h1 id=\"settings\">Settings</h1>\n\n<h2 id=\"printing\">Printing</h2>\n\n<p>In interactive environments the default output of an equation is a\nhuman readable string with the two sides connected by an equals\nsign or a typeset equation with the two sides connected by an equals sign.\n<code>print(Eqn)</code> or <code>str(Eqn)</code> will return this human readable text version of\nthe equation as well. This is consistent with python standards, but not\nsympy, where <code>str()</code> is supposed to return something that can be\ncopy-pasted into code. If the equation has a declared name as in <code>eq1 =\nEqn(a,b/c)</code> the name will be displayed to the right of the equation in\nparentheses (eg. <code>a = b/c    (eq1)</code>). Use <code>print(repr(Eqn))</code> instead of\n<code>print(Eqn)</code> or <code>repr(Eqn)</code> instead of <code>str(Eqn)</code> to get a code\ncompatible version of the equation.</p>\n\n<p>You can adjust this behvior using some flags that impact output:</p>\n\n<ul>\n<li><code>algwsym_config.output.show_code</code> default is <code>False</code>.</li>\n<li><code>algwsym_config.output.human_text</code> default is <code>False</code>.</li>\n<li><code>algwsym_config.output.label</code> default is <code>True</code>.</li>\n</ul>\n\n<p>In interactive environments you can get both types of output by setting\nthe <code>algwsym_config.output.show_code</code> flag. If this flag is true\ncalls to <code>latex</code> and <code>str</code> will also print an additional line \"code\nversion: <code>repr(Eqn)</code>\". Thus in Jupyter you will get a line of typeset\nmathematics output preceded by the code version that can be copy-pasted.\nDefault is <code>False</code>.</p>\n\n<p>A second flag <code>algwsym_config.output.human_text</code> is useful in\ntext-based interactive environments such as command line python or\nipython. If this flag is true <code>repr</code> will return <code>str</code>. Thus the human\nreadable text will be printed as the output of a line that is an\nexpression containing an equation.\nDefault is <code>False</code>.</p>\n\n<p>Setting both of these flags to true in a command line or ipython\nenvironment will show both the code version and the human readable text.\nThese flags impact the behavior of the <code>print(Eqn)</code> statement.</p>\n\n<p>The third flag <code>algwsym_config.output.label</code> has a default value of\n<code>True</code>. Setting this to <code>False</code> suppresses the labeling of an equation\nwith its python name off to the right of the equation.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output", "type": "class", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.__init__", "type": "function", "doc": "<p>This holds settings that impact output.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.show_code", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.show_code", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.human_text", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.human_text", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.label", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.label", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation", "type": "class", "doc": "<p>This class defines an equation with a left-hand-side (lhs) and a right-\nhand-side (rhs) connected by the \"=\" operator (e.g. <code>p*V = n*R*T</code>).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This class defines relations that all high school and college students\nwould recognize as mathematical equations. At present only the \"=\" relation\noperator is recognized.</p>\n\n<p>This class is intended to allow using the mathematical tools in SymPy to\nrearrange equations and perform algebra in a stepwise fashion. In this\nway more people can successfully perform algebraic rearrangements without\nstumbling over missed details such as a negative sign.</p>\n\n<p>Create an equation with the call <code>Equation(lhs,rhs)</code>, where <code>lhs</code> and\n<code>rhs</code> are any valid Sympy expression. <code>Eqn(...)</code> is a synonym for\n<code>Equation(...)</code>.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>lhs: sympy expression, <code>class Expr</code>.\nrhs: sympy expression, <code>class Expr</code>.\nkwargs:</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">algebra_with_sympy</span> <span class=\"kn\">import</span> <span class=\"o\">*</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;a b c x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Equation</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">*</span><span class=\"n\">c</span>\n<span class=\"go\">Equation(a*c, b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a*c, b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(exp(a), exp(b/c))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(a, b/c)</span>\n</code></pre></div>\n\n<p>Simplification and Expansion</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span>\n<span class=\"go\">Equation(x**2 - 1, c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">simplify</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(x - 1, c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(x - 1, c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factor</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"p\">)</span>\n<span class=\"go\">Equation((x - 1)*(x + 1), c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span><span class=\"o\">.</span><span class=\"n\">factor</span><span class=\"p\">()</span>\n<span class=\"go\">Equation((x - 1)*(x + 1), c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f2</span> <span class=\"o\">=</span> <span class=\"n\">f</span><span class=\"o\">+</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">+</span><span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span><span class=\"n\">c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f2</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">f2</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>\n</code></pre></div>\n\n<p>Apply operation to only one side</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applyrhs</span><span class=\"p\">(</span><span class=\"n\">factor</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">factor</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">collect</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>\n</code></pre></div>\n\n<p><code>.apply...</code> also works with user defined python functions</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">def</span> <span class=\"nf\">addsquare</span><span class=\"p\">(</span><span class=\"n\">eqn</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span>    <span class=\"k\">return</span> <span class=\"n\">eqn</span><span class=\"o\">+</span><span class=\"n\">eqn</span><span class=\"o\">**</span><span class=\"mi\">2</span>\n<span class=\"gp\">...</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">apply</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">applyrhs</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">apply</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">,</span> <span class=\"n\">side</span> <span class=\"o\">=</span> <span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">addsquare</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b**2/c**2 + b/c)</span>\n</code></pre></div>\n\n<p>Inaddition to <code>.apply...</code> there is also the less general <code>.do</code>,\n<code>.dolhs</code>, <code>.dorhs</code>, which only works for operations defined on the\n<code>Expr</code> class (e.g.<code>.collect(), .factor(), .expand()</code>, etc...).</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dolhs</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">do</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">factor</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>\n</code></pre></div>\n\n<p><code>poly.do.exp()</code> or other sympy math functions will raise an error.</p>\n\n<p>Rearranging an equation (simple example made complicated as illustration)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span><span class=\"p\">,</span> <span class=\"n\">V</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">R</span><span class=\"p\">,</span> <span class=\"n\">T</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;p V n R T&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq1</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">p</span><span class=\"o\">*</span><span class=\"n\">V</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"o\">*</span><span class=\"n\">R</span><span class=\"o\">*</span><span class=\"n\">T</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq1</span>\n<span class=\"go\">Equation(V*p, R*T*n)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span> <span class=\"o\">=</span><span class=\"n\">eq1</span><span class=\"o\">/</span><span class=\"n\">V</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span>\n<span class=\"go\">Equation(p, R*T*n/V)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq3</span> <span class=\"o\">=</span> <span class=\"n\">eq2</span><span class=\"o\">/</span><span class=\"n\">R</span><span class=\"o\">/</span><span class=\"n\">T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq3</span>\n<span class=\"go\">Equation(p/(R*T), n/V)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq4</span> <span class=\"o\">=</span> <span class=\"n\">eq3</span><span class=\"o\">*</span><span class=\"n\">R</span><span class=\"o\">/</span><span class=\"n\">p</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq4</span>\n<span class=\"go\">Equation(1/T, R*n/(V*p))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">eq4</span>\n<span class=\"go\">Equation(T, V*p/(R*n))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq5</span> <span class=\"o\">=</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">eq4</span> <span class=\"o\">-</span> <span class=\"n\">T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq5</span>\n<span class=\"go\">Equation(0, -T + V*p/(R*n))</span>\n</code></pre></div>\n\n<p>Substitution (#'s and units)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span><span class=\"p\">,</span> <span class=\"n\">atm</span><span class=\"p\">,</span> <span class=\"n\">mol</span><span class=\"p\">,</span> <span class=\"n\">K</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;L atm mol K&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span> <span class=\"c1\"># units</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">R</span><span class=\"p\">:</span><span class=\"mf\">0.08206</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"o\">*</span><span class=\"n\">atm</span><span class=\"o\">/</span><span class=\"n\">mol</span><span class=\"o\">/</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">T</span><span class=\"p\">:</span><span class=\"mi\">273</span><span class=\"o\">*</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"p\">:</span><span class=\"mf\">1.00</span><span class=\"o\">*</span><span class=\"n\">mol</span><span class=\"p\">,</span><span class=\"n\">V</span><span class=\"p\">:</span><span class=\"mf\">24.0</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"p\">})</span>\n<span class=\"go\">Equation(p, 0.9334325*atm)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">R</span><span class=\"p\">:</span><span class=\"mf\">0.08206</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"o\">*</span><span class=\"n\">atm</span><span class=\"o\">/</span><span class=\"n\">mol</span><span class=\"o\">/</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">T</span><span class=\"p\">:</span><span class=\"mi\">273</span><span class=\"o\">*</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"p\">:</span><span class=\"mf\">1.00</span><span class=\"o\">*</span><span class=\"n\">mol</span><span class=\"p\">,</span><span class=\"n\">V</span><span class=\"p\">:</span><span class=\"mf\">24.0</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"p\">})</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(p, 0.9334*atm)</span>\n</code></pre></div>\n\n<p>Combining equations (Math with equations: lhs with lhs and rhs with rhs)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a*c, b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">+</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a*c + a, b/c + b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">/</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(c, 1/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">**</span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a**(a*c), (b/c)**(b/c**2))</span>\n</code></pre></div>\n\n<p>Utility operations</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">reversed</span>\n<span class=\"go\">Equation(b/c, a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">swap</span>\n<span class=\"go\">Equation(b/c, a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">lhs</span>\n<span class=\"go\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">rhs</span>\n<span class=\"go\">b/c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">as_Boolean</span><span class=\"p\">()</span>\n<span class=\"go\">Eq(a, b/c)</span>\n</code></pre></div>\n\n<p><code>.check()</code> convenience method for <code>.as_Boolean().simplify()</code></p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Equation</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">+</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">+</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">check</span><span class=\"p\">()</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">a</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">check</span><span class=\"p\">()</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<p>Differentiation\nDifferentiation is applied to both sides if the wrt variable appears on\nboth sides.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a*c, b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b), c**(-2))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, -2*b/c**3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">),</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(log(a*c), b), 1/b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a, c), 6*b/c**4)</span>\n</code></pre></div>\n\n<p>If you specify multiple differentiation all at once the assumption\nis order of differentiation matters and the lhs will not be\nevaluated.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b, c), -2/c**3)</span>\n</code></pre></div>\n\n<p>To overcome this specify the order of operations.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">),</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a, b), -2/c**3)</span>\n</code></pre></div>\n\n<p>But the reverse order returns an unevaulated lhs (a may depend on b).</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">),</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b, c), -2/c**3)</span>\n</code></pre></div>\n\n<p>Integration can only be performed on one side at a time.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">b**2/(2*c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;lhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">a*b*c</span>\n</code></pre></div>\n\n<p>Make a pretty statement of integration from an equation</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">Integral</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"o\">.</span><span class=\"n\">lhs</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">),</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(Integral(a*c, b), b**2/(2*c))</span>\n</code></pre></div>\n\n<p>Integration of each side with respect to different variables</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">b</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">dolhs</span><span class=\"o\">.</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2*c/2, b**2/(2*c))</span>\n</code></pre></div>\n\n<p>SymPy's solvers do not understand these equations. They expect an\nexpression that the solver assumes = 0. Thus to use the solver the\nequation must be rearranged so that all non-zero symbols are on one side.\nThen just the non-zero symbolic side is passed to <code>solve()</code>.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t2</span> <span class=\"o\">=</span> <span class=\"n\">t</span><span class=\"o\">-</span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">rhs</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t2</span>\n<span class=\"go\">Equation(a - b/c, 0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">solve</span><span class=\"p\">(</span><span class=\"n\">t2</span><span class=\"o\">.</span><span class=\"n\">lhs</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">[b/a]</span>\n</code></pre></div>\n", "bases": "sympy.core.basic.Basic, sympy.core.evalf.EvalfMixin"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.lhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.lhs", "type": "variable", "doc": "<p>Returns the lhs of the equation.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.rhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.rhs", "type": "variable", "doc": "<p>Returns the rhs of the equation.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.as_Boolean", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.as_Boolean", "type": "function", "doc": "<p>Converts the equation to an Equality.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.check", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.check", "type": "function", "doc": "<p>Forces simplification and casts as <code>Equality</code> to check validity.</p>\n\n<h2 id=\"parameters\">Parameters</h2>\n\n<p>kwargs any appropriate for <code>Equality</code>.</p>\n\n<h2 id=\"returns\">Returns</h2>\n\n<p>True, False or an unevaluated <code>Equality</code> if truth cannot be determined.</p>\n", "signature": "(self, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.reversed", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.reversed", "type": "variable", "doc": "<p>Swaps the lhs and the rhs.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.swap", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.swap", "type": "variable", "doc": "<p>Synonym for <code>.reversed</code></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.apply", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.apply", "type": "function", "doc": "<p>Apply an operation/function/method to the equation returning the\nresulting equation.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>func: object\n    object to apply usually a function</p>\n\n<p>args: as necessary for the function</p>\n\n<p>side: 'both', 'lhs', 'rhs', optional\n    Specifies which side of the equation the operation will be applied\n    to. Default is 'both'.</p>\n\n<p>kwargs: as necessary for the function</p>\n", "signature": "(self, func, *args, side='both', **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.applylhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.applylhs", "type": "function", "doc": "<p>If lhs side of the equation has a defined subfunction (attribute) of\nname <code>func</code>, that will be applied instead of the global function.\nThe operation is applied to only the lhs.</p>\n", "signature": "(self, func, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.applyrhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.applyrhs", "type": "function", "doc": "<p>If rhs side of the equation has a defined subfunction (attribute) of\nname <code>func</code>, that will be applied instead of the global function.\nThe operation is applied to only the rhs.</p>\n", "signature": "(self, func, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.do", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.do", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.dolhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.dolhs", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.dorhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.dorhs", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.expand", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.expand", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.simplify", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.simplify", "type": "function", "doc": "<p>See the simplify function in sympy.simplify</p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.factor", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.factor", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.collect", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.collect", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.evalf", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.evalf", "type": "function", "doc": "<p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>subs : dict, optional\n    Substitute numerical values for symbols, e.g.\n    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a\n    dictionary.</p>\n\n<p>maxn : int, optional\n    Allow a maximum temporary working precision of maxn digits.</p>\n\n<p>chop : bool or number, optional\n    Specifies how to replace tiny real or imaginary parts in\n    subresults by exact zeros.</p>\n\n<pre><code>When ``True`` the chop value defaults to standard precision.\n\nOtherwise the chop value is used to determine the\nmagnitude of \"small\" for purposes of chopping.\n\n&gt;&gt;&gt; from sympy import N\n&gt;&gt;&gt; x = 1e-4\n&gt;&gt;&gt; N(x, chop=True)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-5)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-4)\n0\n</code></pre>\n\n<p>strict : bool, optional\n    Raise <code>PrecisionExhausted</code> if any subresult fails to\n    evaluate to full accuracy, given the available maxprec.</p>\n\n<p>quad : str, optional\n    Choose algorithm for numerical quadrature. By default,\n    tanh-sinh quadrature is used. For oscillatory\n    integrals on an infinite interval, try <code>quad='osc'</code>.</p>\n\n<p>verbose : bool, optional\n    Print debug information.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>When Floats are naively substituted into an expression,\nprecision errors may adversely affect the result. For example,\nadding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is\nthen subtracted, the result will be 0.\nThat is exactly what happens in the following:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">values</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">}</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Using the subs argument for evalf is the accurate way to\nevaluate such an expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"n\">subs</span><span class=\"o\">=</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000000000000</span>\n</code></pre></div>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.n", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.n", "type": "function", "doc": "<p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>subs : dict, optional\n    Substitute numerical values for symbols, e.g.\n    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a\n    dictionary.</p>\n\n<p>maxn : int, optional\n    Allow a maximum temporary working precision of maxn digits.</p>\n\n<p>chop : bool or number, optional\n    Specifies how to replace tiny real or imaginary parts in\n    subresults by exact zeros.</p>\n\n<pre><code>When ``True`` the chop value defaults to standard precision.\n\nOtherwise the chop value is used to determine the\nmagnitude of \"small\" for purposes of chopping.\n\n&gt;&gt;&gt; from sympy import N\n&gt;&gt;&gt; x = 1e-4\n&gt;&gt;&gt; N(x, chop=True)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-5)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-4)\n0\n</code></pre>\n\n<p>strict : bool, optional\n    Raise <code>PrecisionExhausted</code> if any subresult fails to\n    evaluate to full accuracy, given the available maxprec.</p>\n\n<p>quad : str, optional\n    Choose algorithm for numerical quadrature. By default,\n    tanh-sinh quadrature is used. For oscillatory\n    integrals on an infinite interval, try <code>quad='osc'</code>.</p>\n\n<p>verbose : bool, optional\n    Print debug information.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>When Floats are naively substituted into an expression,\nprecision errors may adversely affect the result. For example,\nadding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is\nthen subtracted, the result will be 0.\nThat is exactly what happens in the following:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">values</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">}</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Using the subs argument for evalf is the accurate way to\nevaluate such an expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"n\">subs</span><span class=\"o\">=</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000000000000</span>\n</code></pre></div>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn", "type": "class", "doc": "<p>This class defines an equation with a left-hand-side (lhs) and a right-\nhand-side (rhs) connected by the \"=\" operator (e.g. <code>p*V = n*R*T</code>).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This class defines relations that all high school and college students\nwould recognize as mathematical equations. At present only the \"=\" relation\noperator is recognized.</p>\n\n<p>This class is intended to allow using the mathematical tools in SymPy to\nrearrange equations and perform algebra in a stepwise fashion. In this\nway more people can successfully perform algebraic rearrangements without\nstumbling over missed details such as a negative sign.</p>\n\n<p>Create an equation with the call <code>Equation(lhs,rhs)</code>, where <code>lhs</code> and\n<code>rhs</code> are any valid Sympy expression. <code>Eqn(...)</code> is a synonym for\n<code>Equation(...)</code>.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>lhs: sympy expression, <code>class Expr</code>.\nrhs: sympy expression, <code>class Expr</code>.\nkwargs:</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">algebra_with_sympy</span> <span class=\"kn\">import</span> <span class=\"o\">*</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;a b c x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Equation</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">*</span><span class=\"n\">c</span>\n<span class=\"go\">Equation(a*c, b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a*c, b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(exp(a), exp(b/c))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(a, b/c)</span>\n</code></pre></div>\n\n<p>Simplification and Expansion</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span>\n<span class=\"go\">Equation(x**2 - 1, c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">simplify</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(x - 1, c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(x - 1, c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factor</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"p\">)</span>\n<span class=\"go\">Equation((x - 1)*(x + 1), c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span><span class=\"o\">.</span><span class=\"n\">factor</span><span class=\"p\">()</span>\n<span class=\"go\">Equation((x - 1)*(x + 1), c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f2</span> <span class=\"o\">=</span> <span class=\"n\">f</span><span class=\"o\">+</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">+</span><span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span><span class=\"n\">c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f2</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">f2</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>\n</code></pre></div>\n\n<p>Apply operation to only one side</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applyrhs</span><span class=\"p\">(</span><span class=\"n\">factor</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">factor</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">collect</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>\n</code></pre></div>\n\n<p><code>.apply...</code> also works with user defined python functions</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">def</span> <span class=\"nf\">addsquare</span><span class=\"p\">(</span><span class=\"n\">eqn</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span>    <span class=\"k\">return</span> <span class=\"n\">eqn</span><span class=\"o\">+</span><span class=\"n\">eqn</span><span class=\"o\">**</span><span class=\"mi\">2</span>\n<span class=\"gp\">...</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">apply</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">applyrhs</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">apply</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">,</span> <span class=\"n\">side</span> <span class=\"o\">=</span> <span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">addsquare</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b**2/c**2 + b/c)</span>\n</code></pre></div>\n\n<p>Inaddition to <code>.apply...</code> there is also the less general <code>.do</code>,\n<code>.dolhs</code>, <code>.dorhs</code>, which only works for operations defined on the\n<code>Expr</code> class (e.g.<code>.collect(), .factor(), .expand()</code>, etc...).</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dolhs</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">do</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">factor</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>\n</code></pre></div>\n\n<p><code>poly.do.exp()</code> or other sympy math functions will raise an error.</p>\n\n<p>Rearranging an equation (simple example made complicated as illustration)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span><span class=\"p\">,</span> <span class=\"n\">V</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">R</span><span class=\"p\">,</span> <span class=\"n\">T</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;p V n R T&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq1</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">p</span><span class=\"o\">*</span><span class=\"n\">V</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"o\">*</span><span class=\"n\">R</span><span class=\"o\">*</span><span class=\"n\">T</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq1</span>\n<span class=\"go\">Equation(V*p, R*T*n)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span> <span class=\"o\">=</span><span class=\"n\">eq1</span><span class=\"o\">/</span><span class=\"n\">V</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span>\n<span class=\"go\">Equation(p, R*T*n/V)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq3</span> <span class=\"o\">=</span> <span class=\"n\">eq2</span><span class=\"o\">/</span><span class=\"n\">R</span><span class=\"o\">/</span><span class=\"n\">T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq3</span>\n<span class=\"go\">Equation(p/(R*T), n/V)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq4</span> <span class=\"o\">=</span> <span class=\"n\">eq3</span><span class=\"o\">*</span><span class=\"n\">R</span><span class=\"o\">/</span><span class=\"n\">p</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq4</span>\n<span class=\"go\">Equation(1/T, R*n/(V*p))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">eq4</span>\n<span class=\"go\">Equation(T, V*p/(R*n))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq5</span> <span class=\"o\">=</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">eq4</span> <span class=\"o\">-</span> <span class=\"n\">T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq5</span>\n<span class=\"go\">Equation(0, -T + V*p/(R*n))</span>\n</code></pre></div>\n\n<p>Substitution (#'s and units)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span><span class=\"p\">,</span> <span class=\"n\">atm</span><span class=\"p\">,</span> <span class=\"n\">mol</span><span class=\"p\">,</span> <span class=\"n\">K</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;L atm mol K&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span> <span class=\"c1\"># units</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">R</span><span class=\"p\">:</span><span class=\"mf\">0.08206</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"o\">*</span><span class=\"n\">atm</span><span class=\"o\">/</span><span class=\"n\">mol</span><span class=\"o\">/</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">T</span><span class=\"p\">:</span><span class=\"mi\">273</span><span class=\"o\">*</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"p\">:</span><span class=\"mf\">1.00</span><span class=\"o\">*</span><span class=\"n\">mol</span><span class=\"p\">,</span><span class=\"n\">V</span><span class=\"p\">:</span><span class=\"mf\">24.0</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"p\">})</span>\n<span class=\"go\">Equation(p, 0.9334325*atm)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">R</span><span class=\"p\">:</span><span class=\"mf\">0.08206</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"o\">*</span><span class=\"n\">atm</span><span class=\"o\">/</span><span class=\"n\">mol</span><span class=\"o\">/</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">T</span><span class=\"p\">:</span><span class=\"mi\">273</span><span class=\"o\">*</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"p\">:</span><span class=\"mf\">1.00</span><span class=\"o\">*</span><span class=\"n\">mol</span><span class=\"p\">,</span><span class=\"n\">V</span><span class=\"p\">:</span><span class=\"mf\">24.0</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"p\">})</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(p, 0.9334*atm)</span>\n</code></pre></div>\n\n<p>Combining equations (Math with equations: lhs with lhs and rhs with rhs)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a*c, b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">+</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a*c + a, b/c + b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">/</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(c, 1/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">**</span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a**(a*c), (b/c)**(b/c**2))</span>\n</code></pre></div>\n\n<p>Utility operations</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">reversed</span>\n<span class=\"go\">Equation(b/c, a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">swap</span>\n<span class=\"go\">Equation(b/c, a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">lhs</span>\n<span class=\"go\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">rhs</span>\n<span class=\"go\">b/c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">as_Boolean</span><span class=\"p\">()</span>\n<span class=\"go\">Eq(a, b/c)</span>\n</code></pre></div>\n\n<p><code>.check()</code> convenience method for <code>.as_Boolean().simplify()</code></p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Equation</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">+</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">+</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">check</span><span class=\"p\">()</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">a</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">check</span><span class=\"p\">()</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<p>Differentiation\nDifferentiation is applied to both sides if the wrt variable appears on\nboth sides.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a*c, b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b), c**(-2))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, -2*b/c**3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">),</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(log(a*c), b), 1/b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a, c), 6*b/c**4)</span>\n</code></pre></div>\n\n<p>If you specify multiple differentiation all at once the assumption\nis order of differentiation matters and the lhs will not be\nevaluated.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b, c), -2/c**3)</span>\n</code></pre></div>\n\n<p>To overcome this specify the order of operations.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">),</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a, b), -2/c**3)</span>\n</code></pre></div>\n\n<p>But the reverse order returns an unevaulated lhs (a may depend on b).</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">),</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b, c), -2/c**3)</span>\n</code></pre></div>\n\n<p>Integration can only be performed on one side at a time.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">b**2/(2*c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;lhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">a*b*c</span>\n</code></pre></div>\n\n<p>Make a pretty statement of integration from an equation</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">Integral</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"o\">.</span><span class=\"n\">lhs</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">),</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(Integral(a*c, b), b**2/(2*c))</span>\n</code></pre></div>\n\n<p>Integration of each side with respect to different variables</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">b</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">dolhs</span><span class=\"o\">.</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2*c/2, b**2/(2*c))</span>\n</code></pre></div>\n\n<p>SymPy's solvers do not understand these equations. They expect an\nexpression that the solver assumes = 0. Thus to use the solver the\nequation must be rearranged so that all non-zero symbols are on one side.\nThen just the non-zero symbolic side is passed to <code>solve()</code>.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t2</span> <span class=\"o\">=</span> <span class=\"n\">t</span><span class=\"o\">-</span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">rhs</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t2</span>\n<span class=\"go\">Equation(a - b/c, 0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">solve</span><span class=\"p\">(</span><span class=\"n\">t2</span><span class=\"o\">.</span><span class=\"n\">lhs</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">[b/a]</span>\n</code></pre></div>\n", "bases": "sympy.core.basic.Basic, sympy.core.evalf.EvalfMixin"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.lhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.lhs", "type": "variable", "doc": "<p>Returns the lhs of the equation.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.rhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.rhs", "type": "variable", "doc": "<p>Returns the rhs of the equation.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.as_Boolean", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.as_Boolean", "type": "function", "doc": "<p>Converts the equation to an Equality.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.check", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.check", "type": "function", "doc": "<p>Forces simplification and casts as <code>Equality</code> to check validity.</p>\n\n<h2 id=\"parameters\">Parameters</h2>\n\n<p>kwargs any appropriate for <code>Equality</code>.</p>\n\n<h2 id=\"returns\">Returns</h2>\n\n<p>True, False or an unevaluated <code>Equality</code> if truth cannot be determined.</p>\n", "signature": "(self, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.reversed", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.reversed", "type": "variable", "doc": "<p>Swaps the lhs and the rhs.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.swap", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.swap", "type": "variable", "doc": "<p>Synonym for <code>.reversed</code></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.apply", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.apply", "type": "function", "doc": "<p>Apply an operation/function/method to the equation returning the\nresulting equation.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>func: object\n    object to apply usually a function</p>\n\n<p>args: as necessary for the function</p>\n\n<p>side: 'both', 'lhs', 'rhs', optional\n    Specifies which side of the equation the operation will be applied\n    to. Default is 'both'.</p>\n\n<p>kwargs: as necessary for the function</p>\n", "signature": "(self, func, *args, side='both', **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.applylhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.applylhs", "type": "function", "doc": "<p>If lhs side of the equation has a defined subfunction (attribute) of\nname <code>func</code>, that will be applied instead of the global function.\nThe operation is applied to only the lhs.</p>\n", "signature": "(self, func, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.applyrhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.applyrhs", "type": "function", "doc": "<p>If rhs side of the equation has a defined subfunction (attribute) of\nname <code>func</code>, that will be applied instead of the global function.\nThe operation is applied to only the rhs.</p>\n", "signature": "(self, func, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.do", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.do", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.dolhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.dolhs", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.dorhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.dorhs", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.expand", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.expand", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.simplify", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.simplify", "type": "function", "doc": "<p>See the simplify function in sympy.simplify</p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.factor", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.factor", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.collect", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.collect", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.evalf", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.evalf", "type": "function", "doc": "<p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>subs : dict, optional\n    Substitute numerical values for symbols, e.g.\n    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a\n    dictionary.</p>\n\n<p>maxn : int, optional\n    Allow a maximum temporary working precision of maxn digits.</p>\n\n<p>chop : bool or number, optional\n    Specifies how to replace tiny real or imaginary parts in\n    subresults by exact zeros.</p>\n\n<pre><code>When ``True`` the chop value defaults to standard precision.\n\nOtherwise the chop value is used to determine the\nmagnitude of \"small\" for purposes of chopping.\n\n&gt;&gt;&gt; from sympy import N\n&gt;&gt;&gt; x = 1e-4\n&gt;&gt;&gt; N(x, chop=True)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-5)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-4)\n0\n</code></pre>\n\n<p>strict : bool, optional\n    Raise <code>PrecisionExhausted</code> if any subresult fails to\n    evaluate to full accuracy, given the available maxprec.</p>\n\n<p>quad : str, optional\n    Choose algorithm for numerical quadrature. By default,\n    tanh-sinh quadrature is used. For oscillatory\n    integrals on an infinite interval, try <code>quad='osc'</code>.</p>\n\n<p>verbose : bool, optional\n    Print debug information.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>When Floats are naively substituted into an expression,\nprecision errors may adversely affect the result. For example,\nadding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is\nthen subtracted, the result will be 0.\nThat is exactly what happens in the following:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">values</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">}</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Using the subs argument for evalf is the accurate way to\nevaluate such an expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"n\">subs</span><span class=\"o\">=</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000000000000</span>\n</code></pre></div>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.n", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.n", "type": "function", "doc": "<p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>subs : dict, optional\n    Substitute numerical values for symbols, e.g.\n    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a\n    dictionary.</p>\n\n<p>maxn : int, optional\n    Allow a maximum temporary working precision of maxn digits.</p>\n\n<p>chop : bool or number, optional\n    Specifies how to replace tiny real or imaginary parts in\n    subresults by exact zeros.</p>\n\n<pre><code>When ``True`` the chop value defaults to standard precision.\n\nOtherwise the chop value is used to determine the\nmagnitude of \"small\" for purposes of chopping.\n\n&gt;&gt;&gt; from sympy import N\n&gt;&gt;&gt; x = 1e-4\n&gt;&gt;&gt; N(x, chop=True)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-5)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-4)\n0\n</code></pre>\n\n<p>strict : bool, optional\n    Raise <code>PrecisionExhausted</code> if any subresult fails to\n    evaluate to full accuracy, given the available maxprec.</p>\n\n<p>quad : str, optional\n    Choose algorithm for numerical quadrature. By default,\n    tanh-sinh quadrature is used. For oscillatory\n    integrals on an infinite interval, try <code>quad='osc'</code>.</p>\n\n<p>verbose : bool, optional\n    Print debug information.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>When Floats are naively substituted into an expression,\nprecision errors may adversely affect the result. For example,\nadding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is\nthen subtracted, the result will be 0.\nThat is exactly what happens in the following:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">values</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">}</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Using the subs argument for evalf is the accurate way to\nevaluate such an expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"n\">subs</span><span class=\"o\">=</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000000000000</span>\n</code></pre></div>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.solve", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "solve", "type": "function", "doc": "<p>Override of sympy <code>solve()</code>. If it is passed an equation it will\noutput the solutions as typeset equations and return\nthe answer as a list of equations that can be accessed for additional\nmanipulations.</p>\n", "signature": "(f, *symbols, **flags)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sqrt", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sqrt", "type": "function", "doc": "<p>Override of sympy convenience function <code>sqrt</code>. Simply divides equations\ninto two sides if <code>arg</code> is an instance of <code>Equation</code>. This avoids an\nissue with the way sympy is delaying specialized applications of _Pow_ on\nobjects that are not basic sympy expressions.\nReturns the principal square root.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>evaluate : bool, optional\n    The parameter determines if the expression should be evaluated.\n    If <code>None</code>, its value is taken from\n    <code>global_parameters.evaluate</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">**</span><span class=\"mi\">2</span>\n<span class=\"go\">x</span>\n</code></pre></div>\n\n<p>Note that sqrt(x**2) does not simplify to x.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(x**2)</span>\n</code></pre></div>\n\n<p>This is because the two are not equal to each other in general.\nFor example, consider x == -1:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Eq</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eq</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<p>This is because sqrt computes the principal square root, so the square may\nput the argument in a different branch.  This identity does hold if x is\npositive:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;y&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">y</span>\n</code></pre></div>\n\n<p>You can force this simplification by using the powdenest() function with\nthe force option set to True:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">powdenest</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">powdenest</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">force</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">x</span>\n</code></pre></div>\n\n<p>To get both branches of the square root you can use the rootof function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">rootof</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">rootof</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">-</span><span class=\"mi\">3</span><span class=\"p\">,</span><span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span><span class=\"mi\">1</span><span class=\"p\">)]</span>\n<span class=\"go\">[-sqrt(3), sqrt(3)]</span>\n</code></pre></div>\n\n<p>Although <code>sqrt</code> is printed, there is no <code>sqrt</code> function so looking for\n<code>sqrt</code> in an expression will fail:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.utilities.misc</span> <span class=\"kn\">import</span> <span class=\"n\">func_name</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">func_name</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">&#39;Pow&#39;</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">has</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<p>To find <code>sqrt</code> look for <code>Pow</code> with an exponent of <code>1/2</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">find</span><span class=\"p\">(</span><span class=\"k\">lambda</span> <span class=\"n\">i</span><span class=\"p\">:</span> <span class=\"n\">i</span><span class=\"o\">.</span><span class=\"n\">is_Pow</span> <span class=\"ow\">and</span> <span class=\"nb\">abs</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"o\">.</span><span class=\"n\">exp</span><span class=\"p\">)</span> <span class=\"ow\">is</span> <span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">Half</span><span class=\"p\">)</span>\n<span class=\"go\">{1/sqrt(x)}</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.polys.rootoftools.rootof, root, real_root</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Square_root\n.. [2] https://en.wikipedia.org/wiki/Principal_value</p>\n", "signature": "(arg, evaluate=None)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.root", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "root", "type": "function", "doc": "<p>Override of sympy convenience function <code>root</code>. Simply divides equations\ninto two sides if <code>arg</code> is an instance of <code>Equation</code>. This avoids an\nissue with the way sympy is delaying specialized applications of _Pow_ on\nobjects that are not basic sympy expressions.\nReturns the <em>k</em>-th <em>n</em>-th root of <code>arg</code>.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>k : int, optional\n    Should be an integer in ${0, 1, ..., n-1}$.\n    Defaults to the principal root if $0$.</p>\n\n<p>evaluate : bool, optional\n    The parameter determines if the expression should be evaluated.\n    If <code>None</code>, its value is taken from\n    <code>global_parameters.evaluate</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">root</span><span class=\"p\">,</span> <span class=\"n\">Rational</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">x**(1/3)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">x**(1/n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">x**(-3/2)</span>\n</code></pre></div>\n\n<p>To get the k-th n-th root, specify k:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-(-1)**(2/3)*2**(1/3)</span>\n</code></pre></div>\n\n<p>To get all n n-th roots you can use the rootof function.\nThe following examples show the roots of unity for n\nequal 2, 3 and 4:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">rootof</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">rootof</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)]</span>\n<span class=\"go\">[-1, 1]</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">rootof</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">rootof</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">4</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)]</span>\n<span class=\"go\">[-1, 1, -I, I]</span>\n</code></pre></div>\n\n<p>SymPy, like other symbolic algebra systems, returns the\ncomplex root of negative numbers. This is the principal\nroot and differs from the text-book result that one might\nbe expecting. For example, the cube root of -8 does not\ncome back as -2:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">8</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">2*(-1)**(1/3)</span>\n</code></pre></div>\n\n<p>The real_root function can be used to either make the principal\nresult real (or simply to return the real root directly):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">real_root</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">real_root</span><span class=\"p\">(</span><span class=\"n\">_</span><span class=\"p\">)</span>\n<span class=\"go\">-2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">real_root</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">32</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">-2</span>\n</code></pre></div>\n\n<p>Alternatively, the n//2-th n-th root of a negative number can be\ncomputed with root:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">32</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"o\">//</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.polys.rootoftools.rootof\nsympy.core.power.integer_nthroot\nsqrt, real_root</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Square_root\n.. [2] https://en.wikipedia.org/wiki/Real_root\n.. [3] https://en.wikipedia.org/wiki/Root_of_unity\n.. [4] https://en.wikipedia.org/wiki/Principal_value\n.. [5] http://mathworld.wolfram.com/CubeRoot.html</p>\n", "signature": "(arg, n, k=0, evaluate=None)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.collect", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "collect", "type": "function", "doc": "<p>Override of sympy <code>collect()</code>.</p>\n", "signature": "(\n    expr,\n    syms,\n    func=None,\n    evaluate=None,\n    exact=False,\n    distribute_order_term=True\n)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.EqnFunction", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "EqnFunction", "type": "class", "doc": "<p>Extension of the sympy Function class to understand equations. Each\nsympy function impacted by this extension is listed in the documentation\nthat follows.</p>\n", "bases": "sympy.core.function.Function"}, {"fullname": "algebra_with_sympy.algebraic_equation.EqnFunction.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "EqnFunction.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.EqnFunction.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "EqnFunction.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial", "type": "class", "doc": "<p>Implementation of factorial function over nonnegative integers.\nBy convention (consistent with the gamma function and the binomial\ncoefficients), factorial of a negative integer is complex infinity.</p>\n\n<p>The factorial is very important in combinatorics where it gives\nthe number of ways in which <code>n</code> objects can be permuted. It also\narises in calculus, probability, number theory, etc.</p>\n\n<p>There is strict relation of factorial with gamma function. In\nfact <code>n! = gamma(n+1)</code> for nonnegative integers. Rewrite of this\nkind is very useful in case of combinatorial simplification.</p>\n\n<p>Computation of the factorial is done using two algorithms. For\nsmall arguments a precomputed look up table is used. However for bigger\ninput algorithm Prime-Swing is used. It is the fastest algorithm\nknown and computes <code>n!</code> via prime factorization of special class\nof numbers, called here the 'Swing Numbers'.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"mi\">7</span><span class=\"p\">)</span>\n<span class=\"go\">5040</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(2*n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(1/2)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial2, RisingFactorial, FallingFactorial</p>\n", "bases": "sympy.functions.combinatorial.factorials.factorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial2", "type": "class", "doc": "<p>The double factorial <code>n!!</code>, not to be confused with <code>(n!)!</code></p>\n\n<p>The double factorial is defined for nonnegative integers and for odd\nnegative integers as:</p>\n\n<p>.. math:: n!! = \\begin{cases} 1 &amp; n = 0 \\\n                n(n-2)(n-4) \\cdots 1 &amp; n\\ \\text{positive odd} \\\n                n(n-2)(n-4) \\cdots 2 &amp; n\\ \\text{positive even} \\\n                (n+2)!!/(n+2) &amp; n\\ \\text{negative odd} \\end{cases}</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Double_factorial</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">factorial2</span><span class=\"p\">,</span> <span class=\"n\">var</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span>\n<span class=\"go\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial2</span><span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">factorial2(n + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial2</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">15</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">1/3</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, RisingFactorial, FallingFactorial</p>\n", "bases": "sympy.functions.combinatorial.factorials.factorial2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.rf", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "rf", "type": "class", "doc": "<p>Rising factorial (also called Pochhammer symbol [1]_) is a double valued\nfunction arising in concrete mathematics, hypergeometric functions\nand series expansions. It is defined by:</p>\n\n<p>.. math:: rf(x,k) = x \\cdot (x+1) \\cdots (x+k-1)</p>\n\n<p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For\nmore information check \"Concrete mathematics\" by Graham, pp. 66\nor visit http://mathworld.wolfram.com/RisingFactorial.html page.</p>\n\n<p>When <code>x</code> is a Poly instance of degree $\\ge 1$ with a single variable,\n<code>rf(x,k) = x(y) \\cdot x(y+1) \\cdots x(y+k-1)</code>, where <code>y</code> is the\nvariable of <code>x</code>. This is as described in [2]_.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">Poly</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">120</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">4</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">Poly</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain=&#39;ZZ&#39;)</span>\n</code></pre></div>\n\n<p>Rewriting is complicated unless the relationship between\nthe arguments is known, but rising factorial can\nbe rewritten in terms of gamma, factorial, binomial,\nand falling factorial.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">R</span> <span class=\"o\">=</span> <span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span> <span class=\"n\">R</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">RisingFactorial(n, n + 2)</span>\n<span class=\"go\">FallingFactorial(2*n + 1, n + 2)</span>\n<span class=\"go\">factorial(2*n + 1)/factorial(n - 1)</span>\n<span class=\"go\">binomial(2*n + 1, n + 2)*factorial(n + 2)</span>\n<span class=\"go\">gamma(2*n + 2)/gamma(n)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, factorial2, FallingFactorial</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol\n.. [2] Peter Paule, \"Greatest Factorial Factorization and Symbolic\n       Summation\", Journal of Symbolic Computation, vol. 20, pp. 235-268,\n       1995.</p>\n", "bases": "sympy.functions.combinatorial.factorials.RisingFactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.rf.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "rf.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.rf.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "rf.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.ff", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ff", "type": "class", "doc": "<p>Falling factorial (related to rising factorial) is a double valued\nfunction arising in concrete mathematics, hypergeometric functions\nand series expansions. It is defined by</p>\n\n<p>.. math:: ff(x,k) = x \\cdot (x-1) \\cdots (x-k+1)</p>\n\n<p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For\nmore information check \"Concrete mathematics\" by Graham, pp. 66\nor visit http://mathworld.wolfram.com/FallingFactorial.html page.</p>\n\n<p>When <code>x</code> is a Poly instance of degree &gt;= 1 with single variable,\n<code>ff(x,k) = x(y) \\cdot x(y-1) \\cdots x(y-k+1)</code>, where <code>y</code> is the\nvariable of <code>x</code>. This is as described in Peter Paule, \"Greatest\nFactorial Factorization and Symbolic Summation\", Journal of\nSymbolic Computation, vol. 20, pp. 235-268, 1995.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">Poly</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">120</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">Poly</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Poly(x**4 - 2*x**3 + x**2, x, domain=&#39;ZZ&#39;)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(n)</span>\n</code></pre></div>\n\n<p>Rewriting is complicated unless the relationship between\nthe arguments is known, but falling factorial can\nbe rewritten in terms of gamma, factorial and binomial\nand rising factorial.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">F</span> <span class=\"o\">=</span> <span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span> <span class=\"n\">F</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">RisingFactorial(3, n - 2)</span>\n<span class=\"go\">FallingFactorial(n, n - 2)</span>\n<span class=\"go\">factorial(n)/2</span>\n<span class=\"go\">binomial(n, n - 2)*factorial(n - 2)</span>\n<span class=\"go\">gamma(n + 1)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, factorial2, RisingFactorial</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://mathworld.wolfram.com/FallingFactorial.html</p>\n", "bases": "sympy.functions.combinatorial.factorials.FallingFactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.ff.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ff.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.ff.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ff.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.binomial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "binomial", "type": "class", "doc": "<p>Implementation of the binomial coefficient. It can be defined\nin two ways depending on its desired interpretation:</p>\n\n<p>.. math:: \\binom{n}{k} = \\frac{n!}{k!(n-k)!}\\ \\text{or}\\\n            \\binom{n}{k} = \\frac{ff(n, k)}{k!}</p>\n\n<p>First, in a strict combinatorial sense it defines the\nnumber of ways we can choose <code>k</code> elements from a set of\n<code>n</code> elements. In this case both arguments are nonnegative\nintegers and binomial is computed using an efficient\nalgorithm based on prime factorization.</p>\n\n<p>The other definition is generalization for arbitrary <code>n</code>,\nhowever <code>k</code> must also be nonnegative. This case is very\nuseful when evaluating summations.</p>\n\n<p>For the sake of convenience for negative integer <code>k</code> this function\nwill return zero no matter what valued is the other argument.</p>\n\n<p>To expand the binomial when <code>n</code> is a symbol, use either\n<code>expand_func()</code> or <code>expand(func=True)</code>. The former will keep\nthe polynomial in factored form while the latter will expand the\npolynomial itself. See examples for details.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">Rational</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"mi\">15</span><span class=\"p\">,</span> <span class=\"mi\">8</span><span class=\"p\">)</span>\n<span class=\"go\">6435</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Rows of Pascal's triangle can be generated with the binomial function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">N</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span>    <span class=\"nb\">print</span><span class=\"p\">([</span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">N</span><span class=\"p\">,</span> <span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">N</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)])</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">[1]</span>\n<span class=\"go\">[1, 1]</span>\n<span class=\"go\">[1, 2, 1]</span>\n<span class=\"go\">[1, 3, 3, 1]</span>\n<span class=\"go\">[1, 4, 6, 4, 1]</span>\n<span class=\"go\">[1, 5, 10, 10, 5, 1]</span>\n<span class=\"go\">[1, 6, 15, 20, 15, 6, 1]</span>\n<span class=\"go\">[1, 7, 21, 35, 35, 21, 7, 1]</span>\n</code></pre></div>\n\n<p>As can a given diagonal, e.g. the 4th diagonal:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">N</span> <span class=\"o\">=</span> <span class=\"o\">-</span><span class=\"mi\">4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">N</span><span class=\"p\">,</span> <span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">-</span> <span class=\"n\">N</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, -4, 10, -20, 35]</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">),</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">-5/128</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">),</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">-195/128</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">binomial(n, 3)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">n**3/6 - n**2/2 + n/3</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">n*(n - 2)*(n - 1)/6</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://www.johndcook.com/blog/binomial_coefficients/</p>\n", "bases": "sympy.functions.combinatorial.factorials.binomial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.binomial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "binomial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.binomial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "binomial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.RisingFactorial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "RisingFactorial", "type": "class", "doc": "<p>Rising factorial (also called Pochhammer symbol [1]_) is a double valued\nfunction arising in concrete mathematics, hypergeometric functions\nand series expansions. It is defined by:</p>\n\n<p>.. math:: rf(x,k) = x \\cdot (x+1) \\cdots (x+k-1)</p>\n\n<p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For\nmore information check \"Concrete mathematics\" by Graham, pp. 66\nor visit http://mathworld.wolfram.com/RisingFactorial.html page.</p>\n\n<p>When <code>x</code> is a Poly instance of degree $\\ge 1$ with a single variable,\n<code>rf(x,k) = x(y) \\cdot x(y+1) \\cdots x(y+k-1)</code>, where <code>y</code> is the\nvariable of <code>x</code>. This is as described in [2]_.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">Poly</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">120</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">4</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">Poly</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain=&#39;ZZ&#39;)</span>\n</code></pre></div>\n\n<p>Rewriting is complicated unless the relationship between\nthe arguments is known, but rising factorial can\nbe rewritten in terms of gamma, factorial, binomial,\nand falling factorial.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">R</span> <span class=\"o\">=</span> <span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span> <span class=\"n\">R</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">RisingFactorial(n, n + 2)</span>\n<span class=\"go\">FallingFactorial(2*n + 1, n + 2)</span>\n<span class=\"go\">factorial(2*n + 1)/factorial(n - 1)</span>\n<span class=\"go\">binomial(2*n + 1, n + 2)*factorial(n + 2)</span>\n<span class=\"go\">gamma(2*n + 2)/gamma(n)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, factorial2, FallingFactorial</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol\n.. [2] Peter Paule, \"Greatest Factorial Factorization and Symbolic\n       Summation\", Journal of Symbolic Computation, vol. 20, pp. 235-268,\n       1995.</p>\n", "bases": "sympy.functions.combinatorial.factorials.RisingFactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.RisingFactorial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "RisingFactorial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.RisingFactorial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "RisingFactorial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.FallingFactorial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "FallingFactorial", "type": "class", "doc": "<p>Falling factorial (related to rising factorial) is a double valued\nfunction arising in concrete mathematics, hypergeometric functions\nand series expansions. It is defined by</p>\n\n<p>.. math:: ff(x,k) = x \\cdot (x-1) \\cdots (x-k+1)</p>\n\n<p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For\nmore information check \"Concrete mathematics\" by Graham, pp. 66\nor visit http://mathworld.wolfram.com/FallingFactorial.html page.</p>\n\n<p>When <code>x</code> is a Poly instance of degree &gt;= 1 with single variable,\n<code>ff(x,k) = x(y) \\cdot x(y-1) \\cdots x(y-k+1)</code>, where <code>y</code> is the\nvariable of <code>x</code>. This is as described in Peter Paule, \"Greatest\nFactorial Factorization and Symbolic Summation\", Journal of\nSymbolic Computation, vol. 20, pp. 235-268, 1995.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">Poly</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">120</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">Poly</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Poly(x**4 - 2*x**3 + x**2, x, domain=&#39;ZZ&#39;)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(n)</span>\n</code></pre></div>\n\n<p>Rewriting is complicated unless the relationship between\nthe arguments is known, but falling factorial can\nbe rewritten in terms of gamma, factorial and binomial\nand rising factorial.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">F</span> <span class=\"o\">=</span> <span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span> <span class=\"n\">F</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">RisingFactorial(3, n - 2)</span>\n<span class=\"go\">FallingFactorial(n, n - 2)</span>\n<span class=\"go\">factorial(n)/2</span>\n<span class=\"go\">binomial(n, n - 2)*factorial(n - 2)</span>\n<span class=\"go\">gamma(n + 1)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, factorial2, RisingFactorial</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://mathworld.wolfram.com/FallingFactorial.html</p>\n", "bases": "sympy.functions.combinatorial.factorials.FallingFactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.FallingFactorial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "FallingFactorial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.FallingFactorial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "FallingFactorial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.subfactorial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "subfactorial", "type": "class", "doc": "<p>The subfactorial counts the derangements of n items and is\ndefined for non-negative integers as:</p>\n\n<p>.. math:: !n = \\begin{cases} 1 &amp; n = 0 \\ 0 &amp; n = 1 \\\n                (n-1)(!(n-1) + !(n-2)) &amp; n &gt; 1 \\end{cases}</p>\n\n<p>It can also be written as <code>int(round(n!/exp(1)))</code> but the\nrecursive definition with caching is implemented for this function.</p>\n\n<p>An interesting analytic expression is the following [2]_</p>\n\n<p>.. math:: !x = \\Gamma(x + 1, -1)/e</p>\n\n<p>which is valid for non-negative integers <code>x</code>. The above formula\nis not very useful incase of non-integers. :math:<code>\\Gamma(x + 1, -1)</code> is\nsingle-valued only for integral arguments <code>x</code>, elsewhere on the positive\nreal axis it has an infinite number of branches none of which are real.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Subfactorial\n.. [2] http://mathworld.wolfram.com/Subfactorial.html</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">subfactorial</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">subfactorial</span><span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">subfactorial(n + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">subfactorial</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">44</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.combinatorial.factorials.factorial,\nsympy.utilities.iterables.generate_derangements,\nsympy.functions.special.gamma_functions.uppergamma</p>\n", "bases": "sympy.functions.combinatorial.factorials.subfactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.subfactorial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "subfactorial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.subfactorial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "subfactorial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.carmichael", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "carmichael", "type": "class", "doc": "<p>Carmichael Numbers:</p>\n\n<p>Certain cryptographic algorithms make use of big prime numbers.\nHowever, checking whether a big number is prime is not so easy.\nRandomized prime number checking tests exist that offer a high degree of confidence of\naccurate determination at low cost, such as the Fermat test.</p>\n\n<p>Let 'a' be a random number between 2 and n - 1, where n is the number whose primality we are testing.\nThen, n is probably prime if it satisfies the modular arithmetic congruence relation :</p>\n\n<p>a^(n-1) = 1(mod n).\n(where mod refers to the modulo operation)</p>\n\n<p>If a number passes the Fermat test several times, then it is prime with a\nhigh probability.</p>\n\n<p>Unfortunately, certain composite numbers (non-primes) still pass the Fermat test\nwith every number smaller than themselves.\nThese numbers are called Carmichael numbers.</p>\n\n<p>A Carmichael number will pass a Fermat primality test to every base b relatively prime to the number,\neven though it is not actually prime. This makes tests based on Fermat's Little Theorem less effective than\nstrong probable prime tests such as the Baillie-PSW primality test and the Miller-Rabin primality test.\nmr functions given in sympy/sympy/ntheory/primetest.py will produce wrong results for each and every\ncarmichael number.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">carmichael</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">find_first_n_carmichaels</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">[561, 1105, 1729, 2465, 2821]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">is_prime</span><span class=\"p\">(</span><span class=\"mi\">2465</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">is_prime</span><span class=\"p\">(</span><span class=\"mi\">1729</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">find_carmichael_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">562</span><span class=\"p\">)</span>\n<span class=\"go\">[561]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">find_carmichael_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span><span class=\"mi\">1000</span><span class=\"p\">)</span>\n<span class=\"go\">[561]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">find_carmichael_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span><span class=\"mi\">2000</span><span class=\"p\">)</span>\n<span class=\"go\">[561, 1105, 1729]</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Carmichael_number\n.. [2] https://en.wikipedia.org/wiki/Fermat_primality_test\n.. [3] https://www.jstor.org/stable/23248683?seq=1#metadata_info_tab_contents</p>\n", "bases": "sympy.functions.combinatorial.numbers.carmichael, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.carmichael.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "carmichael.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.carmichael.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "carmichael.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.fibonacci", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fibonacci", "type": "class", "doc": "<p>Fibonacci numbers / Fibonacci polynomials</p>\n\n<p>The Fibonacci numbers are the integer sequence defined by the\ninitial terms <code>F_0 = 0</code>, <code>F_1 = 1</code> and the two-term recurrence\nrelation <code>F_n = F_{n-1} + F_{n-2}</code>.  This definition\nextended to arbitrary real and complex arguments using\nthe formula</p>\n\n<p>.. math :: F_z = \\frac{\\phi^z - \\cos(\\pi z) \\phi^{-z}}{\\sqrt 5}</p>\n\n<p>The Fibonacci polynomials are defined by <code>F_1(x) = 1</code>,\n<code>F_2(x) = x</code>, and <code>F_n(x) = x*F_{n-1}(x) + F_{n-2}(x)</code> for <code>n &gt; 2</code>.\nFor all positive integers <code>n</code>, <code>F_n(1) = F_n</code>.</p>\n\n<ul>\n<li><code>fibonacci(n)</code> gives the <code>n^{th}</code> Fibonacci number, <code>F_n</code></li>\n<li><code>fibonacci(n, x)</code> gives the <code>n^{th}</code> Fibonacci polynomial in <code>x</code>, <code>F_n(x)</code></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">fibonacci</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">fibonacci</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fibonacci</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;t&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">t**4 + 3*t**2 + 1</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, harmonic, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Fibonacci_number\n.. [2] http://mathworld.wolfram.com/FibonacciNumber.html</p>\n", "bases": "sympy.functions.combinatorial.numbers.fibonacci, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.fibonacci.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fibonacci.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.fibonacci.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fibonacci.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.lucas", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lucas", "type": "class", "doc": "<p>Lucas numbers</p>\n\n<p>Lucas numbers satisfy a recurrence relation similar to that of\nthe Fibonacci sequence, in which each term is the sum of the\npreceding two. They are generated by choosing the initial\nvalues <code>L_0 = 2</code> and <code>L_1 = 1</code>.</p>\n\n<ul>\n<li><code>lucas(n)</code> gives the <code>n^{th}</code> Lucas number</li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lucas</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">lucas</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, harmonic, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Lucas_number\n.. [2] http://mathworld.wolfram.com/LucasNumber.html</p>\n", "bases": "sympy.functions.combinatorial.numbers.lucas, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.lucas.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lucas.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.lucas.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lucas.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.motzkin", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "motzkin", "type": "class", "doc": "<p>The nth Motzkin number is the number\nof ways of drawing non-intersecting chords\nbetween n points on a circle (not necessarily touching\nevery point by a chord). The Motzkin numbers are named\nafter Theodore Motzkin and have diverse applications\nin geometry, combinatorics and number theory.</p>\n\n<p>Motzkin numbers are the integer sequence defined by the\ninitial terms <code>M_0 = 1</code>, <code>M_1 = 1</code> and the two-term recurrence relation\n<code>M_n = \frac{2*n + 1}{n + 2} * M_{n-1} + \frac{3n - 3}{n + 2} * M_{n-2}</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">motzkin</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">motzkin</span><span class=\"o\">.</span><span class=\"n\">is_motzkin</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">motzkin</span><span class=\"o\">.</span><span class=\"n\">find_motzkin_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span><span class=\"mi\">300</span><span class=\"p\">)</span>\n<span class=\"go\">[2, 4, 9, 21, 51, 127]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">motzkin</span><span class=\"o\">.</span><span class=\"n\">find_motzkin_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span><span class=\"mi\">900</span><span class=\"p\">)</span>\n<span class=\"go\">[2, 4, 9, 21, 51, 127, 323, 835]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">motzkin</span><span class=\"o\">.</span><span class=\"n\">find_first_n_motzkins</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)</span>\n<span class=\"go\">[1, 1, 2, 4, 9, 21, 51, 127, 323, 835]</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Motzkin_number\n.. [2] https://mathworld.wolfram.com/MotzkinNumber.html</p>\n", "bases": "sympy.functions.combinatorial.numbers.motzkin, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.motzkin.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "motzkin.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.motzkin.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "motzkin.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.tribonacci", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tribonacci", "type": "class", "doc": "<p>Tribonacci numbers / Tribonacci polynomials</p>\n\n<p>The Tribonacci numbers are the integer sequence defined by the\ninitial terms <code>T_0 = 0</code>, <code>T_1 = 1</code>, <code>T_2 = 1</code> and the three-term\nrecurrence relation <code>T_n = T_{n-1} + T_{n-2} + T_{n-3}</code>.</p>\n\n<p>The Tribonacci polynomials are defined by <code>T_0(x) = 0</code>, <code>T_1(x) = 1</code>,\n<code>T_2(x) = x^2</code>, and <code>T_n(x) = x^2 T_{n-1}(x) + x T_{n-2}(x) + T_{n-3}(x)</code>\nfor <code>n &gt; 2</code>.  For all positive integers <code>n</code>, <code>T_n(1) = T_n</code>.</p>\n\n<ul>\n<li><code>tribonacci(n)</code> gives the <code>n^{th}</code> Tribonacci number, <code>T_n</code></li>\n<li><code>tribonacci(n, x)</code> gives the <code>n^{th}</code> Tribonacci polynomial in <code>x</code>, <code>T_n(x)</code></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">tribonacci</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">tribonacci</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tribonacci</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;t&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">t**8 + 3*t**5 + 3*t**2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers\n.. [2] http://mathworld.wolfram.com/TribonacciNumber.html\n.. [3] https://oeis.org/A000073</p>\n", "bases": "sympy.functions.combinatorial.numbers.tribonacci, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.tribonacci.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tribonacci.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.tribonacci.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tribonacci.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.harmonic", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "harmonic", "type": "class", "doc": "<p>Harmonic numbers</p>\n\n<p>The nth harmonic number is given by <code>\\operatorname{H}_{n} =\n1 + \\frac{1}{2} + \\frac{1}{3} + \\ldots + \\frac{1}{n}</code>.</p>\n\n<p>More generally:</p>\n\n<p>.. math:: \\operatorname{H}_{n,m} = \\sum_{k=1}^{n} \\frac{1}{k^m}</p>\n\n<p>As <code>n \\rightarrow \\infty</code>, <code>\\operatorname{H}_{n,m} \\rightarrow \\zeta(m)</code>,\nthe Riemann zeta function.</p>\n\n<ul>\n<li><p><code>harmonic(n)</code> gives the nth harmonic number, <code>\\operatorname{H}_n</code></p></li>\n<li><p><code>harmonic(n, m)</code> gives the nth generalized harmonic number\nof order <code>m</code>, <code>\\operatorname{H}_{n,m}</code>, where\n<code>harmonic(n) == harmonic(n, 1)</code></p></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">harmonic</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">)]</span>\n<span class=\"go\">[0, 1, 3/2, 11/6, 25/12, 137/60]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">)]</span>\n<span class=\"go\">[0, 1, 5/4, 49/36, 205/144, 5269/3600]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">pi**2/6</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">Sum</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Sum</span><span class=\"p\">)</span>\n<span class=\"go\">Sum(1/_k, (_k, 1, n))</span>\n</code></pre></div>\n\n<p>We can evaluate harmonic numbers for all integral and positive\nrational arguments:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">simplify</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">)</span>\n<span class=\"go\">761/280</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)</span>\n<span class=\"go\">83711/27720</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">H</span> <span class=\"o\">=</span> <span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">H</span>\n<span class=\"go\">harmonic(1/3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span> <span class=\"o\">=</span> <span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">H</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span>\n<span class=\"go\">-log(6) - sqrt(3)*pi/6 + 2*Sum(log(sin(_k*pi/3))*cos(2*_k*pi/3), (_k, 1, 1))</span>\n<span class=\"go\">                       + 3*Sum(1/(3*_k + 1), (_k, 0, 0))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">()</span>\n<span class=\"go\">-log(6) - sqrt(3)*pi/6 - log(sqrt(3)/2) + 3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">H</span> <span class=\"o\">=</span> <span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">25</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">7</span><span class=\"p\">))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span> <span class=\"o\">=</span> <span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">H</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">())</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span>\n<span class=\"go\">log(sin(2*pi/7)**(2*cos(16*pi/7))/(14*sin(pi/7)**(2*cos(pi/7))*cos(pi/14)**(2*sin(pi/14)))) + pi*tan(pi/14)/2 + 30247/9900</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">)</span>\n<span class=\"go\">1.983697455232980674869851942390639915940</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">25</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">7</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">)</span>\n<span class=\"go\">1.983697455232980674869851942390639915940</span>\n</code></pre></div>\n\n<p>We can rewrite harmonic numbers in terms of polygamma functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">digamma</span><span class=\"p\">,</span> <span class=\"n\">polygamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">m</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;m&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">digamma</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(0, n + 1) + EulerGamma</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(0, n + 1) + EulerGamma</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(2, n + 1)/2 - polygamma(2, 1)/2</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)</span>\n</code></pre></div>\n\n<p>Integer offsets in the argument can be pulled out:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">+</span><span class=\"mi\">4</span><span class=\"p\">))</span>\n<span class=\"go\">harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">-</span><span class=\"mi\">4</span><span class=\"p\">))</span>\n<span class=\"go\">harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n</span>\n</code></pre></div>\n\n<p>Some limits can be computed as well:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">limit</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">limit</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">limit</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">pi**2/6</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">limit</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-polygamma(2, 1)/2</span>\n</code></pre></div>\n\n<p>However we cannot compute the general relation yet:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">limit</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">harmonic(oo, m)</span>\n</code></pre></div>\n\n<p>which equals <code>zeta(m)</code> for <code>m &gt; 1</code>.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Harmonic_number\n.. [2] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber/\n.. [3] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/</p>\n", "bases": "sympy.functions.combinatorial.numbers.harmonic, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.harmonic.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "harmonic.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.harmonic.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "harmonic.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.bernoulli", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bernoulli", "type": "class", "doc": "<p>Bernoulli numbers / Bernoulli polynomials</p>\n\n<p>The Bernoulli numbers are a sequence of rational numbers\ndefined by <code>B_0 = 1</code> and the recursive relation (<code>n &gt; 0</code>):</p>\n\n<p>.. math :: 0 = \\sum_{k=0}^n \\binom{n+1}{k} B_k</p>\n\n<p>They are also commonly defined by their exponential generating\nfunction, which is <code>\\frac{x}{e^x - 1}</code>. For odd indices &gt; 1, the\nBernoulli numbers are zero.</p>\n\n<p>The Bernoulli polynomials satisfy the analogous formula:</p>\n\n<p>.. math :: B_n(x) = \\sum_{k=0}^n \\binom{n}{k} B_k x^{n-k}</p>\n\n<p>Bernoulli numbers and Bernoulli polynomials are related as\n<code>B_n(0) = B_n</code>.</p>\n\n<p>We compute Bernoulli numbers using Ramanujan's formula:</p>\n\n<p>.. math :: B_n = \\frac{A(n) - S(n)}{\\binom{n+3}{n}}</p>\n\n<p>where:</p>\n\n<p>.. math :: A(n) = \\begin{cases} \\frac{n+3}{3} &amp;\n    n \\equiv 0\\ \\text{or}\\ 2 \\pmod{6} \\\n    -\\frac{n+3}{6} &amp; n \\equiv 4 \\pmod{6} \\end{cases}</p>\n\n<p>and:</p>\n\n<p>.. math :: S(n) = \\sum_{k=1}^{[n/6]} \\binom{n+3}{n-6k} B_{n-6k}</p>\n\n<p>This formula is similar to the sum given in the definition, but\ncuts 2/3 of the terms. For Bernoulli polynomials, we use the\nformula in the definition.</p>\n\n<ul>\n<li><code>bernoulli(n)</code> gives the nth Bernoulli number, <code>B_n</code></li>\n<li><code>bernoulli(n, x)</code> gives the nth Bernoulli polynomial in <code>x</code>, <code>B_n(x)</code></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">bernoulli</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">bernoulli</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">bernoulli</span><span class=\"p\">(</span><span class=\"mi\">1000001</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Bernoulli_number\n.. [2] https://en.wikipedia.org/wiki/Bernoulli_polynomial\n.. [3] http://mathworld.wolfram.com/BernoulliNumber.html\n.. [4] http://mathworld.wolfram.com/BernoulliPolynomial.html</p>\n", "bases": "sympy.functions.combinatorial.numbers.bernoulli, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.bernoulli.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bernoulli.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.bernoulli.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bernoulli.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.bell", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bell", "type": "class", "doc": "<p>Bell numbers / Bell polynomials</p>\n\n<p>The Bell numbers satisfy <code>B_0 = 1</code> and</p>\n\n<p>.. math:: B_n = \\sum_{k=0}^{n-1} \\binom{n-1}{k} B_k.</p>\n\n<p>They are also given by:</p>\n\n<p>.. math:: B_n = \\frac{1}{e} \\sum_{k=0}^{\\infty} \\frac{k^n}{k!}.</p>\n\n<p>The Bell polynomials are given by <code>B_0(x) = 1</code> and</p>\n\n<p>.. math:: B_n(x) = x \\sum_{k=1}^{n-1} \\binom{n-1}{k-1} B_{k-1}(x).</p>\n\n<p>The second kind of Bell polynomials (are sometimes called \"partial\" Bell\npolynomials or incomplete Bell polynomials) are defined as</p>\n\n<p>.. math:: B_{n,k}(x_1, x_2,\\dotsc x_{n-k+1}) =\n        \\sum_{j_1+j_2+j_2+\\dotsb=k \\atop j_1+2j_2+3j_2+\\dotsb=n}\n            \\frac{n!}{j_1!j_2!\\dotsb j_{n-k+1}!}\n            \\left(\\frac{x_1}{1!} \\right)^{j_1}\n            \\left(\\frac{x_2}{2!} \\right)^{j_2} \\dotsb\n            \\left(\\frac{x_{n-k+1}}{(n-k+1)!} \\right) ^{j_{n-k+1}}.</p>\n\n<ul>\n<li><code>bell(n)</code> gives the <code>n^{th}</code> Bell number, <code>B_n</code>.</li>\n<li><code>bell(n, x)</code> gives the <code>n^{th}</code> Bell polynomial, <code>B_n(x)</code>.</li>\n<li><code>bell(n, k, (x1, x2, ...))</code> gives Bell polynomials of the second kind,\n<code>B_{n,k}(x_1, x_2, \\dotsc, x_{n-k+1})</code>.</li>\n</ul>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>Not to be confused with Bernoulli numbers and Bernoulli polynomials,\nwhich use the same notation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">bell</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">symbols</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">bell</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">bell</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">846749014511809332450147</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">bell</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;t&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">t**4 + 6*t**3 + 7*t**2 + t</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">bell</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;x:6&#39;</span><span class=\"p\">)[</span><span class=\"mi\">1</span><span class=\"p\">:])</span>\n<span class=\"go\">6*x1*x5 + 15*x2*x4 + 10*x3**2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Bell_number\n.. [2] http://mathworld.wolfram.com/BellNumber.html\n.. [3] http://mathworld.wolfram.com/BellPolynomial.html</p>\n", "bases": "sympy.functions.combinatorial.numbers.bell, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.bell.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bell.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.bell.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bell.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.euler", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "euler", "type": "class", "doc": "<p>Euler numbers / Euler polynomials</p>\n\n<p>The Euler numbers are given by:</p>\n\n<p>.. math:: E_{2n} = I \\sum_{k=1}^{2n+1} \\sum_{j=0}^k \\binom{k}{j}\n    \\frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k}</p>\n\n<p>.. math:: E_{2n+1} = 0</p>\n\n<p>Euler numbers and Euler polynomials are related by</p>\n\n<p>.. math:: E_n = 2^n E_n\\left(\\frac{1}{2}\\right).</p>\n\n<p>We compute symbolic Euler polynomials using [5]_</p>\n\n<p>.. math:: E_n(x) = \\sum_{k=0}^n \\binom{n}{k} \\frac{E_k}{2^k}\n                   \\left(x - \\frac{1}{2}\\right)^{n-k}.</p>\n\n<p>However, numerical evaluation of the Euler polynomial is computed\nmore efficiently (and more accurately) using the mpmath library.</p>\n\n<ul>\n<li><code>euler(n)</code> gives the <code>n^{th}</code> Euler number, <code>E_n</code>.</li>\n<li><code>euler(n, x)</code> gives the <code>n^{th}</code> Euler polynomial, <code>E_n(x)</code>.</li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">euler</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">euler(3*n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;x&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">euler(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x - 1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**2 - x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**3 - 3*x**2/2 + 1/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**4 - 2*x**3 + x</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">12</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">Half</span><span class=\"p\">)</span>\n<span class=\"go\">2702765/4096</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">12</span><span class=\"p\">)</span>\n<span class=\"go\">2702765</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, fibonacci, harmonic, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Euler_numbers\n.. [2] http://mathworld.wolfram.com/EulerNumber.html\n.. [3] https://en.wikipedia.org/wiki/Alternating_permutation\n.. [4] http://mathworld.wolfram.com/AlternatingPermutation.html\n.. [5] http://dlmf.nist.gov/24.2#ii</p>\n", "bases": "sympy.functions.combinatorial.numbers.euler, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.euler.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "euler.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.euler.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "euler.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.catalan", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "catalan", "type": "class", "doc": "<p>Catalan numbers</p>\n\n<p>The <code>n^{th}</code> catalan number is given by:</p>\n\n<p>.. math :: C_n = \\frac{1}{n+1} \\binom{2n}{n}</p>\n\n<ul>\n<li><code>catalan(n)</code> gives the <code>n^{th}</code> Catalan number, <code>C_n</code></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"p\">(</span><span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">,</span> <span class=\"n\">hyper</span><span class=\"p\">,</span>\n<span class=\"gp\">... </span>    <span class=\"n\">catalan</span><span class=\"p\">,</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">combsimp</span><span class=\"p\">,</span> <span class=\"n\">Rational</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"mi\">10</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, 2, 5, 14, 42, 132, 429, 1430, 4862]</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">catalan(n)</span>\n</code></pre></div>\n\n<p>Catalan numbers can be transformed into several other, identical\nexpressions involving other mathematical functions</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">binomial</span><span class=\"p\">)</span>\n<span class=\"go\">binomial(2*n, n)/(n + 1)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">)</span>\n<span class=\"go\">4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">hyper((1 - n, -n), (2,), 1)</span>\n</code></pre></div>\n\n<p>For some non-integer values of n we can get closed form\nexpressions by rewriting in terms of gamma functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">)</span>\n<span class=\"go\">8/(3*pi)</span>\n</code></pre></div>\n\n<p>We can differentiate the Catalan numbers C(n) interpreted as a\ncontinuous real function in n:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">(polygamma(0, n + 1/2) - polygamma(0, n + 2) + log(4))*catalan(n)</span>\n</code></pre></div>\n\n<p>As a more advanced example consider the following ratio\nbetween consecutive numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">combsimp</span><span class=\"p\">((</span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">binomial</span><span class=\"p\">))</span>\n<span class=\"go\">2*(2*n + 1)/(n + 2)</span>\n</code></pre></div>\n\n<p>The Catalan numbers can be generalized to complex numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">)</span>\n<span class=\"go\">4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))</span>\n</code></pre></div>\n\n<p>and evaluated with arbitrary precision:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">0.39764993382373624267 - 0.020884341620842555705*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci\nsympy.functions.combinatorial.factorials.binomial</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Catalan_number\n.. [2] http://mathworld.wolfram.com/CatalanNumber.html\n.. [3] http://functions.wolfram.com/GammaBetaErf/CatalanNumber/\n.. [4] http://geometer.org/mathcircles/catalan.pdf</p>\n", "bases": "sympy.functions.combinatorial.numbers.catalan, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.catalan.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "catalan.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.catalan.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "catalan.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.genocchi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "genocchi", "type": "class", "doc": "<p>Genocchi numbers</p>\n\n<p>The Genocchi numbers are a sequence of integers <code>G_n</code> that satisfy the\nrelation:</p>\n\n<p>.. math:: \\frac{2t}{e^t + 1} = \\sum_{n=1}^\\infty \\frac{G_n t^n}{n!}</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">genocchi</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">genocchi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">9</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, -1, 0, 1, 0, -3, 0, 17]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">genocchi</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Genocchi_number\n.. [2] http://mathworld.wolfram.com/GenocchiNumber.html</p>\n", "bases": "sympy.functions.combinatorial.numbers.genocchi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.genocchi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "genocchi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.genocchi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "genocchi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.partition", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "partition", "type": "class", "doc": "<p>Partition numbers</p>\n\n<p>The Partition numbers are a sequence of integers <code>p_n</code> that represent the\nnumber of distinct ways of representing <code>n</code> as a sum of natural numbers\n(with order irrelevant). The generating function for <code>p_n</code> is given by:</p>\n\n<p>.. math:: \\sum_{n=0}^\\infty p_n x^n = \\prod_{k=1}^\\infty (1 - x^k)^{-1}</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">partition</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">partition</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">9</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, 1, 2, 3, 5, 7, 11, 15, 22]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">negative</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">partition</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Partition_(number_theory%29\n.. [2] https://en.wikipedia.org/wiki/Pentagonal_number_theorem</p>\n", "bases": "sympy.functions.combinatorial.numbers.partition, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.partition.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "partition.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.partition.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "partition.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Rem", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Rem", "type": "class", "doc": "<p>Returns the remainder when <code>p</code> is divided by <code>q</code> where <code>p</code> is finite\nand <code>q</code> is not equal to zero. The result, <code>p - int(p/q)*q</code>, has the same sign\nas the divisor.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>p : Expr\n    Dividend.</p>\n\n<p>q : Expr\n    Divisor.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p><code>Rem</code> corresponds to the <code>%</code> operator in C.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Rem</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Rem</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">Rem(x**3, y)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Rem</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"o\">-</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">3</span><span class=\"p\">})</span>\n<span class=\"go\">-2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Mod</p>\n", "bases": "sympy.functions.elementary.miscellaneous.Rem, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Rem.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Rem.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Rem.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Rem.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.re", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re", "type": "class", "doc": "<p>Returns real part of expression. This function performs only\nelementary analysis and so it will fail to decompose properly\nmore complicated expressions. If completely simplified result\nis needed then use <code>Basic.as_real_imag()</code> or perform complex\nexpansion on instance of this function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">re</span><span class=\"p\">,</span> <span class=\"n\">im</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"n\">symbols</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;x y&#39;</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">E</span><span class=\"p\">)</span>\n<span class=\"go\">2*E</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">17</span><span class=\"p\">)</span>\n<span class=\"go\">17</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"mi\">5</span> <span class=\"o\">+</span> <span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">7</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>expr : Expr\n    Real part of expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>im</p>\n", "bases": "sympy.functions.elementary.complexes.re, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'extended_real': True, 'commutative': True, 'imaginary': False}"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.im", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im", "type": "class", "doc": "<p>Returns imaginary part of expression. This function performs only\nelementary analysis and so it will fail to decompose properly more\ncomplicated expressions. If completely simplified result is needed then\nuse <code>Basic.as_real_imag()</code> or perform complex expansion on instance of\nthis function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">re</span><span class=\"p\">,</span> <span class=\"n\">im</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">E</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">17</span><span class=\"p\">)</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">re(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">im(y)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">3</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>expr : Expr\n    Imaginary part of expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>re</p>\n", "bases": "sympy.functions.elementary.complexes.im, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'extended_real': True, 'commutative': True, 'imaginary': False}"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign", "type": "class", "doc": "<p>Returns the complex sign of an expression:</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>If the expression is real the sign will be:</p>\n\n<pre><code>* $1$ if expression is positive\n* $0$ if expression is equal to zero\n* $-1$ if expression is negative\n</code></pre>\n\n<p>If the expression is imaginary the sign will be:</p>\n\n<pre><code>* $I$ if im(expression) is positive\n* $-I$ if im(expression) is negative\n</code></pre>\n\n<p>Otherwise an unevaluated expression will be returned. When evaluated, the\nresult (in general) will be <code>cos(arg(expr)) + I*sin(arg(expr))</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sign</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sign</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sign</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sign</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">-I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sign</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">sign(1 + I)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">_</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">()</span>\n<span class=\"go\">0.707106781186548 + 0.707106781186548*I</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or imaginary expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>expr : Expr\n    Complex sign of expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Abs, conjugate</p>\n", "bases": "sympy.functions.elementary.complexes.sign, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'complex': True, 'finite': True, 'commutative': True, 'infinite': False}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs", "type": "class", "doc": "<p>Return the absolute value of the argument.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This is an extension of the built-in function <code>abs()</code> to accept symbolic\nvalues.  If you pass a SymPy expression to the built-in <code>abs()</code>, it will\npass it automatically to <code>Abs()</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Abs</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Abs(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">x**2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">abs</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"c1\"># The Python built-in</span>\n<span class=\"go\">Abs(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(9*x**2 + 4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">8</span>\n</code></pre></div>\n\n<p>Note that the Python built-in will return either an Expr or int depending on\nthe argument::</p>\n\n<pre><code>&gt;&gt;&gt; type(abs(-1))\n&lt;... 'int'&gt;\n&gt;&gt;&gt; type(abs(S.NegativeOne))\n&lt;class 'sympy.core.numbers.One'&gt;\n</code></pre>\n\n<p>Abs will always return a SymPy object.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>expr : Expr\n    Absolute value returned can be an expression or integer depending on\n    input arg.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sign, conjugate</p>\n", "bases": "sympy.functions.elementary.complexes.Abs, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_extended_negative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_extended_negative", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_extended_nonnegative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_extended_nonnegative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'extended_real': True, 'commutative': True, 'imaginary': False, 'extended_nonnegative': True, 'extended_negative': False, 'negative': False}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_negative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_negative", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.conjugate", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "conjugate", "type": "class", "doc": "<p>Returns the <em>complex conjugate</em> [1]_ of an argument.\nIn mathematics, the complex conjugate of a complex number\nis given by changing the sign of the imaginary part.</p>\n\n<p>Thus, the conjugate of the complex number\n:math:<code>a + ib</code> (where $a$ and $b$ are real numbers) is :math:<code>a - ib</code></p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">-I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">3 - 2*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"mi\">5</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">5 + I</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>arg : Expr\n    Complex conjugate of arg as real, imaginary or mixed expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sign, Abs</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Complex_conjugation</p>\n", "bases": "sympy.functions.elementary.complexes.conjugate, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.conjugate.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "conjugate.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.conjugate.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "conjugate.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg", "type": "class", "doc": "<p>Returns the argument (in radians) of a complex number. The argument is\nevaluated in consistent convention with <code>atan2</code> where the branch-cut is\ntaken along the negative real axis and <code>arg(z)</code> is in the interval\n$(-\\pi,\\pi]$. For a positive number, the argument is always 0; the\nargument of a negative number is $\\pi$; and the argument of 0\nis undefined and returns <code>nan</code>. So the <code>arg</code> function will never nest\ngreater than 3 levels since at the 4th application, the result must be\nnan; for a real number, nan is returned on the 3rd application.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">arg</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">Dummy</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"mf\">2.0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">pi/6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"mi\">4</span> <span class=\"o\">+</span> <span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">atan(3/4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"mf\">0.8</span> <span class=\"o\">+</span> <span class=\"mf\">0.6</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">0.643501108793284</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))))</span>\n<span class=\"go\">nan</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">real</span> <span class=\"o\">=</span> <span class=\"n\">Dummy</span><span class=\"p\">(</span><span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">real</span><span class=\"p\">)))</span>\n<span class=\"go\">nan</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>value : Expr\n    Returns arc tangent of arg measured in radians.</p>\n", "bases": "sympy.functions.elementary.complexes.arg, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'extended_real': True, 'commutative': True, 'imaginary': False, 'real': True, 'hermitian': True, 'complex': True, 'infinite': False, 'finite': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.polar_lift", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polar_lift", "type": "class", "doc": "<p>Lift argument to the Riemann surface of the logarithm, using the\nstandard branch.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">polar_lift</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;p&#39;</span><span class=\"p\">,</span> <span class=\"n\">polar</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">4*exp_polar(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">4*exp_polar(I*pi)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">exp_polar(-I*pi/2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polar_lift(2 + I)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">4*polar_lift(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">p</span><span class=\"p\">)</span>\n<span class=\"go\">4*p</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.exponential.exp_polar\nperiodic_argument</p>\n", "bases": "sympy.functions.elementary.complexes.polar_lift, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.polar_lift.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polar_lift.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.polar_lift.is_polar", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polar_lift.is_polar", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.polar_lift.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polar_lift.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'polar': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.periodic_argument", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "periodic_argument", "type": "class", "doc": "<p>Represent the argument on a quotient of the Riemann surface of the\nlogarithm. That is, given a period $P$, always return a value in\n$(-P/2, P/2]$, by using $\\exp(PI) = 1$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">periodic_argument</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">periodic_argument</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">-pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>ar : Expr\n    A polar number.</p>\n\n<p>period : Expr\n    The period $P$.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.exponential.exp_polar\npolar_lift : Lift argument to the Riemann surface of the logarithm\nprincipal_branch</p>\n", "bases": "sympy.functions.elementary.complexes.periodic_argument, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.periodic_argument.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "periodic_argument.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.periodic_argument.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "periodic_argument.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.principal_branch", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "principal_branch", "type": "class", "doc": "<p>Represent a polar number reduced to its principal branch on a quotient\nof the Riemann surface of the logarithm.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This is a function of two arguments. The first argument is a polar\nnumber <code>z</code>, and the second one a positive real number or infinity, <code>p</code>.\nThe result is <code>z mod exp_polar(I*p)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">principal_branch</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">principal_branch</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">principal_branch</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">3*exp_polar(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">principal_branch</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">3*principal_branch(z, 2*pi)</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>x : Expr\n    A polar number.</p>\n\n<p>period : Expr\n    Positive real number or infinity.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.exponential.exp_polar\npolar_lift : Lift argument to the Riemann surface of the logarithm\nperiodic_argument</p>\n", "bases": "sympy.functions.elementary.complexes.principal_branch, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.principal_branch.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "principal_branch.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.principal_branch.is_polar", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "principal_branch.is_polar", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.principal_branch.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "principal_branch.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'polar': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.transpose", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "transpose", "type": "class", "doc": "<p>Linear map transposition.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">transpose</span><span class=\"p\">,</span> <span class=\"n\">Matrix</span><span class=\"p\">,</span> <span class=\"n\">MatrixSymbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">A</span> <span class=\"o\">=</span> <span class=\"n\">MatrixSymbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;A&#39;</span><span class=\"p\">,</span> <span class=\"mi\">25</span><span class=\"p\">,</span> <span class=\"mi\">9</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">transpose</span><span class=\"p\">(</span><span class=\"n\">A</span><span class=\"p\">)</span>\n<span class=\"go\">A.T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">B</span> <span class=\"o\">=</span> <span class=\"n\">MatrixSymbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;B&#39;</span><span class=\"p\">,</span> <span class=\"mi\">9</span><span class=\"p\">,</span> <span class=\"mi\">22</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">transpose</span><span class=\"p\">(</span><span class=\"n\">B</span><span class=\"p\">)</span>\n<span class=\"go\">B.T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">transpose</span><span class=\"p\">(</span><span class=\"n\">A</span><span class=\"o\">*</span><span class=\"n\">B</span><span class=\"p\">)</span>\n<span class=\"go\">B.T*A.T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">M</span> <span class=\"o\">=</span> <span class=\"n\">Matrix</span><span class=\"p\">([[</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">90</span><span class=\"p\">,</span> <span class=\"mi\">12</span><span class=\"p\">]])</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">M</span>\n<span class=\"go\">Matrix([</span>\n<span class=\"go\">[ 4,  5],</span>\n<span class=\"go\">[ 2,  1],</span>\n<span class=\"go\">[90, 12]])</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">transpose</span><span class=\"p\">(</span><span class=\"n\">M</span><span class=\"p\">)</span>\n<span class=\"go\">Matrix([</span>\n<span class=\"go\">[4, 2, 90],</span>\n<span class=\"go\">[5, 1, 12]])</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Matrix\n     Matrix or matrix expression to take the transpose of.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>value : Matrix\n    Transpose of arg.</p>\n", "bases": "sympy.functions.elementary.complexes.transpose, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.transpose.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "transpose.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.transpose.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "transpose.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.adjoint", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "adjoint", "type": "class", "doc": "<p>Conjugate transpose or Hermite conjugation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">adjoint</span><span class=\"p\">,</span> <span class=\"n\">MatrixSymbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">A</span> <span class=\"o\">=</span> <span class=\"n\">MatrixSymbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;A&#39;</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">adjoint</span><span class=\"p\">(</span><span class=\"n\">A</span><span class=\"p\">)</span>\n<span class=\"go\">Adjoint(A)</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Matrix\n    Matrix or matrix expression to take the adjoint of.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>value : Matrix\n    Represents the conjugate transpose or Hermite\n    conjugation of arg.</p>\n", "bases": "sympy.functions.elementary.complexes.adjoint, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.adjoint.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "adjoint.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.adjoint.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "adjoint.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sin", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sin", "type": "class", "doc": "<p>The sine function.</p>\n\n<p>Returns the sine of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function will evaluate automatically in the\ncase $x/\\pi$ is some rational number [4]_.  For example,\nif $x$ is a multiple of $\\pi$, $\\pi/2$, $\\pi/3$, $\\pi/4$, and $\\pi/6$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sin</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x*cos(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">6</span><span class=\"p\">)</span>\n<span class=\"go\">1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">12</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(2)/4 + sqrt(6)/4</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>csc, cos, sec, tan, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.14\n.. [3] http://functions.wolfram.com/ElementaryFunctions/Sin\n.. [4] http://mathworld.wolfram.com/TrigonometryAngles.html</p>\n", "bases": "sympy.functions.elementary.trigonometric.sin, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sin.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sin.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sin.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sin.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.cos", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cos", "type": "class", "doc": "<p>The cosine function.</p>\n\n<p>Returns the cosine of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See :func:<code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">cos</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*x*sin(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">-1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">12</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)/4 + sqrt(6)/4</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, sec, tan, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.14\n.. [3] http://functions.wolfram.com/ElementaryFunctions/Cos</p>\n", "bases": "sympy.functions.elementary.trigonometric.cos, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.cos.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cos.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.cos.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cos.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.tan", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tan", "type": "class", "doc": "<p>The tangent function.</p>\n\n<p>Returns the tangent of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See :class:<code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">tan</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tan</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x*(tan(x**2)**2 + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tan</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tan</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">8</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">()</span>\n<span class=\"go\">-1 + sqrt(2)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.14\n.. [3] http://functions.wolfram.com/ElementaryFunctions/Tan</p>\n", "bases": "sympy.functions.elementary.trigonometric.tan, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.tan.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tan.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.tan.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tan.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sec", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sec", "type": "class", "doc": "<p>The secant function.</p>\n\n<p>Returns the secant of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See :class:<code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sec</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sec</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x*tan(x**2)*sec(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sec</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, tan, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.14\n.. [3] http://functions.wolfram.com/ElementaryFunctions/Sec</p>\n", "bases": "sympy.functions.elementary.trigonometric.sec, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sec.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sec.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sec.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sec.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.csc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csc", "type": "class", "doc": "<p>The cosecant function.</p>\n\n<p>Returns the cosecant of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See :func:<code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">csc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">csc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*x*cot(x**2)*csc(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">csc</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, cos, sec, tan, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.14\n.. [3] http://functions.wolfram.com/ElementaryFunctions/Csc</p>\n", "bases": "sympy.functions.elementary.trigonometric.csc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.csc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.csc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.cot", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cot", "type": "class", "doc": "<p>The cotangent function.</p>\n\n<p>Returns the cotangent of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See :class:<code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">cot</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cot</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x*(-cot(x**2)**2 - 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cot</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cot</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">12</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(3) + 2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.14\n.. [3] http://functions.wolfram.com/ElementaryFunctions/Cot</p>\n", "bases": "sympy.functions.elementary.trigonometric.cot, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.cot.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cot.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.cot.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cot.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinc", "type": "class", "doc": "<p>Represents an unnormalized sinc function:</p>\n\n<p>.. math::</p>\n\n<pre><code>\\operatorname{sinc}(x) =\n\\begin{cases}\n  \\frac{\\sin x}{x} &amp; \\qquad x \\neq 0 \\\\\n  1 &amp; \\qquad x = 0\n\\end{cases}\n</code></pre>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sinc</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">jn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sinc(x)</span>\n</code></pre></div>\n\n<ul>\n<li>Automated Evaluation</li>\n</ul>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<ul>\n<li>Differentiation</li>\n</ul>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">()</span>\n<span class=\"go\">cos(x)/x - sin(x)/x**2</span>\n</code></pre></div>\n\n<ul>\n<li>Series Expansion</li>\n</ul>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">()</span>\n<span class=\"go\">1 - x**2/6 + x**4/120 + O(x**6)</span>\n</code></pre></div>\n\n<ul>\n<li>As zero'th order spherical Bessel Function</li>\n</ul>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">)</span>\n<span class=\"go\">jn(0, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See also</h1>\n\n<p>sin</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Sinc_function</p>\n", "bases": "sympy.functions.elementary.trigonometric.sinc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.asin", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asin", "type": "class", "doc": "<p>The inverse sine function.</p>\n\n<p>Returns the arcsine of x in radians.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>asin(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$ and for some instances when the\nresult is a rational multiple of $\\pi$ (see the <code>eval</code> class method).</p>\n\n<p>A purely imaginary argument will lead to an asinh expression.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">asin</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asin</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asin</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asin</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asin</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-oo*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nacsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.23\n.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSin</p>\n", "bases": "sympy.functions.elementary.trigonometric.asin, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.asin.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asin.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.asin.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asin.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acos", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acos", "type": "class", "doc": "<p>The inverse cosine function.</p>\n\n<p>Returns the arc cosine of x (measured in radians).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p><code>acos(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$ and for some instances when\nthe result is a rational multiple of $\\pi$ (see the eval class method).</p>\n\n<p><code>acos(zoo)</code> evaluates to <code>zoo</code>\n(see note in :class:<code>sympy.functions.elementary.trigonometric.asec</code>)</p>\n\n<p>A purely imaginary argument will be rewritten to asinh.</p>\n\n<h1 id=\"examples-2\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acos</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acos</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acos</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acos</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.23\n.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCos</p>\n", "bases": "sympy.functions.elementary.trigonometric.acos, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acos.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acos.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acos.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acos.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan", "type": "class", "doc": "<p>The inverse tangent function.</p>\n\n<p>Returns the arc tangent of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>atan(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$ and for some instances when the\nresult is a rational multiple of $\\pi$ (see the eval class method).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atan</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, acos, asec, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.23\n.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan</p>\n", "bases": "sympy.functions.elementary.trigonometric.atan, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.asec", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asec", "type": "class", "doc": "<p>The inverse secant function.</p>\n\n<p>Returns the arc secant of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>asec(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$ and for some instances when the\nresult is a rational multiple of $\\pi$ (see the eval class method).</p>\n\n<p><code>asec(x)</code> has branch cut in the interval $[-1, 1]$. For complex arguments,\nit can be defined [4]_ as</p>\n\n<p>.. math::\n    \\operatorname{sec^{-1}}(z) = -i\\frac{\\log\\left(\\sqrt{1 - z^2} + 1\\right)}{z}</p>\n\n<p>At <code>x = 0</code>, for positive branch cut, the limit evaluates to <code>zoo</code>. For\nnegative branch cut, the limit</p>\n\n<p>.. math::\n    \\lim_{z \\to 0}-i\\frac{\\log\\left(-\\sqrt{1 - z^2} + 1\\right)}{z}</p>\n\n<p>simplifies to :math:<code>-i\\log\\left(z/2 + O\\left(z^3\\right)\\right)</code> which\nultimately evaluates to <code>zoo</code>.</p>\n\n<p>As <code>acos(x) = asec(1/x)</code>, a similar argument can be given for\n<code>acos(x)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">asec</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asec</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asec</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asec</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asec</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, acos, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.23\n.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec\n.. [4] http://reference.wolfram.com/language/ref/ArcSec.html</p>\n", "bases": "sympy.functions.elementary.trigonometric.asec, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.asec.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asec.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.asec.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asec.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsc", "type": "class", "doc": "<p>The inverse cosecant function.</p>\n\n<p>Returns the arc cosecant of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>acsc(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$` and for some instances when the\nresult is a rational multiple of $\\pi$ (see the <code>eval</code> class method).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acsc</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions\n.. [2] http://dlmf.nist.gov/4.23\n.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsc</p>\n", "bases": "sympy.functions.elementary.trigonometric.acsc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acot", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acot", "type": "class", "doc": "<p>The inverse cotangent function.</p>\n\n<p>Returns the arc cotangent of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>acot(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, \\tilde{\\infty}, 0, 1, -1}$\nand for some instances when the result is a rational multiple of $\\pi$\n(see the eval class method).</p>\n\n<p>A purely imaginary argument will lead to an <code>acoth</code> expression.</p>\n\n<p><code>acot(x)</code> has a branch cut along $(-i, i)$, hence it is discontinuous\nat 0. Its range for real $x$ is $(-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acot</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acot</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acot</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acot</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-5*pi/12</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, acos, asec, atan, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://dlmf.nist.gov/4.23\n.. [2] http://functions.wolfram.com/ElementaryFunctions/ArcCot</p>\n", "bases": "sympy.functions.elementary.trigonometric.acot, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acot.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acot.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acot.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acot.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan2", "type": "class", "doc": "<p>The function <code>atan2(y, x)</code> computes <code>\\operatorname{atan}(y/x)</code> taking\ntwo arguments <code>y</code> and <code>x</code>.  Signs of both <code>y</code> and <code>x</code> are considered to\ndetermine the appropriate quadrant of <code>\\operatorname{atan}(y/x)</code>.\nThe range is <code>(-\\pi, \\pi]</code>. The complete definition reads as follows:</p>\n\n<p>.. math::</p>\n\n<pre><code>\\operatorname{atan2}(y, x) =\n\\begin{cases}\n  \\arctan\\left(\\frac y x\\right) &amp; \\qquad x &gt; 0 \\\\\n  \\arctan\\left(\\frac y x\\right) + \\pi&amp; \\qquad y \\ge 0, x &lt; 0 \\\\\n  \\arctan\\left(\\frac y x\\right) - \\pi&amp; \\qquad y &lt; 0, x &lt; 0 \\\\\n  +\\frac{\\pi}{2} &amp; \\qquad y &gt; 0, x = 0 \\\\\n  -\\frac{\\pi}{2} &amp; \\qquad y &lt; 0, x = 0 \\\\\n  \\text{undefined} &amp; \\qquad y = 0, x = 0\n\\end{cases}\n</code></pre>\n\n<p>Attention: Note the role reversal of both arguments. The <code>y</code>-coordinate\nis the first argument and the <code>x</code>-coordinate the second.</p>\n\n<p>If either <code>x</code> or <code>y</code> is complex:</p>\n\n<p>.. math::</p>\n\n<pre><code>\\operatorname{atan2}(y, x) =\n    -i\\log\\left(\\frac{x + iy}{\\sqrt{x^2 + y^2}}\\right)\n</code></pre>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Going counter-clock wise around the origin we find the\nfollowing angles:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atan2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">3*pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-3*pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/4</span>\n</code></pre></div>\n\n<p>which are all correct. Compare this to the results of the ordinary\n<code>\\operatorname{atan}</code> function for the point <code>(x, y) = (-1, 1)</code></p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atan</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">3*pi/4</span>\n</code></pre></div>\n\n<p>where only the <code>\\operatorname{atan2}</code> function reurns what we expect.\nWe can differentiate the function with respect to both arguments:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-y/(x**2 + y**2)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">x/(x**2 + y**2)</span>\n</code></pre></div>\n\n<p>We can express the <code>\\operatorname{atan2}</code> function in terms of\ncomplex logarithms:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">log</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">)</span>\n<span class=\"go\">-I*log((x + I*y)/sqrt(x**2 + y**2))</span>\n</code></pre></div>\n\n<p>and in terms of <code>\\operatorname(atan)</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atan</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">atan</span><span class=\"p\">)</span>\n<span class=\"go\">Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) &lt; 0), (0, Ne(x, 0)), (nan, True))</span>\n</code></pre></div>\n\n<p>but note that this form is undefined on the negative real axis.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, acos, asec, atan, acot</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions\n.. [2] https://en.wikipedia.org/wiki/Atan2\n.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan2</p>\n", "bases": "sympy.functions.elementary.trigonometric.atan2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp_polar", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp_polar", "type": "class", "doc": "<p>Represent a <em>polar number</em> (see g-function Sphinx documentation).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>exp_polar</code> represents the function\n<code>Exp: \\mathbb{C} \\rightarrow \\mathcal{S}</code>, sending the complex number\n<code>z = a + bi</code> to the polar number <code>r = exp(a), \\theta = b</code>. It is one of\nthe main functions to construct polar numbers.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">exp</span>\n</code></pre></div>\n\n<p>The main difference is that polar numbers don't \"wrap around\" at <code>2 \\pi</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">exp_polar(2*I*pi)</span>\n</code></pre></div>\n\n<p>apart from that they behave mostly like classical complex numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">exp_polar(5)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.simplify.powsimp.powsimp\npolar_lift\nperiodic_argument\nprincipal_branch</p>\n", "bases": "sympy.functions.elementary.exponential.exp_polar, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp_polar.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp_polar.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp_polar.is_polar", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp_polar.is_polar", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp_polar.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp_polar.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'polar': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp", "type": "class", "doc": "<p>The exponential function, :math:<code>e^x</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">exp(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">exp(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>log</p>\n", "bases": "sympy.functions.elementary.exponential.exp, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.ln", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ln", "type": "class", "doc": "<p>The natural logarithm function <code>\\ln(x)</code> or <code>\\log(x)</code>.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Logarithms are taken with the natural base, <code>e</code>. To get\na logarithm of a different base <code>b</code>, use <code>log(x, b)</code>,\nwhich is essentially short-hand for <code>log(x)/log(b)</code>.</p>\n\n<p><code>log</code> represents the principal branch of the natural\nlogarithm. As such it has a branch cut along the negative\nreal axis and returns values having a complex argument in\n<code>(-\\pi, \\pi]</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">log</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-log(3)/log(2) + 3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">log(2) + 2*I*pi/3</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>exp</p>\n", "bases": "sympy.functions.elementary.exponential.log, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.ln.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ln.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.ln.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ln.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.log", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "log", "type": "class", "doc": "<p>The natural logarithm function <code>\\ln(x)</code> or <code>\\log(x)</code>.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Logarithms are taken with the natural base, <code>e</code>. To get\na logarithm of a different base <code>b</code>, use <code>log(x, b)</code>,\nwhich is essentially short-hand for <code>log(x)/log(b)</code>.</p>\n\n<p><code>log</code> represents the principal branch of the natural\nlogarithm. As such it has a branch cut along the negative\nreal axis and returns values having a complex argument in\n<code>(-\\pi, \\pi]</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">log</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-log(3)/log(2) + 3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">log(2) + 2*I*pi/3</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>exp</p>\n", "bases": "sympy.functions.elementary.exponential.log, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.log.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "log.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.log.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "log.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.LambertW", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LambertW", "type": "class", "doc": "<p>The Lambert W function $W(z)$ is defined as the inverse\nfunction of $w \\exp(w)$ [1]_.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>In other words, the value of $W(z)$ is such that $z = W(z) \\exp(W(z))$\nfor any complex number $z$.  The Lambert W function is a multivalued\nfunction with infinitely many branches $W_k(z)$, indexed by\n$k \\in \\mathbb{Z}$.  Each branch gives a different solution $w$\nof the equation $z = w \\exp(w)$.</p>\n\n<p>The Lambert W function has two partially real branches: the\nprincipal branch ($k = 0$) is real for real $z &gt; -1/e$, and the\n$k = -1$ branch is real for $-1/e &lt; z &lt; 0$. All branches except\n$k = 0$ have a logarithmic singularity at $z = 0$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">LambertW</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LambertW</span><span class=\"p\">(</span><span class=\"mf\">1.2</span><span class=\"p\">)</span>\n<span class=\"go\">0.635564016364870</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LambertW</span><span class=\"p\">(</span><span class=\"mf\">1.2</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">()</span>\n<span class=\"go\">-1.34747534407696 - 4.41624341514535*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LambertW</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">is_real</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Lambert_W_function</p>\n", "bases": "sympy.functions.elementary.exponential.LambertW, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.LambertW.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LambertW.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.LambertW.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LambertW.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinh", "type": "class", "doc": "<p><code>sinh(x)</code> is the hyperbolic sine of <code>x</code>.</p>\n\n<p>The hyperbolic sine function is $\\frac{e^x - e^{-x}}{2}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sinh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sinh(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>cosh, tanh, asinh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.sinh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.cosh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cosh", "type": "class", "doc": "<p><code>cosh(x)</code> is the hyperbolic cosine of <code>x</code>.</p>\n\n<p>The hyperbolic cosine function is $\\frac{e^x + e^{-x}}{2}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">cosh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cosh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">cosh(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, tanh, acosh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.cosh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.cosh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cosh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.cosh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cosh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.tanh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tanh", "type": "class", "doc": "<p><code>tanh(x)</code> is the hyperbolic tangent of <code>x</code>.</p>\n\n<p>The hyperbolic tangent function is $\\frac{\\sinh(x)}{\\cosh(x)}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">tanh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tanh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">tanh(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, cosh, atanh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.tanh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.tanh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tanh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.tanh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tanh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.coth", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "coth", "type": "class", "doc": "<p><code>coth(x)</code> is the hyperbolic cotangent of <code>x</code>.</p>\n\n<p>The hyperbolic cotangent function is $\\frac{\\cosh(x)}{\\sinh(x)}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">coth</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">coth</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">coth(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, cosh, acoth</p>\n", "bases": "sympy.functions.elementary.hyperbolic.coth, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.coth.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "coth.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.coth.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "coth.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sech", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sech", "type": "class", "doc": "<p><code>sech(x)</code> is the hyperbolic secant of <code>x</code>.</p>\n\n<p>The hyperbolic secant function is $\\frac{2}{e^x + e^{-x}}$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sech</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sech</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sech(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, cosh, tanh, coth, csch, asinh, acosh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.sech, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sech.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sech.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sech.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sech.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.csch", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csch", "type": "class", "doc": "<p><code>csch(x)</code> is the hyperbolic cosecant of <code>x</code>.</p>\n\n<p>The hyperbolic cosecant function is $\\frac{2}{e^x - e^{-x}}$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">csch</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">csch</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">csch(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, cosh, tanh, sech, asinh, acosh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.csch, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.csch.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csch.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.csch.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csch.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.asinh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asinh", "type": "class", "doc": "<p><code>asinh(x)</code> is the inverse hyperbolic sine of <code>x</code>.</p>\n\n<p>The inverse hyperbolic sine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">asinh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asinh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1/sqrt(x**2 + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asinh</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">log(1 + sqrt(2))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>acosh, atanh, sinh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.asinh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.asinh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asinh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.asinh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asinh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acosh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acosh", "type": "class", "doc": "<p><code>acosh(x)</code> is the inverse hyperbolic cosine of <code>x</code>.</p>\n\n<p>The inverse hyperbolic cosine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acosh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acosh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1/sqrt(x**2 - 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acosh</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh, atanh, cosh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.acosh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acosh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acosh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acosh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acosh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.atanh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atanh", "type": "class", "doc": "<p><code>atanh(x)</code> is the inverse hyperbolic tangent of <code>x</code>.</p>\n\n<p>The inverse hyperbolic tangent function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atanh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atanh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1/(1 - x**2)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh, acosh, tanh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.atanh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.atanh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atanh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.atanh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atanh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acoth", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acoth", "type": "class", "doc": "<p><code>acoth(x)</code> is the inverse hyperbolic cotangent of <code>x</code>.</p>\n\n<p>The inverse hyperbolic cotangent function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acoth</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acoth</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1/(1 - x**2)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh, acosh, coth</p>\n", "bases": "sympy.functions.elementary.hyperbolic.acoth, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acoth.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acoth.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acoth.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acoth.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.asech", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asech", "type": "class", "doc": "<p><code>asech(x)</code> is the inverse hyperbolic secant of <code>x</code>.</p>\n\n<p>The inverse hyperbolic secant function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">asech</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-1/(x*sqrt(1 - x**2))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">I*pi/3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">3*I*pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">((</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)))</span>\n<span class=\"go\">I*pi/12</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh, atanh, cosh, acoth</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function\n.. [2] http://dlmf.nist.gov/4.37\n.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSech/</p>\n", "bases": "sympy.functions.elementary.hyperbolic.asech, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.asech.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asech.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.asech.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asech.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsch", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsch", "type": "class", "doc": "<p><code>acsch(x)</code> is the inverse hyperbolic cosecant of <code>x</code>.</p>\n\n<p>The inverse hyperbolic cosecant function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acsch</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-1/(x**2*sqrt(1 + x**(-2)))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">log(1 + sqrt(2))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">ImaginaryUnit</span><span class=\"p\">)</span>\n<span class=\"go\">-I*pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">ImaginaryUnit</span><span class=\"p\">)</span>\n<span class=\"go\">I*pi/6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">ImaginaryUnit</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)))</span>\n<span class=\"go\">-5*I*pi/12</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function\n.. [2] http://dlmf.nist.gov/4.37\n.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsch/</p>\n", "bases": "sympy.functions.elementary.hyperbolic.acsch, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsch.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsch.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsch.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsch.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.floor", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "floor", "type": "class", "doc": "<p>Floor is a univariate function which returns the largest integer\nvalue not greater than its argument. This implementation\ngeneralizes floor to complex numbers by taking the floor of the\nreal and imaginary parts separately.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">floor</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">Float</span><span class=\"p\">,</span> <span class=\"n\">Rational</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"mi\">17</span><span class=\"p\">)</span>\n<span class=\"go\">17</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">23</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">))</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">E</span><span class=\"p\">)</span>\n<span class=\"go\">5</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">Float</span><span class=\"p\">(</span><span class=\"mf\">0.567</span><span class=\"p\">))</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">2 + 2*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.integers.ceiling</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] \"Concrete mathematics\" by Graham, pp. 87\n.. [2] http://mathworld.wolfram.com/FloorFunction.html</p>\n", "bases": "sympy.functions.elementary.integers.floor, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.floor.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "floor.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.floor.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "floor.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.ceiling", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ceiling", "type": "class", "doc": "<p>Ceiling is a univariate function which returns the smallest integer\nvalue not less than its argument. This implementation\ngeneralizes ceiling to complex numbers by taking the ceiling of the\nreal and imaginary parts separately.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">ceiling</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">Float</span><span class=\"p\">,</span> <span class=\"n\">Rational</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"mi\">17</span><span class=\"p\">)</span>\n<span class=\"go\">17</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">23</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">))</span>\n<span class=\"go\">3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">E</span><span class=\"p\">)</span>\n<span class=\"go\">6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">Float</span><span class=\"p\">(</span><span class=\"mf\">0.567</span><span class=\"p\">))</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">3 + 3*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.integers.floor</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] \"Concrete mathematics\" by Graham, pp. 87\n.. [2] http://mathworld.wolfram.com/CeilingFunction.html</p>\n", "bases": "sympy.functions.elementary.integers.ceiling, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.ceiling.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ceiling.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.ceiling.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ceiling.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.frac", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "frac", "type": "class", "doc": "<p>Represents the fractional part of x</p>\n\n<p>For real numbers it is defined [1]_ as</p>\n\n<p>.. math::\n    x - \\left\\lfloor{x}\\right\\rfloor</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">frac</span><span class=\"p\">,</span> <span class=\"n\">Rational</span><span class=\"p\">,</span> <span class=\"n\">floor</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">1/3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">2/3</span>\n</code></pre></div>\n\n<p>returns zero for integer arguments</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>rewrite as floor</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">floor</span><span class=\"p\">)</span>\n<span class=\"go\">x - floor(x)</span>\n</code></pre></div>\n\n<p>for complex arguments</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">r</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;r&#39;</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;t&#39;</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"n\">t</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">r</span><span class=\"p\">)</span>\n<span class=\"go\">I*frac(r) + frac(t)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.integers.floor\nsympy.functions.elementary.integers.ceiling</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Fractional_part\n.. [2] http://mathworld.wolfram.com/FractionalPart.html</p>\n", "bases": "sympy.functions.elementary.integers.frac, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.frac.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "frac.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.frac.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "frac.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Piecewise", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Piecewise", "type": "class", "doc": "<p>Represents a piecewise function.</p>\n\n<p>Usage:</p>\n\n<p>Piecewise( (expr,cond), (expr,cond), ... )\n    - Each argument is a 2-tuple defining an expression and condition\n    - The conds are evaluated in turn returning the first that is True.\n      If any of the evaluated conds are not explicitly False,\n      e.g. <code>x &lt; 1</code>, the function is returned in symbolic form.\n    - If the function is evaluated at a place where all conditions are False,\n      nan will be returned.\n    - Pairs where the cond is explicitly False, will be removed and no pair\n      appearing after a True condition will ever be retained. If a single\n      pair with a True condition remains, it will be returned, even when\n      evaluation is False.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Piecewise</span><span class=\"p\">,</span> <span class=\"n\">log</span><span class=\"p\">,</span> <span class=\"n\">piecewise_fold</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span> <span class=\"o\">=</span> <span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span> <span class=\"o\">=</span> <span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span> <span class=\"o\">=</span> <span class=\"n\">Piecewise</span><span class=\"p\">((</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">&lt;</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"n\">f</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">&lt;=</span> <span class=\"mi\">1</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"n\">g</span><span class=\"p\">,</span> <span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">log(5)</span>\n</code></pre></div>\n\n<p>Booleans can contain Piecewise elements:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cond</span> <span class=\"o\">=</span> <span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">&lt;</span> <span class=\"n\">y</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">Piecewise</span><span class=\"p\">((</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">&lt;</span> <span class=\"mi\">0</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"kc\">True</span><span class=\"p\">)));</span> <span class=\"n\">cond</span>\n<span class=\"go\">Piecewise((2, x &lt; 0), (3, True)) &lt; y</span>\n</code></pre></div>\n\n<p>The folded version of this results in a Piecewise whose\nexpressions are Booleans:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">folded_cond</span> <span class=\"o\">=</span> <span class=\"n\">piecewise_fold</span><span class=\"p\">(</span><span class=\"n\">cond</span><span class=\"p\">);</span> <span class=\"n\">folded_cond</span>\n<span class=\"go\">Piecewise((2 &lt; y, x &lt; 0), (3 &lt; y, True))</span>\n</code></pre></div>\n\n<p>When a Boolean containing Piecewise (like cond) or a Piecewise\nwith Boolean expressions (like folded_cond) is used as a condition,\nit is converted to an equivalent :class:<code>~.ITE</code> object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Piecewise</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">folded_cond</span><span class=\"p\">))</span>\n<span class=\"go\">Piecewise((1, ITE(x &lt; 0, y &gt; 2, y &gt; 3)))</span>\n</code></pre></div>\n\n<p>When a condition is an <code>ITE</code>, it will be converted to a simplified\nBoolean expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">piecewise_fold</span><span class=\"p\">(</span><span class=\"n\">_</span><span class=\"p\">)</span>\n<span class=\"go\">Piecewise((1, ((x &gt;= 0) | (y &gt; 2)) &amp; ((y &gt; 3) | (x &lt; 0))))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>piecewise_fold, ITE</p>\n", "bases": "sympy.functions.elementary.piecewise.Piecewise, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Piecewise.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Piecewise.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Piecewise.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Piecewise.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf", "type": "class", "doc": "<p>The Gauss error function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erf}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2} \\mathrm{d}t.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">erf</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-oo*I</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1 and $I$ from the argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-erf(z)</span>\n</code></pre></div>\n\n<p>The error function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">erf(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">2*exp(-z**2)/sqrt(pi)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the error function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.999999984582742099719981147840</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">-1296959.73071763923152794095062*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erfc: Complementary error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Error_function\n.. [2] http://dlmf.nist.gov/7\n.. [3] http://mathworld.wolfram.com/Erf.html\n.. [4] http://functions.wolfram.com/GammaBetaErf/Erf</p>\n", "bases": "sympy.functions.special.error_functions.erf, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfc", "type": "class", "doc": "<p>Complementary Error Function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erfc}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_x^\\infty e^{-t^2} \\mathrm{d}t</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">erfc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-oo*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo*I</span>\n</code></pre></div>\n\n<p>The error function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">erfc(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-2*exp(-z**2)/sqrt(pi)</span>\n</code></pre></div>\n\n<p>It also follows</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">2 - erfc(z)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the complementary error function to arbitrary\nprecision on the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.0000000154172579002800188521596734869</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">1.0 - 1296959.73071763923152794095062*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Error_function\n.. [2] http://dlmf.nist.gov/7\n.. [3] http://mathworld.wolfram.com/Erfc.html\n.. [4] http://functions.wolfram.com/GammaBetaErf/Erfc</p>\n", "bases": "sympy.functions.special.error_functions.erfc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfi", "type": "class", "doc": "<p>Imaginary error function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The function erfi is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erfi}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{t^2} \\mathrm{d}t</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">erfi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-I</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1 and $I$ from the argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-erfi(z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">erfi(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">2*exp(z**2)/sqrt(pi)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the imaginary error function to arbitrary\nprecision on the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">18.5648024145755525987042919132</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">-0.995322265018952734162069256367*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Error_function\n.. [2] http://mathworld.wolfram.com/Erfi.html\n.. [3] http://functions.wolfram.com/GammaBetaErf/Erfi</p>\n", "bases": "sympy.functions.special.error_functions.erfi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2", "type": "class", "doc": "<p>Two-argument error function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erf2}(x, y) = \\frac{2}{\\sqrt{\\pi}} \\int_x^y e^{-t^2} \\mathrm{d}t</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">erf2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">1 - erf(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-erf(x) - 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">erf(y) - 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">erf(y) + 1</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">-erf2(x, y)</span>\n</code></pre></div>\n\n<p>The error function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">))</span>\n<span class=\"go\">erf2(conjugate(x), conjugate(y))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$, $y$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*exp(-x**2)/sqrt(pi)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">2*exp(-y**2)/sqrt(pi)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerfi: Imaginary error function.\nerfinv: Inverse error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://functions.wolfram.com/GammaBetaErf/Erf2/</p>\n", "bases": "sympy.functions.special.error_functions.erf2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfinv", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfinv", "type": "class", "doc": "<p>Inverse Error Function. The erfinv function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erf}(x) = y \\quad \\Rightarrow \\quad \\mathrm{erfinv}(y) = x</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">erfinv</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfinv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfinv</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erfinv</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)*exp(erfinv(x)**2)/2</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the inverse error function to arbitrary\nprecision on [-1, 1]:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfinv</span><span class=\"p\">(</span><span class=\"mf\">0.2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.179143454621291692285822705344</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions\n.. [2] http://functions.wolfram.com/GammaBetaErf/InverseErf/</p>\n", "bases": "sympy.functions.special.error_functions.erfinv, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfinv.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfinv.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfinv.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfinv.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfcinv", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfcinv", "type": "class", "doc": "<p>Inverse Complementary Error Function. The erfcinv function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erfc}(x) = y \\quad \\Rightarrow \\quad \\mathrm{erfcinv}(y) = x</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">erfcinv</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfcinv</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfcinv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erfcinv</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(pi)*exp(erfcinv(x)**2)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions\n.. [2] http://functions.wolfram.com/GammaBetaErf/InverseErfc/</p>\n", "bases": "sympy.functions.special.error_functions.erfcinv, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfcinv.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfcinv.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfcinv.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfcinv.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2inv", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2inv", "type": "class", "doc": "<p>Two-argument Inverse error function. The erf2inv function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erf2}(x, w) = y \\quad \\Rightarrow \\quad \\mathrm{erf2inv}(x, y) = w</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">erf2inv</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">erfinv(y)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">erfcinv(-y)</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ and $y$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">exp(-x**2 + erf2inv(x, y)**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)*exp(erf2inv(x, y)**2)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerfcinv: Inverse complementary error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://functions.wolfram.com/GammaBetaErf/InverseErf2/</p>\n", "bases": "sympy.functions.special.error_functions.erf2inv, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2inv.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2inv.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2inv.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2inv.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ei", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ei", "type": "class", "doc": "<p>The classical exponential integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For use in SymPy, this function is defined as</p>\n\n<p>.. math:: \\operatorname{Ei}(x) = \\sum_{n=1}^\\infty \\frac{x^n}{n\\, n!}\n                                 + \\log(x) + \\gamma,</p>\n\n<p>where $\\gamma$ is the Euler-Mascheroni constant.</p>\n\n<p>If $x$ is a polar number, this defines an analytic function on the\nRiemann surface of the logarithm. Otherwise this defines an analytic\nfunction in the cut plane $\\mathbb{C} \\setminus (-\\infty, 0]$.</p>\n\n<p><strong>Background</strong></p>\n\n<p>The name exponential integral comes from the following statement:</p>\n\n<p>.. math:: \\operatorname{Ei}(x) = \\int_{-\\infty}^x \\frac{e^t}{t} \\mathrm{d}t</p>\n\n<p>If the integral is interpreted as a Cauchy principal value, this statement\nholds for $x &gt; 0$ and $\\operatorname{Ei}(x)$ as defined above.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ei</span><span class=\"p\">,</span> <span class=\"n\">polar_lift</span><span class=\"p\">,</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">Ei(-1)</span>\n</code></pre></div>\n\n<p>This yields a real value:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">(</span><span class=\"n\">chop</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">-0.219383934395520</span>\n</code></pre></div>\n\n<p>On the other hand the analytic continuation is not real:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">(</span><span class=\"n\">chop</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">-0.21938393439552 + 3.14159265358979*I</span>\n</code></pre></div>\n\n<p>The exponential integral has a logarithmic branch point at the origin:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Ei(x) + 2*I*pi</span>\n</code></pre></div>\n\n<p>Differentiation is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">exp(x)/x</span>\n</code></pre></div>\n\n<p>The exponential integral is related to many other special functions.\nFor example:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span><span class=\"p\">,</span> <span class=\"n\">Shi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">-expint(1, x*exp_polar(I*pi)) - I*pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Shi</span><span class=\"p\">)</span>\n<span class=\"go\">Chi(x) + Shi(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>expint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.\nSi: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.\nuppergamma: Upper incomplete gamma function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://dlmf.nist.gov/6.6\n.. [2] https://en.wikipedia.org/wiki/Exponential_integral\n.. [3] Abramowitz &amp; Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm</p>\n", "bases": "sympy.functions.special.error_functions.Ei, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ei.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ei.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ei.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ei.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.expint", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "expint", "type": "class", "doc": "<p>Generalized exponential integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>.. math:: \\operatorname{E}_\\nu(z) = z^{\\nu - 1} \\Gamma(1 - \\nu, z),</p>\n\n<p>where $\\Gamma(1 - \\nu, z)$ is the upper incomplete gamma function\n(<code>uppergamma</code>).</p>\n\n<p>Hence for $z$ with positive real part we have</p>\n\n<p>.. math:: \\operatorname{E}_\\nu(z)\n          =   \\int_1^\\infty \\frac{e^{-zt}}{t^\\nu} \\mathrm{d}t,</p>\n\n<p>which explains the name.</p>\n\n<p>The representation as an incomplete gamma function provides an analytic\ncontinuation for $\\operatorname{E}_\\nu(z)$. If $\\nu$ is a\nnon-positive integer, the exponential integral is thus an unbranched\nfunction of $z$, otherwise there is a branch point at the origin.\nRefer to the incomplete gamma function documentation for details of the\nbranching behavior.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Differentiation is supported. Differentiation with respect to $z$ further\nexplains the name: for integral orders, the exponential integral is an\niterated integral of the exponential function.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-expint(nu - 1, z)</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $\\nu$ has no classical expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">)</span>\n<span class=\"go\">-z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z)</span>\n</code></pre></div>\n\n<p>At non-postive integer orders, the exponential integral reduces to the\nexponential function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">exp(-z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">exp(-z)/z + exp(-z)/z**2</span>\n</code></pre></div>\n\n<p>At half-integers it reduces to error functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)*erfc(sqrt(z))/sqrt(z)</span>\n</code></pre></div>\n\n<p>At positive integer orders it can be rewritten in terms of exponentials\nand <code>expint(1, z)</code>. Use <code>expand_func()</code> to do this:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24</span>\n</code></pre></div>\n\n<p>The generalised exponential integral is essentially equivalent to the\nincomplete gamma function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">uppergamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">uppergamma</span><span class=\"p\">)</span>\n<span class=\"go\">z**(nu - 1)*uppergamma(1 - nu, z)</span>\n</code></pre></div>\n\n<p>As such it is branched at the origin:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">))</span>\n<span class=\"go\">I*pi*z**3/3 + expint(4, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">))</span>\n<span class=\"go\">z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Ei: Another related function called exponential integral.\nE1: The classical case, returns expint(1, z).\nli: Logarithmic integral.\nLi: Offset logarithmic integral.\nSi: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.\nuppergamma</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://dlmf.nist.gov/8.19\n.. [2] http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/\n.. [3] https://en.wikipedia.org/wiki/Exponential_integral</p>\n", "bases": "sympy.functions.special.error_functions.expint, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.expint.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "expint.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.expint.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "expint.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.li", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "li", "type": "class", "doc": "<p>The classical logarithmic integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For use in SymPy, this function is defined as</p>\n\n<p>.. math:: \\operatorname{li}(x) = \\int_0^x \\frac{1}{\\log(t)} \\mathrm{d}t \\,.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">li</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">1/log(z)</span>\n</code></pre></div>\n\n<p>Defining the <code>li</code> function via an integral:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">integrate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">z*li(z) - Ei(2*log(z))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z*li(z) - Ei(2*log(z))</span>\n</code></pre></div>\n\n<p>The logarithmic integral can also be defined in terms of <code>Ei</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ei</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Ei</span><span class=\"p\">)</span>\n<span class=\"go\">Ei(log(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Ei</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">1/log(z)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the logarithmic integral to arbitrary precision\non the whole complex plane (except the singular points):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">1.04516378011749278484458888919</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">1.0652795784357498247001125598 + 3.08346052231061726610939702133*I</span>\n</code></pre></div>\n\n<p>We can even compute Soldner's constant by the help of mpmath:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">mpmath</span> <span class=\"kn\">import</span> <span class=\"n\">findroot</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">findroot</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">1.45136923488338</span>\n</code></pre></div>\n\n<p>Further transformations include rewriting <code>li</code> in terms of\nthe trigonometric integrals <code>Si</code>, <code>Ci</code>, <code>Shi</code> and <code>Chi</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Si</span><span class=\"p\">,</span> <span class=\"n\">Ci</span><span class=\"p\">,</span> <span class=\"n\">Shi</span><span class=\"p\">,</span> <span class=\"n\">Chi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Si</span><span class=\"p\">)</span>\n<span class=\"go\">-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Ci</span><span class=\"p\">)</span>\n<span class=\"go\">-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Shi</span><span class=\"p\">)</span>\n<span class=\"go\">-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Chi</span><span class=\"p\">)</span>\n<span class=\"go\">-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Li: Offset logarithmic integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nSi: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Logarithmic_integral\n.. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html\n.. [3] http://dlmf.nist.gov/6\n.. [4] http://mathworld.wolfram.com/SoldnersConstant.html</p>\n", "bases": "sympy.functions.special.error_functions.li, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.li.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "li.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.li.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "li.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Li", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Li", "type": "class", "doc": "<p>The offset logarithmic integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For use in SymPy, this function is defined as</p>\n\n<p>.. math:: \\operatorname{Li}(x) = \\operatorname{li}(x) - \\operatorname{li}(2)</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Li</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The following special value is known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">1/log(z)</span>\n</code></pre></div>\n\n<p>The shifted logarithmic integral can be written in terms of $li(z)$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">li</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">)</span>\n<span class=\"go\">li(z) - li(2)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the logarithmic integral to arbitrary precision\non the whole complex plane (except the singular points):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">1.92242131492155809316615998938</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>li: Logarithmic integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nSi: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Logarithmic_integral\n.. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html\n.. [3] http://dlmf.nist.gov/6</p>\n", "bases": "sympy.functions.special.error_functions.Li, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Li.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Li.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Li.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Li.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Si", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Si", "type": "class", "doc": "<p>Sine integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined by</p>\n\n<p>.. math:: \\operatorname{Si}(z) = \\int_0^z \\frac{\\sin{t}}{t} \\mathrm{d}t.</p>\n\n<p>It is an entire function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Si</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The sine integral is an antiderivative of $sin(z)/z$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sin(z)/z</span>\n</code></pre></div>\n\n<p>It is unbranched:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Si(z)</span>\n</code></pre></div>\n\n<p>Sine integral behaves much like ordinary sine under multiplication by <code>I</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">I*Shi(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-Si(z)</span>\n</code></pre></div>\n\n<p>It can also be expressed in terms of exponential integrals, but beware\nthat the latter is branched:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">-I*(-expint(1, z*exp_polar(-I*pi/2))/2 +</span>\n<span class=\"go\">     expint(1, z*exp_polar(I*pi/2))/2) + pi/2</span>\n</code></pre></div>\n\n<p>It can be rewritten in the form of sinc function (by definition):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sinc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">sinc</span><span class=\"p\">)</span>\n<span class=\"go\">Integral(sinc(t), (t, 0, z))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Ci: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nsinc: unnormalized sinc function\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral</p>\n", "bases": "sympy.functions.special.error_functions.Si, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Si.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Si.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Si.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Si.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ci", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ci", "type": "class", "doc": "<p>Cosine integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined for positive $x$ by</p>\n\n<p>.. math:: \\operatorname{Ci}(x) = \\gamma + \\log{x}\n                     + \\int_0^x \\frac{\\cos{t} - 1}{t} \\mathrm{d}t\n       = -\\int_x^\\infty \\frac{\\cos{t}}{t} \\mathrm{d}t,</p>\n\n<p>where $\\gamma$ is the Euler-Mascheroni constant.</p>\n\n<p>We have</p>\n\n<p>.. math:: \\operatorname{Ci}(z) =\n    -\\frac{\\operatorname{E}_1\\left(e^{i\\pi/2} z\\right)\n           + \\operatorname{E}_1\\left(e^{-i \\pi/2} z\\right)}{2}</p>\n\n<p>which holds for all polar $z$ and thus provides an analytic\ncontinuation to the Riemann surface of the logarithm.</p>\n\n<p>The formula also holds as stated\nfor $z \\in \\mathbb{C}$ with $\\Re(z) &gt; 0$.\nBy lifting to the principal branch, we obtain an analytic function on the\ncut complex plane.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ci</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The cosine integral is a primitive of $\\cos(z)/z$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">cos(z)/z</span>\n</code></pre></div>\n\n<p>It has a logarithmic branch point at the origin:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Ci(z) + 2*I*pi</span>\n</code></pre></div>\n\n<p>The cosine integral behaves somewhat like ordinary $\\cos$ under\nmultiplication by $i$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">polar_lift</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">Chi(z) + I*pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">Ci(z) + I*pi</span>\n</code></pre></div>\n\n<p>It can also be expressed in terms of exponential integrals:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">-expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Si: Sine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral</p>\n", "bases": "sympy.functions.special.error_functions.Ci, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ci.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ci.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ci.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ci.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Shi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Shi", "type": "class", "doc": "<p>Sinh integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined by</p>\n\n<p>.. math:: \\operatorname{Shi}(z) = \\int_0^z \\frac{\\sinh{t}}{t} \\mathrm{d}t.</p>\n\n<p>It is an entire function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Shi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The Sinh integral is a primitive of $\\sinh(z)/z$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sinh(z)/z</span>\n</code></pre></div>\n\n<p>It is unbranched:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Shi(z)</span>\n</code></pre></div>\n\n<p>The $\\sinh$ integral behaves much like ordinary $\\sinh$ under\nmultiplication by $i$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">I*Si(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-Shi(z)</span>\n</code></pre></div>\n\n<p>It can also be expressed in terms of exponential integrals, but beware\nthat the latter is branched:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Si: Sine integral.\nCi: Cosine integral.\nChi: Hyperbolic cosine integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral</p>\n", "bases": "sympy.functions.special.error_functions.Shi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Shi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Shi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Shi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Shi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Chi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Chi", "type": "class", "doc": "<p>Cosh integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined for positive $x$ by</p>\n\n<p>.. math:: \\operatorname{Chi}(x) = \\gamma + \\log{x}\n                     + \\int_0^x \\frac{\\cosh{t} - 1}{t} \\mathrm{d}t,</p>\n\n<p>where $\\gamma$ is the Euler-Mascheroni constant.</p>\n\n<p>We have</p>\n\n<p>.. math:: \\operatorname{Chi}(z) = \\operatorname{Ci}\\left(e^{i \\pi/2}z\\right)\n                     - i\\frac{\\pi}{2},</p>\n\n<p>which holds for all polar $z$ and thus provides an analytic\ncontinuation to the Riemann surface of the logarithm.\nBy lifting to the principal branch we obtain an analytic function on the\ncut complex plane.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Chi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The $\\cosh$ integral is a primitive of $\\cosh(z)/z$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">cosh(z)/z</span>\n</code></pre></div>\n\n<p>It has a logarithmic branch point at the origin:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Chi(z) + 2*I*pi</span>\n</code></pre></div>\n\n<p>The $\\cosh$ integral behaves somewhat like ordinary $\\cosh$ under\nmultiplication by $i$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">polar_lift</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">Ci(z) + I*pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">Chi(z) + I*pi</span>\n</code></pre></div>\n\n<p>It can also be expressed in terms of exponential integrals:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">-expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Si: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral</p>\n", "bases": "sympy.functions.special.error_functions.Chi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Chi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Chi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Chi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Chi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnels", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnels", "type": "class", "doc": "<p>Fresnel integral S.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined by</p>\n\n<p>.. math:: \\operatorname{S}(z) = \\int_0^z \\sin{\\frac{\\pi}{2} t^2} \\mathrm{d}t.</p>\n\n<p>It is an entire function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">fresnels</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-I/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">I/2</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1 and $i$ from the argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-fresnels(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-I*fresnels(z)</span>\n</code></pre></div>\n\n<p>The Fresnel S integral obeys the mirror symmetry\n$\\overline{S(z)} = S(\\bar{z})$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">fresnels(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sin(pi*z**2/2)</span>\n</code></pre></div>\n\n<p>Defining the Fresnel functions via an integral:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">integrate</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">sin</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">fresnels(z)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Fresnel integral to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.343415678363698242195300815958</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.343415678363698242195300815958*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>fresnelc: Fresnel cosine integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Fresnel_integral\n.. [2] http://dlmf.nist.gov/7\n.. [3] http://mathworld.wolfram.com/FresnelIntegrals.html\n.. [4] http://functions.wolfram.com/GammaBetaErf/FresnelS\n.. [5] The converging factors for the fresnel integrals\n        by John W. Wrench Jr. and Vicki Alley</p>\n", "bases": "sympy.functions.special.error_functions.fresnels, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnels.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnels.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnels.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnels.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnelc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnelc", "type": "class", "doc": "<p>Fresnel integral C.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined by</p>\n\n<p>.. math:: \\operatorname{C}(z) = \\int_0^z \\cos{\\frac{\\pi}{2} t^2} \\mathrm{d}t.</p>\n\n<p>It is an entire function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">fresnelc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">I/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-I/2</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1 and $i$ from the argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-fresnelc(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">I*fresnelc(z)</span>\n</code></pre></div>\n\n<p>The Fresnel C integral obeys the mirror symmetry\n$\\overline{C(z)} = C(\\bar{z})$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">fresnelc(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">cos(pi*z**2/2)</span>\n</code></pre></div>\n\n<p>Defining the Fresnel functions via an integral:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">integrate</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">cos</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">fresnelc(z)*gamma(1/4)/(4*gamma(5/4))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">fresnelc(z)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Fresnel integral to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.488253406075340754500223503357</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">-0.488253406075340754500223503357*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>fresnels: Fresnel sine integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Fresnel_integral\n.. [2] http://dlmf.nist.gov/7\n.. [3] http://mathworld.wolfram.com/FresnelIntegrals.html\n.. [4] http://functions.wolfram.com/GammaBetaErf/FresnelC\n.. [5] The converging factors for the fresnel integrals\n        by John W. Wrench Jr. and Vicki Alley</p>\n", "bases": "sympy.functions.special.error_functions.fresnelc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnelc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnelc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnelc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnelc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.gamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gamma", "type": "class", "doc": "<p>The gamma function</p>\n\n<p>.. math::\n    \\Gamma(x) := \\int^{\\infty}_{0} t^{x-1} e^{-t} \\mathrm{d}t.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The <code>gamma</code> function implements the function which passes through the\nvalues of the factorial function (i.e., $\\Gamma(n) = (n - 1)!$ when n is\nan integer). More generally, $\\Gamma(z)$ is defined in the whole complex\nplane except at the negative integers where there are simple poles.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">gamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)/2</span>\n</code></pre></div>\n\n<p>The <code>gamma</code> function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">gamma(conjugate(x))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">gamma(x)*polygamma(0, x)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the <code>gamma</code> function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">)</span>\n<span class=\"go\">2.288037795340032417959588909060233922890</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"o\">+</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">0.49801566811835604271 - 0.15494982830181068512*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>lowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Gamma_function\n.. [2] http://dlmf.nist.gov/5\n.. [3] http://mathworld.wolfram.com/GammaFunction.html\n.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma/</p>\n", "bases": "sympy.functions.special.gamma_functions.gamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.gamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.gamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.lowergamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lowergamma", "type": "class", "doc": "<p>The lower incomplete gamma function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>It can be defined as the meromorphic continuation of</p>\n\n<p>.. math::\n    \\gamma(s, x) := \\int_0^x t^{s-1} e^{-t} \\mathrm{d}t = \\Gamma(s) - \\Gamma(s, x).</p>\n\n<p>This can be shown to be the same as</p>\n\n<p>.. math::\n    \\gamma(s, x) = \\frac{x^s}{s} {}_1F_1\\left({s \\atop s+1} \\middle| -x\\right),</p>\n\n<p>where ${}_1F_1$ is the (confluent) hypergeometric function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lowergamma</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lowergamma</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">lowergamma(s, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lowergamma</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*(x**2/2 + x + 1)*exp(-x) + 2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lowergamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function\n.. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6,\n       Section 5, Handbook of Mathematical Functions with Formulas, Graphs,\n       and Mathematical Tables\n.. [3] http://dlmf.nist.gov/8\n.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma2/\n.. [5] http://functions.wolfram.com/GammaBetaErf/Gamma3/</p>\n", "bases": "sympy.functions.special.gamma_functions.lowergamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.lowergamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lowergamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.lowergamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lowergamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.uppergamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "uppergamma", "type": "class", "doc": "<p>The upper incomplete gamma function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>It can be defined as the meromorphic continuation of</p>\n\n<p>.. math::\n    \\Gamma(s, x) := \\int_x^\\infty t^{s-1} e^{-t} \\mathrm{d}t = \\Gamma(s) - \\gamma(s, x).</p>\n\n<p>where $\\gamma(s, x)$ is the lower incomplete gamma function,\n:class:<code>lowergamma</code>. This can be shown to be the same as</p>\n\n<p>.. math::\n    \\Gamma(s, x) = \\Gamma(s) - \\frac{x^s}{s} {}_1F_1\\left({s \\atop s+1} \\middle| -x\\right),</p>\n\n<p>where ${}_1F_1$ is the (confluent) hypergeometric function.</p>\n\n<p>The upper incomplete gamma function is also essentially equivalent to the\ngeneralized exponential integral:</p>\n\n<p>.. math::\n    \\operatorname{E}_{n}(x) = \\int_{1}^{\\infty}{\\frac{e^{-xt}}{t^n} \\, dt} = x^{n-1}\\Gamma(1-n,x).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">uppergamma</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uppergamma</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">uppergamma(s, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uppergamma</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*(x**2/2 + x + 1)*exp(-x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uppergamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uppergamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">expint(3, x)/x**2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function\n.. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6,\n       Section 5, Handbook of Mathematical Functions with Formulas, Graphs,\n       and Mathematical Tables\n.. [3] http://dlmf.nist.gov/8\n.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma2/\n.. [5] http://functions.wolfram.com/GammaBetaErf/Gamma3/\n.. [6] https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions</p>\n", "bases": "sympy.functions.special.gamma_functions.uppergamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.uppergamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "uppergamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.uppergamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "uppergamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.polygamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polygamma", "type": "class", "doc": "<p>The function <code>polygamma(n, z)</code> returns <code>log(gamma(z)).diff(n + 1)</code>.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>It is a meromorphic function on $\\mathbb{C}$ and defined as the $(n+1)$-th\nderivative of the logarithm of the gamma function:</p>\n\n<p>.. math::\n    \\psi^{(n)} (z) := \\frac{\\mathrm{d}^{n+1}}{\\mathrm{d} z^{n+1}} \\log\\Gamma(z).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">polygamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">-2*log(2) - EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">-log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">))</span>\n<span class=\"go\">-pi/2 - log(4) - log(2) - EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">1 - EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">23</span><span class=\"p\">)</span>\n<span class=\"go\">19093197/5173168 - EulerGamma</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;x&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(1, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(2, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(3, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(2, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(3, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(3, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(4, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(n + 1, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(n + 2, x)</span>\n</code></pre></div>\n\n<p>We can rewrite <code>polygamma</code> functions in terms of harmonic numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">harmonic</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">)</span>\n<span class=\"go\">harmonic(x - 1) - EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">)</span>\n<span class=\"go\">2*harmonic(x - 1, 3) - 2*zeta(3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ni</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"n\">ni</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Polygamma_function\n.. [2] http://mathworld.wolfram.com/PolygammaFunction.html\n.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma/\n.. [4] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/</p>\n", "bases": "sympy.functions.special.gamma_functions.polygamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.polygamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polygamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.polygamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polygamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.loggamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "loggamma", "type": "class", "doc": "<p>The <code>loggamma</code> function implements the logarithm of the\ngamma function (i.e., $\\log\\Gamma(x)$).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Several special values are known. For numerical integral\narguments we have:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">loggamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">log(2)</span>\n</code></pre></div>\n\n<p>And for symbolic values:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">log(gamma(n))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>For half-integral values:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">log(3*sqrt(pi)/4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">log(2**(1 - n)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2))</span>\n</code></pre></div>\n\n<p>And general rational arguments:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span> <span class=\"o\">=</span> <span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">16</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">L</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">()</span>\n<span class=\"go\">-5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span> <span class=\"o\">=</span> <span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">19</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">L</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">()</span>\n<span class=\"go\">-4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span> <span class=\"o\">=</span> <span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">23</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">7</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">L</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">()</span>\n<span class=\"go\">-3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16)</span>\n</code></pre></div>\n\n<p>The <code>loggamma</code> function has the following limits towards infinity:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n</code></pre></div>\n\n<p>The <code>loggamma</code> function obeys the mirror symmetry\nif $x \\in \\mathbb{C} \\setminus {-\\infty, 0}$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">loggamma(conjugate(x))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(0, x)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">cancel</span><span class=\"p\">()</span>\n<span class=\"go\">-log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the <code>gamma</code> function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">3.17805383034794561964694160130</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">-0.65092319930185633889 - 1.8724366472624298171*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Gamma_function\n.. [2] http://dlmf.nist.gov/5\n.. [3] http://mathworld.wolfram.com/LogGammaFunction.html\n.. [4] http://functions.wolfram.com/GammaBetaErf/LogGamma/</p>\n", "bases": "sympy.functions.special.gamma_functions.loggamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.loggamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "loggamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.loggamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "loggamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.digamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "digamma", "type": "class", "doc": "<p>The <code>digamma</code> function is the first derivative of the <code>loggamma</code>\nfunction</p>\n\n<p>.. math::\n    \\psi(x) := \\frac{\\mathrm{d}}{\\mathrm{d} z} \\log\\Gamma(z)\n            = \\frac{\\Gamma'(z)}{\\Gamma(z) }.</p>\n\n<p>In this case, <code>digamma(z) = polygamma(0, z)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">digamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">digamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;z&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">digamma</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(0, z)</span>\n</code></pre></div>\n\n<p>To retain <code>digamma</code> as it is:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">digamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">evaluate</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">digamma(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">digamma</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">evaluate</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">digamma(z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Digamma_function\n.. [2] http://mathworld.wolfram.com/DigammaFunction.html\n.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/</p>\n", "bases": "sympy.functions.special.gamma_functions.digamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.digamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "digamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.digamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "digamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.trigamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "trigamma", "type": "class", "doc": "<p>The <code>trigamma</code> function is the second derivative of the <code>loggamma</code>\nfunction</p>\n\n<p>.. math::\n    \\psi^{(1)}(z) := \\frac{\\mathrm{d}^{2}}{\\mathrm{d} z^{2}} \\log\\Gamma(z).</p>\n\n<p>In this case, <code>trigamma(z) = polygamma(1, z)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">trigamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">trigamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;z&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">trigamma</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(1, z)</span>\n</code></pre></div>\n\n<p>To retain <code>trigamma</code> as it is:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">trigamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">evaluate</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">trigamma(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">trigamma</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">evaluate</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">trigamma(z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Trigamma_function\n.. [2] http://mathworld.wolfram.com/TrigammaFunction.html\n.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/</p>\n", "bases": "sympy.functions.special.gamma_functions.trigamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.trigamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "trigamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.trigamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "trigamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.multigamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "multigamma", "type": "class", "doc": "<p>The multivariate gamma function is a generalization of the gamma function</p>\n\n<p>.. math::\n    \\Gamma_p(z) = \\pi^{p(p-1)/4}\\prod_{k=1}^p \\Gamma[z + (1 - k)/2].</p>\n\n<p>In a special case, <code>multigamma(x, 1) = gamma(x)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">multigamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;p&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">p</span><span class=\"p\">)</span>\n<span class=\"go\">pi**(p*(p - 1)/4)*Product(gamma(-_k/2 + x + 1/2), (_k, 1, p))</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)/2</span>\n</code></pre></div>\n\n<p>Writing <code>multigamma</code> in terms of the <code>gamma</code> function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">gamma(x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)*gamma(x)*gamma(x - 1/2)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">pi**(3/2)*gamma(x)*gamma(x - 1)*gamma(x - 1/2)</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>p : order or dimension of the multivariate gamma function</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma, lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma,\nbeta</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Multivariate_gamma_function</p>\n", "bases": "sympy.functions.special.gamma_functions.multigamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.multigamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "multigamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.multigamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "multigamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.dirichlet_eta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "dirichlet_eta", "type": "class", "doc": "<p>Dirichlet eta function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For $\\operatorname{Re}(s) &gt; 0$, this function is defined as</p>\n\n<p>.. math:: \\eta(s) = \\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n^s}.</p>\n\n<p>It admits a unique analytic continuation to all of $\\mathbb{C}$.\nIt is an entire, unbranched function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>The Dirichlet eta function is closely related to the Riemann zeta function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">dirichlet_eta</span><span class=\"p\">,</span> <span class=\"n\">zeta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">dirichlet_eta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">zeta</span><span class=\"p\">)</span>\n<span class=\"go\">(1 - 2**(1 - s))*zeta(s)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>zeta</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Dirichlet_eta_function</p>\n", "bases": "sympy.functions.special.zeta_functions.dirichlet_eta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.dirichlet_eta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "dirichlet_eta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.dirichlet_eta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "dirichlet_eta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.zeta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "zeta", "type": "class", "doc": "<p>Hurwitz zeta function (or Riemann zeta function).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For $\\operatorname{Re}(a) &gt; 0$ and $\\operatorname{Re}(s) &gt; 1$, this\nfunction is defined as</p>\n\n<p>.. math:: \\zeta(s, a) = \\sum_{n=0}^\\infty \\frac{1}{(n + a)^s},</p>\n\n<p>where the standard choice of argument for $n + a$ is used. For fixed\n$a$ with $\\operatorname{Re}(a) &gt; 0$ the Hurwitz zeta function admits a\nmeromorphic continuation to all of $\\mathbb{C}$, it is an unbranched\nfunction with a simple pole at $s = 1$.</p>\n\n<p>Analytic continuation to other $a$ is possible under some circumstances,\nbut this is not typically done.</p>\n\n<p>The Hurwitz zeta function is a special case of the Lerch transcendent:</p>\n\n<p>.. math:: \\zeta(s, a) = \\Phi(1, s, a).</p>\n\n<p>This formula defines an analytic continuation for all possible values of\n$s$ and $a$ (also $\\operatorname{Re}(a) &lt; 0$), see the documentation of\n:class:<code>lerchphi</code> for a description of the branching behavior.</p>\n\n<p>If no value is passed for $a$, by this function assumes a default value\nof $a = 1$, yielding the Riemann zeta function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>For $a = 1$ the Hurwitz zeta function reduces to the famous Riemann\nzeta function:</p>\n\n<p>.. math:: \\zeta(s, 1) = \\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">zeta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">zeta(s)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n<span class=\"go\">zeta(s)</span>\n</code></pre></div>\n\n<p>The Riemann zeta function can also be expressed using the Dirichlet eta\nfunction:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">dirichlet_eta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">dirichlet_eta</span><span class=\"p\">)</span>\n<span class=\"go\">dirichlet_eta(s)/(1 - 2**(1 - s))</span>\n</code></pre></div>\n\n<p>The Riemann zeta function at positive even integer and negative odd integer\nvalues is related to the Bernoulli numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">pi**2/6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">pi**4/90</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-1/12</span>\n</code></pre></div>\n\n<p>The specific formulae are:</p>\n\n<p>.. math:: \\zeta(2n) = (-1)^{n+1} \\frac{B_{2n} (2\\pi)^{2n}}{2(2n)!}\n.. math:: \\zeta(-n) = -\\frac{B_{n+1}}{n+1}</p>\n\n<p>At negative even integers the Riemann zeta function is zero:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>No closed-form expressions are known at positive odd integers, but\nnumerical evaluation is possible:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">()</span>\n<span class=\"go\">1.20205690315959</span>\n</code></pre></div>\n\n<p>The derivative of $\\zeta(s, a)$ with respect to $a$ can be computed:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">-s*zeta(s + 1, a)</span>\n</code></pre></div>\n\n<p>However the derivative with respect to $s$ has no useful closed form\nexpression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n<span class=\"go\">Derivative(zeta(s, a), s)</span>\n</code></pre></div>\n\n<p>The Hurwitz zeta function can be expressed in terms of the Lerch\ntranscendent, :class:<code>~.lerchphi</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lerchphi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">)</span>\n<span class=\"go\">lerchphi(1, s, a)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>dirichlet_eta, lerchphi, polylog</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://dlmf.nist.gov/25.11\n.. [2] https://en.wikipedia.org/wiki/Hurwitz_zeta_function</p>\n", "bases": "sympy.functions.special.zeta_functions.zeta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.zeta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "zeta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.zeta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "zeta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.lerchphi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lerchphi", "type": "class", "doc": "<p>Lerch transcendent (Lerch phi function).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For $\\operatorname{Re}(a) &gt; 0$, $|z| &lt; 1$ and $s \\in \\mathbb{C}$, the\nLerch transcendent is defined as</p>\n\n<p>.. math :: \\Phi(z, s, a) = \\sum_{n=0}^\\infty \\frac{z^n}{(n + a)^s},</p>\n\n<p>where the standard branch of the argument is used for $n + a$,\nand by analytic continuation for other values of the parameters.</p>\n\n<p>A commonly used related function is the Lerch zeta function, defined by</p>\n\n<p>.. math:: L(q, s, a) = \\Phi(e^{2\\pi i q}, s, a).</p>\n\n<p><strong>Analytic Continuation and Branching Behavior</strong></p>\n\n<p>It can be shown that</p>\n\n<p>.. math:: \\Phi(z, s, a) = z\\Phi(z, s, a+1) + a^{-s}.</p>\n\n<p>This provides the analytic continuation to $\\operatorname{Re}(a) \\le 0$.</p>\n\n<p>Assume now $\\operatorname{Re}(a) &gt; 0$. The integral representation</p>\n\n<p>.. math:: \\Phi_0(z, s, a) = \\int_0^\\infty \\frac{t^{s-1} e^{-at}}{1 - ze^{-t}}\n                            \\frac{\\mathrm{d}t}{\\Gamma(s)}</p>\n\n<p>provides an analytic continuation to $\\mathbb{C} - [1, \\infty)$.\nFinally, for $x \\in (1, \\infty)$ we find</p>\n\n<p>.. math:: \\lim_{\\epsilon \\to 0^+} \\Phi_0(x + i\\epsilon, s, a)\n         -\\lim_{\\epsilon \\to 0^+} \\Phi_0(x - i\\epsilon, s, a)\n         = \\frac{2\\pi i \\log^{s-1}{x}}{x^a \\Gamma(s)},</p>\n\n<p>using the standard branch for both $\\log{x}$ and\n$\\log{\\log{x}}$ (a branch of $\\log{\\log{x}}$ is needed to\nevaluate $\\log{x}^{s-1}$).\nThis concludes the analytic continuation. The Lerch transcendent is thus\nbranched at $z \\in {0, 1, \\infty}$ and\n$a \\in \\mathbb{Z}_{\\le 0}$. For fixed $z, a$ outside these\nbranch points, it is an entire function of $s$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>The Lerch transcendent is a fairly general function, for this reason it does\nnot automatically evaluate to simpler functions. Use <code>expand_func()</code> to\nachieve this.</p>\n\n<p>If $z=1$, the Lerch transcendent reduces to the Hurwitz zeta function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lerchphi</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">))</span>\n<span class=\"go\">zeta(s, a)</span>\n</code></pre></div>\n\n<p>More generally, if $z$ is a root of unity, the Lerch transcendent\nreduces to a sum of Hurwitz zeta functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">))</span>\n<span class=\"go\">zeta(s, a/2)/2**s - zeta(s, a/2 + 1/2)/2**s</span>\n</code></pre></div>\n\n<p>If $a=1$, the Lerch transcendent reduces to the polylogarithm:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">polylog(s, z)/z</span>\n</code></pre></div>\n\n<p>More generally, if $a$ is rational, the Lerch transcendent reduces\nto a sum of polylogarithms:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -</span>\n<span class=\"go\">            polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">-2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -</span>\n<span class=\"go\">                      polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z</span>\n</code></pre></div>\n\n<p>The derivatives with respect to $z$ and $a$ can be computed in\nclosed form:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">(-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">-s*lerchphi(z, s + 1, a)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>polylog, zeta</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions,\n       Vol. I, New York: McGraw-Hill. Section 1.11.\n.. [2] http://dlmf.nist.gov/25.14\n.. [3] https://en.wikipedia.org/wiki/Lerch_transcendent</p>\n", "bases": "sympy.functions.special.zeta_functions.lerchphi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.lerchphi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lerchphi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.lerchphi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lerchphi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.polylog", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polylog", "type": "class", "doc": "<p>Polylogarithm function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For $|z| &lt; 1$ and $s \\in \\mathbb{C}$, the polylogarithm is\ndefined by</p>\n\n<p>.. math:: \\operatorname{Li}_s(z) = \\sum_{n=1}^\\infty \\frac{z^n}{n^s},</p>\n\n<p>where the standard branch of the argument is used for $n$. It admits\nan analytic continuation which is branched at $z=1$ (notably not on the\nsheet of initial definition), $z=0$ and $z=\\infty$.</p>\n\n<p>The name polylogarithm comes from the fact that for $s=1$, the\npolylogarithm is related to the ordinary logarithm (see examples), and that</p>\n\n<p>.. math:: \\operatorname{Li}_{s+1}(z) =\n                \\int_0^z \\frac{\\operatorname{Li}_s(t)}{t} \\mathrm{d}t.</p>\n\n<p>The polylogarithm is a special case of the Lerch transcendent:</p>\n\n<p>.. math:: \\operatorname{Li}_{s}(z) = z \\Phi(z, s, 1).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>For $z \\in {0, 1, -1}$, the polylogarithm is automatically expressed\nusing other functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">polylog</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">zeta(s)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-dirichlet_eta(s)</span>\n</code></pre></div>\n\n<p>If $s$ is a negative integer, $0$ or $1$, the polylogarithm can be\nexpressed using elementary functions. This can be done using\n<code>expand_func()</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">-log(1 - z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">z/(1 - z)</span>\n</code></pre></div>\n\n<p>The derivative with respect to $z$ can be computed in closed form:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">polylog(s - 1, z)/z</span>\n</code></pre></div>\n\n<p>The polylogarithm can be expressed in terms of the lerch transcendent:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lerchphi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">)</span>\n<span class=\"go\">z*lerchphi(z, s, 1)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>zeta, lerchphi</p>\n", "bases": "sympy.functions.special.zeta_functions.polylog, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.polylog.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polylog.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.polylog.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polylog.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.stieltjes", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "stieltjes", "type": "class", "doc": "<p>Represents Stieltjes constants, $\\gamma_{k}$ that occur in\nLaurent Series expansion of the Riemann zeta function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">stieltjes</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">stieltjes(n)</span>\n</code></pre></div>\n\n<p>The zero'th stieltjes constant:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">EulerGamma</span>\n</code></pre></div>\n\n<p>For generalized stieltjes constants:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">)</span>\n<span class=\"go\">stieltjes(n, m)</span>\n</code></pre></div>\n\n<p>Constants are only defined for integers &gt;= 0:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Stieltjes_constants</p>\n", "bases": "sympy.functions.special.zeta_functions.stieltjes, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.stieltjes.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "stieltjes.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.stieltjes.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "stieltjes.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita", "type": "class", "doc": "<p>Represent the Levi-Civita symbol.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For even permutations of indices it returns 1, for odd permutations -1, and\nfor everything else (a repeated index) it returns 0.</p>\n\n<p>Thus it represents an alternating pseudotensor.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">LeviCivita</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">,</span> <span class=\"n\">k</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">,</span> <span class=\"n\">k</span><span class=\"p\">)</span>\n<span class=\"go\">LeviCivita(i, j, k)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">,</span> <span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Eijk</p>\n", "bases": "sympy.functions.special.tensor_functions.LeviCivita, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_integer", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_integer", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'integer': True, 'hermitian': True, 'extended_real': True, 'commutative': True, 'complex': True, 'real': True, 'transcendental': False, 'infinite': False, 'imaginary': False, 'algebraic': True, 'finite': True, 'rational': True, 'noninteger': False, 'irrational': False}"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_transcendental", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_transcendental", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_algebraic", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_algebraic", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_rational", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_rational", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_noninteger", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_noninteger", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_irrational", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_irrational", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta", "type": "class", "doc": "<p>The discrete, or Kronecker, delta function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>A function that takes in two integers $i$ and $j$. It returns $0$ if $i$\nand $j$ are not equal, or it returns $1$ if $i$ and $j$ are equal.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>An example with integer indices:</p>\n\n<pre><code>&gt;&gt;&gt; from sympy import KroneckerDelta\n&gt;&gt;&gt; KroneckerDelta(1, 2)\n0\n&gt;&gt;&gt; KroneckerDelta(3, 3)\n1\n</code></pre>\n\n<p>Symbolic indices:</p>\n\n<pre><code>&gt;&gt;&gt; from sympy.abc import i, j, k\n&gt;&gt;&gt; KroneckerDelta(i, j)\nKroneckerDelta(i, j)\n&gt;&gt;&gt; KroneckerDelta(i, i)\n1\n&gt;&gt;&gt; KroneckerDelta(i, i + 1)\n0\n&gt;&gt;&gt; KroneckerDelta(i, i + 1 + k)\nKroneckerDelta(i, i + k + 1)\n</code></pre>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>i : Number, Symbol\n    The first index of the delta function.\nj : Number, Symbol\n    The second index of the delta function.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>eval\nDiracDelta</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Kronecker_delta</p>\n", "bases": "sympy.functions.special.tensor_functions.KroneckerDelta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_integer", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_integer", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'integer': True, 'hermitian': True, 'extended_real': True, 'commutative': True, 'complex': True, 'real': True, 'transcendental': False, 'infinite': False, 'imaginary': False, 'algebraic': True, 'finite': True, 'rational': True, 'noninteger': False, 'irrational': False}"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_transcendental", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_transcendental", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_algebraic", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_algebraic", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_rational", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_rational", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_noninteger", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_noninteger", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_irrational", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_irrational", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction", "type": "class", "doc": "<p>Singularity functions are a class of discontinuous functions.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Singularity functions take a variable, an offset, and an exponent as\narguments. These functions are represented using Macaulay brackets as:</p>\n\n<p>SingularityFunction(x, a, n) := <x - a>^n</p>\n\n<p>The singularity function will automatically evaluate to\n<code>Derivative(DiracDelta(x - a), x, -n - 1)</code> if <code>n &lt; 0</code>\nand <code>(x - a)**n*Heaviside(x - a)</code> if <code>n &gt;= 0</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">,</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">Piecewise</span><span class=\"p\">,</span> <span class=\"n\">DiracDelta</span><span class=\"p\">,</span> <span class=\"n\">Heaviside</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">SingularityFunction(x, a, n)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;y&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">nonnegative</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">(y + 10)**n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;y&#39;</span><span class=\"p\">,</span> <span class=\"n\">negative</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">243</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">SingularityFunction(x, 4, -2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Piecewise</span><span class=\"p\">)</span>\n<span class=\"go\">Piecewise(((x - 4)**5, x &gt; 4), (0, True))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span> <span class=\"o\">=</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;y&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">nonnegative</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">:</span> <span class=\"o\">-</span><span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">:</span> <span class=\"n\">n</span><span class=\"p\">})</span>\n<span class=\"go\">(y + 10)**n</span>\n</code></pre></div>\n\n<p>The methods <code>rewrite(DiracDelta)</code>, <code>rewrite(Heaviside)</code>, and\n<code>rewrite('HeavisideDiracDelta')</code> returns the same output. One can use any\nof these methods according to their choice.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span> <span class=\"o\">=</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Heaviside</span><span class=\"p\">)</span>\n<span class=\"go\">(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">DiracDelta</span><span class=\"p\">)</span>\n<span class=\"go\">(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"s1\">&#39;HeavisideDiracDelta&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>DiracDelta, Heaviside</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Singularity_function</p>\n", "bases": "sympy.functions.special.singularity_functions.SingularityFunction, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'real': True, 'hermitian': True, 'commutative': True, 'extended_real': True, 'complex': True, 'infinite': False, 'imaginary': False, 'finite': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta", "type": "class", "doc": "<p>The DiracDelta function and its derivatives.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>DiracDelta is not an ordinary function. It can be rigorously defined either\nas a distribution or as a measure.</p>\n\n<p>DiracDelta only makes sense in definite integrals, and in particular,\nintegrals of the form <code>Integral(f(x)*DiracDelta(x - x0), (x, a, b))</code>,\nwhere it equals <code>f(x0)</code> if <code>a &lt;= x0 &lt;= b</code> and <code>0</code> otherwise. Formally,\nDiracDelta acts in some ways like a function that is <code>0</code> everywhere except\nat <code>0</code>, but in many ways it also does not. It can often be useful to treat\nDiracDelta in formal ways, building up and manipulating expressions with\ndelta functions (which may eventually be integrated), but care must be taken\nto not treat it as a real function. SymPy's <code>oo</code> is similar. It only\ntruly makes sense formally in certain contexts (such as integration limits),\nbut SymPy allows its use everywhere, and it tries to be consistent with\noperations on it (like <code>1/oo</code>), but it is easy to get into trouble and get\nwrong results if <code>oo</code> is treated too much like a number. Similarly, if\nDiracDelta is treated too much like a function, it is easy to get wrong or\nnonsensical results.</p>\n\n<p>DiracDelta function has the following properties:</p>\n\n<p>1) $\\frac{d}{d x} \\theta(x) = \\delta(x)$\n2) $\\int_{-\\infty}^\\infty \\delta(x - a)f(x)\\, dx = f(a)$ and $\\int_{a-\n   \\epsilon}^{a+\\epsilon} \\delta(x - a)f(x)\\, dx = f(a)$\n3) $\\delta(x) = 0$ for all $x \\neq 0$\n4) $\\delta(g(x)) = \\sum_i \\frac{\\delta(x - x_i)}{\\|g'(x_i)\\|}$ where $x_i$\n   are the roots of $g$\n5) $\\delta(-x) = \\delta(x)$</p>\n\n<p>Derivatives of <code>k</code>-th order of DiracDelta have the following properties:</p>\n\n<p>6) $\\delta(x, k) = 0$ for all $x \\neq 0$\n7) $\\delta(-x, k) = -\\delta(x, k)$ for odd $k$\n8) $\\delta(-x, k) = \\delta(x, k)$ for even $k$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">DiracDelta</span><span class=\"p\">,</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">DiracDelta(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">DiracDelta(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">DiracDelta(x, 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">),</span><span class=\"n\">x</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">DiracDelta(x - 1, 2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">),</span><span class=\"n\">x</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">2*(2*x**2*DiracDelta(x**2 - 1, 2) + DiracDelta(x**2 - 1, 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">is_simple</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">is_simple</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">((</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">y</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">diracdelta</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">wrt</span><span class=\"o\">=</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">DiracDelta(x - 1)/(2*Abs(y)) + DiracDelta(x + 1)/(2*Abs(y))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Heaviside\nsympy.simplify.simplify.simplify, is_simple\nsympy.functions.special.tensor_functions.KroneckerDelta</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://mathworld.wolfram.com/DeltaFunction.html</p>\n", "bases": "sympy.functions.special.delta_functions.DiracDelta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'real': True, 'hermitian': True, 'commutative': True, 'extended_real': True, 'complex': True, 'infinite': False, 'imaginary': False, 'finite': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside", "type": "class", "doc": "<p>Heaviside step function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Heaviside step function has the following properties:</p>\n\n<p>1) $\\frac{d}{d x} \\theta(x) = \\delta(x)$\n2) $\\theta(x) = \\begin{cases} 0 &amp; \\text{for}\\: x &lt; 0 \\ \\frac{1}{2} &amp;\n   \\text{for}\\: x = 0 \\1 &amp; \\text{for}\\: x &gt; 0 \\end{cases}$\n3) $\\frac{d}{d x} \\max(x, 0) = \\theta(x)$</p>\n\n<p>Heaviside(x) is printed as $\\theta(x)$ with the SymPy LaTeX printer.</p>\n\n<p>The value at 0 is set differently in different fields. SymPy uses 1/2,\nwhich is a convention from electronics and signal processing, and is\nconsistent with solving improper integrals by Fourier transform and\nconvolution.</p>\n\n<p>To specify a different value of Heaviside at <code>x=0</code>, a second argument\ncan be given. Using <code>Heaviside(x, nan)</code> gives an expression that will\nevaluate to nan for x=0.</p>\n\n<p>.. versionchanged:: 1.9 <code>Heaviside(0)</code> now returns 1/2 (before: undefined)</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Heaviside</span><span class=\"p\">,</span> <span class=\"n\">nan</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"mi\">9</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">9</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">nan</span><span class=\"p\">)</span>\n<span class=\"go\">nan</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">replace</span><span class=\"p\">(</span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Heaviside(x, 1) + 1</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>DiracDelta</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://mathworld.wolfram.com/HeavisideStepFunction.html\n.. [2] http://dlmf.nist.gov/1.16#iv</p>\n", "bases": "sympy.functions.special.delta_functions.Heaviside, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'real': True, 'hermitian': True, 'commutative': True, 'extended_real': True, 'complex': True, 'infinite': False, 'imaginary': False, 'finite': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselj", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselj", "type": "class", "doc": "<p>Bessel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Bessel $J$ function of order $\\nu$ is defined to be the function\nsatisfying Bessel's differential equation</p>\n\n<p>.. math ::\n    z^2 \\frac{\\mathrm{d}^2 w}{\\mathrm{d}z^2}\n    + z \\frac{\\mathrm{d}w}{\\mathrm{d}z} + (z^2 - \\nu^2) w = 0,</p>\n\n<p>with Laurent expansion</p>\n\n<p>.. math ::\n    J_\\nu(z) = z^\\nu \\left(\\frac{1}{\\Gamma(\\nu + 1) 2^\\nu} + O(z^2) \\right),</p>\n\n<p>if $\\nu$ is not a negative integer. If $\\nu=-n \\in \\mathbb{Z}_{&lt;0}$\n<em>is</em> a negative integer, then the definition is</p>\n\n<p>.. math ::\n    J_{-n}(z) = (-1)^n J_n(z).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create a Bessel function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">besselj</span><span class=\"p\">,</span> <span class=\"n\">jn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span> <span class=\"o\">=</span> <span class=\"n\">besselj</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<p>Differentiate it:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">besselj(n - 1, z)/2 - besselj(n + 1, z)/2</span>\n</code></pre></div>\n\n<p>Rewrite in terms of spherical Bessel functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)</span>\n</code></pre></div>\n\n<p>Access the parameter and argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">order</span>\n<span class=\"go\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">argument</span>\n<span class=\"go\">z</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bessely, besseli, besselk</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), \"Chapter 9\",\n       Handbook of Mathematical Functions with Formulas, Graphs, and\n       Mathematical Tables\n.. [2] Luke, Y. L. (1969), The Special Functions and Their\n       Approximations, Volume 1\n.. [3] https://en.wikipedia.org/wiki/Bessel_function\n.. [4] http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/</p>\n", "bases": "sympy.functions.special.bessel.besselj, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselj.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselj.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselj.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselj.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.bessely", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bessely", "type": "class", "doc": "<p>Bessel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Bessel $Y$ function of order $\\nu$ is defined as</p>\n\n<p>.. math ::\n    Y_\\nu(z) = \\lim_{\\mu \\to \\nu} \\frac{J_\\mu(z) \\cos(\\pi \\mu)\n                                        - J_{-\\mu}(z)}{\\sin(\\pi \\mu)},</p>\n\n<p>where $J_\\mu(z)$ is the Bessel function of the first kind.</p>\n\n<p>It is a solution to Bessel's equation, and linearly independent from\n$J_\\nu$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">bessely</span><span class=\"p\">,</span> <span class=\"n\">yn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span> <span class=\"o\">=</span> <span class=\"n\">bessely</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">bessely(n - 1, z)/2 - bessely(n + 1, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, besseli, besselk</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/</p>\n", "bases": "sympy.functions.special.bessel.bessely, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.bessely.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bessely.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.bessely.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bessely.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.besseli", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besseli", "type": "class", "doc": "<p>Modified Bessel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Bessel $I$ function is a solution to the modified Bessel equation</p>\n\n<p>.. math ::\n    z^2 \\frac{\\mathrm{d}^2 w}{\\mathrm{d}z^2}\n    + z \\frac{\\mathrm{d}w}{\\mathrm{d}z} + (z^2 + \\nu^2)^2 w = 0.</p>\n\n<p>It can be defined as</p>\n\n<p>.. math ::\n    I_\\nu(z) = i^{-\\nu} J_\\nu(iz),</p>\n\n<p>where $J_\\nu(z)$ is the Bessel function of the first kind.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">besseli</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">besseli</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">besseli(n - 1, z)/2 + besseli(n + 1, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, bessely, besselk</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/</p>\n", "bases": "sympy.functions.special.bessel.besseli, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.besseli.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besseli.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.besseli.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besseli.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselk", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselk", "type": "class", "doc": "<p>Modified Bessel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Bessel $K$ function of order $\\nu$ is defined as</p>\n\n<p>.. math ::\n    K_\\nu(z) = \\lim_{\\mu \\to \\nu} \\frac{\\pi}{2}\n               \\frac{I_{-\\mu}(z) -I_\\mu(z)}{\\sin(\\pi \\mu)},</p>\n\n<p>where $I_\\mu(z)$ is the modified Bessel function of the first kind.</p>\n\n<p>It is a solution of the modified Bessel equation, and linearly independent\nfrom $Y_\\nu$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">besselk</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">besselk</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-besselk(n - 1, z)/2 - besselk(n + 1, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, besseli, bessely</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/</p>\n", "bases": "sympy.functions.special.bessel.besselk, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselk.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselk.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselk.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselk.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel1", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel1", "type": "class", "doc": "<p>Hankel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>.. math ::\n    H_\\nu^{(1)} = J_\\nu(z) + iY_\\nu(z),</p>\n\n<p>where $J_\\nu(z)$ is the Bessel function of the first kind, and\n$Y_\\nu(z)$ is the Bessel function of the second kind.</p>\n\n<p>It is a solution to Bessel's equation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hankel1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hankel1</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hankel2, besselj, bessely</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/</p>\n", "bases": "sympy.functions.special.bessel.hankel1, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel1.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel1.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel1.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel1.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel2", "type": "class", "doc": "<p>Hankel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>.. math ::\n    H_\\nu^{(2)} = J_\\nu(z) - iY_\\nu(z),</p>\n\n<p>where $J_\\nu(z)$ is the Bessel function of the first kind, and\n$Y_\\nu(z)$ is the Bessel function of the second kind.</p>\n\n<p>It is a solution to Bessel's equation, and linearly independent from\n$H_\\nu^{(1)}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hankel2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hankel2</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hankel1, besselj, bessely</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/</p>\n", "bases": "sympy.functions.special.bessel.hankel2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.jn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jn", "type": "class", "doc": "<p>Spherical Bessel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is a solution to the spherical Bessel equation</p>\n\n<p>.. math ::\n    z^2 \\frac{\\mathrm{d}^2 w}{\\mathrm{d}z^2}\n      + 2z \\frac{\\mathrm{d}w}{\\mathrm{d}z} + (z^2 - \\nu(\\nu + 1)) w = 0.</p>\n\n<p>It can be defined as</p>\n\n<p>.. math ::\n    j_\\nu(z) = \\sqrt{\\frac{\\pi}{2z}} J_{\\nu + \\frac{1}{2}}(z),</p>\n\n<p>where $J_\\nu(z)$ is the Bessel function of the first kind.</p>\n\n<p>The spherical Bessel functions of integral order are\ncalculated using the formula:</p>\n\n<p>.. math:: j_n(z) = f_n(z) \\sin{z} + (-1)^{n+1} f_{-n-1}(z) \\cos{z},</p>\n\n<p>where the coefficients $f_n(z)$ are available as\n:func:<code>sympy.polys.orthopolys.spherical_bessel_fn</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">jn</span><span class=\"p\">,</span> <span class=\"n\">sin</span><span class=\"p\">,</span> <span class=\"n\">cos</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">besselj</span><span class=\"p\">,</span> <span class=\"n\">bessely</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;z&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">nu</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;nu&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">sin(z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span> <span class=\"o\">==</span> <span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">z</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">(-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">besselj</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">bessely</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mf\">5.2</span><span class=\"o\">+</span><span class=\"mf\">0.3</span><span class=\"n\">j</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">0.099419756723640344491 - 0.054525080242173562897*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, bessely, besselk, yn</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://dlmf.nist.gov/10.47</p>\n", "bases": "sympy.functions.special.bessel.jn, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.jn.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jn.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.jn.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jn.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.yn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "yn", "type": "class", "doc": "<p>Spherical Bessel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is another solution to the spherical Bessel equation, and\nlinearly independent from $j_n$. It can be defined as</p>\n\n<p>.. math ::\n    y_\\nu(z) = \\sqrt{\\frac{\\pi}{2z}} Y_{\\nu + \\frac{1}{2}}(z),</p>\n\n<p>where $Y_\\nu(z)$ is the Bessel function of the second kind.</p>\n\n<p>For integral orders $n$, $y_n$ is calculated using the formula:</p>\n\n<p>.. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z)</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">yn</span><span class=\"p\">,</span> <span class=\"n\">sin</span><span class=\"p\">,</span> <span class=\"n\">cos</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">besselj</span><span class=\"p\">,</span> <span class=\"n\">bessely</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;z&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">nu</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;nu&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">-cos(z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span> <span class=\"o\">==</span> <span class=\"o\">-</span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">-</span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">z</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">besselj</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">bessely</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mf\">5.2</span><span class=\"o\">+</span><span class=\"mf\">0.3</span><span class=\"n\">j</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">0.18525034196069722536 + 0.014895573969924817587*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, bessely, besselk, jn</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://dlmf.nist.gov/10.47</p>\n", "bases": "sympy.functions.special.bessel.yn, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.yn.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "yn.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.yn.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "yn.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn1", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn1", "type": "class", "doc": "<p>Spherical Hankel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>.. math:: h_\\nu^(1)(z) = j_\\nu(z) + i y_\\nu(z),</p>\n\n<p>where $j_\\nu(z)$ and $y_\\nu(z)$ are the spherical\nBessel function of the first and second kinds.</p>\n\n<p>For integral orders $n$, $h_n^(1)$ is calculated using the formula:</p>\n\n<p>.. math:: h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z)</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">hn1</span><span class=\"p\">,</span> <span class=\"n\">hankel1</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">yn</span><span class=\"p\">,</span> <span class=\"n\">jn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;z&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">nu</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;nu&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">jn(nu, z) + I*yn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">sin(z)/z - I*cos(z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">-I*sin(z)/z - cos(z)/z + sin(z)/z**2 - I*cos(z)/z**2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**nu*yn(-nu - 1, z) + I*yn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hankel1</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel1(nu, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hn2, jn, yn, hankel1, hankel2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://dlmf.nist.gov/10.47</p>\n", "bases": "sympy.functions.special.bessel.hn1, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn1.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn1.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn1.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn1.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn2", "type": "class", "doc": "<p>Spherical Hankel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>.. math:: h_\\nu^(2)(z) = j_\\nu(z) - i y_\\nu(z),</p>\n\n<p>where $j_\\nu(z)$ and $y_\\nu(z)$ are the spherical\nBessel function of the first and second kinds.</p>\n\n<p>For integral orders $n$, $h_n^(2)$ is calculated using the formula:</p>\n\n<p>.. math:: h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z)</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">hn2</span><span class=\"p\">,</span> <span class=\"n\">hankel2</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">jn</span><span class=\"p\">,</span> <span class=\"n\">yn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;z&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">nu</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;nu&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">jn(nu, z) - I*yn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">sin(z)/z + I*cos(z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">I*sin(z)/z - cos(z)/z + sin(z)/z**2 + I*cos(z)/z**2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hankel2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel2(nu, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">)</span>\n<span class=\"go\">-(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**nu*yn(-nu - 1, z) - I*yn(nu, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hn1, jn, yn, hankel1, hankel2</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] http://dlmf.nist.gov/10.47</p>\n", "bases": "sympy.functions.special.bessel.hn2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyai", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyai", "type": "class", "doc": "<p>The Airy function $\\operatorname{Ai}$ of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Airy function $\\operatorname{Ai}(z)$ is defined to be the function\nsatisfying Airy's differential equation</p>\n\n<p>.. math::\n    \\frac{\\mathrm{d}^2 w(z)}{\\mathrm{d}z^2} - z w(z) = 0.</p>\n\n<p>Equivalently, for real $z$</p>\n\n<p>.. math::\n    \\operatorname{Ai}(z) := \\frac{1}{\\pi}\n    \\int_0^\\infty \\cos\\left(\\frac{t^3}{3} + z t\\right) \\mathrm{d}t.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create an Airy function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">airyai</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airyai(z)</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/3)/(3*gamma(2/3))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>The Airy function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">airyai(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airyaiprime(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">z*airyai(z)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Airy function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">50</span><span class=\"p\">)</span>\n<span class=\"go\">0.22740742820168557599192443603787379946077222541710</span>\n</code></pre></div>\n\n<p>Rewrite $\\operatorname{Ai}(z)$ in terms of hypergeometric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">-3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>airybi: Airy function of the second kind.\nairyaiprime: Derivative of the Airy function of the first kind.\nairybiprime: Derivative of the Airy function of the second kind.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Airy_function\n.. [2] http://dlmf.nist.gov/9\n.. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions\n.. [4] http://mathworld.wolfram.com/AiryFunctions.html</p>\n", "bases": "sympy.functions.special.bessel.airyai, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyai.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyai.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyai.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyai.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybi", "type": "class", "doc": "<p>The Airy function $\\operatorname{Bi}$ of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Airy function $\\operatorname{Bi}(z)$ is defined to be the function\nsatisfying Airy's differential equation</p>\n\n<p>.. math::\n    \\frac{\\mathrm{d}^2 w(z)}{\\mathrm{d}z^2} - z w(z) = 0.</p>\n\n<p>Equivalently, for real $z$</p>\n\n<p>.. math::\n    \\operatorname{Bi}(z) := \\frac{1}{\\pi}\n             \\int_0^\\infty\n               \\exp\\left(-\\frac{t^3}{3} + z t\\right)\n               + \\sin\\left(\\frac{t^3}{3} + z t\\right) \\mathrm{d}t.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create an Airy function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">airybi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airybi(z)</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">3**(5/6)/(3*gamma(2/3))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>The Airy function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">airybi(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airybiprime(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">z*airybi(z)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Airy function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">50</span><span class=\"p\">)</span>\n<span class=\"go\">-0.41230258795639848808323405461146104203453483447240</span>\n</code></pre></div>\n\n<p>Rewrite $\\operatorname{Bi}(z)$ in terms of hypergeometric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>airyai: Airy function of the first kind.\nairyaiprime: Derivative of the Airy function of the first kind.\nairybiprime: Derivative of the Airy function of the second kind.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Airy_function\n.. [2] http://dlmf.nist.gov/9\n.. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions\n.. [4] http://mathworld.wolfram.com/AiryFunctions.html</p>\n", "bases": "sympy.functions.special.bessel.airybi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyaiprime", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyaiprime", "type": "class", "doc": "<p>The derivative $\\operatorname{Ai}^\\prime$ of the Airy function of the first\nkind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Airy function $\\operatorname{Ai}^\\prime(z)$ is defined to be the\nfunction</p>\n\n<p>.. math::\n    \\operatorname{Ai}^\\prime(z) := \\frac{\\mathrm{d} \\operatorname{Ai}(z)}{\\mathrm{d} z}.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create an Airy function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">airyaiprime</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airyaiprime(z)</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">-3**(2/3)/(3*gamma(1/3))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>The Airy function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">airyaiprime(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z*airyai(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">z*airyaiprime(z) + airyai(z)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">-3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Airy function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">50</span><span class=\"p\">)</span>\n<span class=\"go\">0.61825902074169104140626429133247528291577794512415</span>\n</code></pre></div>\n\n<p>Rewrite $\\operatorname{Ai}^\\prime(z)$ in terms of hypergeometric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>airyai: Airy function of the first kind.\nairybi: Airy function of the second kind.\nairybiprime: Derivative of the Airy function of the second kind.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Airy_function\n.. [2] http://dlmf.nist.gov/9\n.. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions\n.. [4] http://mathworld.wolfram.com/AiryFunctions.html</p>\n", "bases": "sympy.functions.special.bessel.airyaiprime, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyaiprime.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyaiprime.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyaiprime.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyaiprime.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybiprime", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybiprime", "type": "class", "doc": "<p>The derivative $\\operatorname{Bi}^\\prime$ of the Airy function of the first\nkind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Airy function $\\operatorname{Bi}^\\prime(z)$ is defined to be the\nfunction</p>\n\n<p>.. math::\n    \\operatorname{Bi}^\\prime(z) := \\frac{\\mathrm{d} \\operatorname{Bi}(z)}{\\mathrm{d} z}.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create an Airy function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">airybiprime</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airybiprime(z)</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/6)/gamma(1/3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>The Airy function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">airybiprime(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z*airybi(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">z*airybiprime(z) + airybi(z)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Airy function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">50</span><span class=\"p\">)</span>\n<span class=\"go\">0.27879516692116952268509756941098324140300059345163</span>\n</code></pre></div>\n\n<p>Rewrite $\\operatorname{Bi}^\\prime(z)$ in terms of hypergeometric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>airyai: Airy function of the first kind.\nairybi: Airy function of the second kind.\nairyaiprime: Derivative of the Airy function of the first kind.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Airy_function\n.. [2] http://dlmf.nist.gov/9\n.. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions\n.. [4] http://mathworld.wolfram.com/AiryFunctions.html</p>\n", "bases": "sympy.functions.special.bessel.airybiprime, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybiprime.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybiprime.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybiprime.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybiprime.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.marcumq", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "marcumq", "type": "class", "doc": "<p>The Marcum Q-function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Marcum Q-function is defined by the meromorphic continuation of</p>\n\n<p>.. math::\n    Q_m(a, b) = a^{- m + 1} \\int_{b}^{\\infty} x^{m} e^{- \\frac{a^{2}}{2} - \\frac{x^{2}}{2}} I_{m - 1}\\left(a x\\right)\\, dx</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">marcumq</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">marcumq(m, a, b)</span>\n</code></pre></div>\n\n<p>Special values:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">uppergamma(m, b**2/2)/gamma(m)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1 - exp(-a**2/2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">1/2 + exp(-a**2)*besseli(0, a**2)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2)</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $a$ and $b$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">),</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">a*(-marcumq(m, a, b) + marcumq(m + 1, a, b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">),</span> <span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">-a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b)</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Marcum_Q-function\n.. [2] http://mathworld.wolfram.com/MarcumQ-Function.html</p>\n", "bases": "sympy.functions.special.bessel.marcumq, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.marcumq.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "marcumq.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.marcumq.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "marcumq.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hyper", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hyper", "type": "class", "doc": "<p>The generalized hypergeometric function is defined by a series where\nthe ratios of successive terms are a rational function of the summation\nindex. When convergent, it is continued analytically to the largest\npossible domain.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The hypergeometric function depends on two vectors of parameters, called\nthe numerator parameters $a_p$, and the denominator parameters\n$b_q$. It also has an argument $z$. The series definition is</p>\n\n<p>.. math ::\n    {}_pF_q\\left(\\begin{matrix} a_1, \\cdots, a_p \\ b_1, \\cdots, b_q \\end{matrix}\n                 \\middle| z \\right)\n    = \\sum_{n=0}^\\infty \\frac{(a_1)_n \\cdots (a_p)_n}{(b_1)_n \\cdots (b_q)_n}\n                        \\frac{z^n}{n!},</p>\n\n<p>where $(a)_n = (a)(a+1)\\cdots(a+n-1)$ denotes the rising factorial.</p>\n\n<p>If one of the $b_q$ is a non-positive integer then the series is\nundefined unless one of the $a_p$ is a larger (i.e., smaller in\nmagnitude) non-positive integer. If none of the $b_q$ is a\nnon-positive integer and one of the $a_p$ is a non-positive\ninteger, then the series reduces to a polynomial. To simplify the\nfollowing discussion, we assume that none of the $a_p$ or\n$b_q$ is a non-positive integer. For more details, see the\nreferences.</p>\n\n<p>The series converges for all $z$ if $p \\le q$, and thus\ndefines an entire single-valued function in this case. If $p =\nq+1$ the series converges for $|z| &lt; 1$, and can be continued\nanalytically into a half-plane. If $p &gt; q+1$ the series is\ndivergent for all $z$.</p>\n\n<p>Please note the hypergeometric function constructor currently does <em>not</em>\ncheck if the parameters actually yield a well-defined function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>The parameters $a_p$ and $b_q$ can be passed as arbitrary\niterables, for example:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyper</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">),</span> <span class=\"p\">[</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">hyper((1, 2, 3), (3, 4), x)</span>\n</code></pre></div>\n\n<p>There is also pretty printing (it looks better using Unicode):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">pprint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">pprint</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">),</span> <span class=\"p\">[</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">use_unicode</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">  _</span>\n<span class=\"go\"> |_  /1, 2, 3 |  \\</span>\n<span class=\"go\"> |   |        | x|</span>\n<span class=\"go\">3  2 \\  3, 4  |  /</span>\n</code></pre></div>\n\n<p>The parameters must always be iterables, even if they are vectors of\nlength one or zero:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyper</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"p\">),</span> <span class=\"p\">[],</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">hyper((1,), (), x)</span>\n</code></pre></div>\n\n<p>But of course they may be variables (but if they depend on $x$ then you\nshould not expect much implemented functionality):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyper</span><span class=\"p\">((</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">hyper((n, a), (n**2,), x)</span>\n</code></pre></div>\n\n<p>The hypergeometric function generalizes many named special functions.\nThe function <code>hyperexpand()</code> tries to express a hypergeometric function\nusing named special functions. For example:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyperexpand</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">([],</span> <span class=\"p\">[],</span> <span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">exp(x)</span>\n</code></pre></div>\n\n<p>You can also use <code>expand_func()</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">hyper</span><span class=\"p\">([</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">log(x + 1)</span>\n</code></pre></div>\n\n<p>More examples:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">([],</span> <span class=\"p\">[</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">4</span><span class=\"p\">))</span>\n<span class=\"go\">cos(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">hyper</span><span class=\"p\">([</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">asin(x)</span>\n</code></pre></div>\n\n<p>We can also sometimes <code>hyperexpand()</code> parametric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">([</span><span class=\"o\">-</span><span class=\"n\">a</span><span class=\"p\">],</span> <span class=\"p\">[],</span> <span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">(1 - x)**a</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.simplify.hyperexpand\ngamma\nmeijerg</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations,\n       Volume 1\n.. [2] https://en.wikipedia.org/wiki/Generalized_hypergeometric_function</p>\n", "bases": "sympy.functions.special.hyper.hyper, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hyper.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hyper.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hyper.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hyper.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.hyper.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hyper.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'commutative': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.meijerg", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "meijerg", "type": "class", "doc": "<p>The Meijer G-function is defined by a Mellin-Barnes type integral that\nresembles an inverse Mellin transform. It generalizes the hypergeometric\nfunctions.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Meijer G-function depends on four sets of parameters. There are\n\"<em>numerator parameters</em>\"\n$a_1, \\ldots, a_n$ and $a_{n+1}, \\ldots, a_p$, and there are\n\"<em>denominator parameters</em>\"\n$b_1, \\ldots, b_m$ and $b_{m+1}, \\ldots, b_q$.\nConfusingly, it is traditionally denoted as follows (note the position\nof $m$, $n$, $p$, $q$, and how they relate to the lengths of the four\nparameter vectors):</p>\n\n<p>.. math ::\n    G_{p,q}^{m,n} \\left(\\begin{matrix}a_1, \\cdots, a_n &amp; a_{n+1}, \\cdots, a_p \\\n                                    b_1, \\cdots, b_m &amp; b_{m+1}, \\cdots, b_q\n                      \\end{matrix} \\middle| z \\right).</p>\n\n<p>However, in SymPy the four parameter vectors are always available\nseparately (see examples), so that there is no need to keep track of the\ndecorating sub- and super-scripts on the G symbol.</p>\n\n<p>The G function is defined as the following integral:</p>\n\n<p>.. math ::\n     \\frac{1}{2 \\pi i} \\int_L \\frac{\\prod_{j=1}^m \\Gamma(b_j - s)\n     \\prod_{j=1}^n \\Gamma(1 - a_j + s)}{\\prod_{j=m+1}^q \\Gamma(1- b_j +s)\n     \\prod_{j=n+1}^p \\Gamma(a_j - s)} z^s \\mathrm{d}s,</p>\n\n<p>where $\\Gamma(z)$ is the gamma function. There are three possible\ncontours which we will not describe in detail here (see the references).\nIf the integral converges along more than one of them, the definitions\nagree. The contours all separate the poles of $\\Gamma(1-a_j+s)$\nfrom the poles of $\\Gamma(b_k-s)$, so in particular the G function\nis undefined if $a_j - b_k \\in \\mathbb{Z}_{&gt;0}$ for some\n$j \\le n$ and $k \\le m$.</p>\n\n<p>The conditions under which one of the contours yields a convergent integral\nare complicated and we do not state them here, see the references.</p>\n\n<p>Please note currently the Meijer G-function constructor does <em>not</em> check any\nconvergence conditions.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>You can pass the parameters either as four separate vectors:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">meijerg</span><span class=\"p\">,</span> <span class=\"n\">Tuple</span><span class=\"p\">,</span> <span class=\"n\">pprint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">pprint</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,),</span> <span class=\"p\">[],</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">use_unicode</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\"> __1, 2 /1, 2  a, 4 |  \\</span>\n<span class=\"go\">/__     |           | x|</span>\n<span class=\"go\">\\_|4, 1 \\ 5         |  /</span>\n</code></pre></div>\n\n<p>Or as two nested vectors:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">pprint</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">([(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">)],</span> <span class=\"p\">([</span><span class=\"mi\">5</span><span class=\"p\">],</span> <span class=\"n\">Tuple</span><span class=\"p\">()),</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">use_unicode</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\"> __1, 2 /1, 2  3, 4 |  \\</span>\n<span class=\"go\">/__     |           | x|</span>\n<span class=\"go\">\\_|4, 1 \\ 5         |  /</span>\n</code></pre></div>\n\n<p>As with the hypergeometric function, the parameters may be passed as\narbitrary iterables. Vectors of length zero and one also have to be\npassed as iterables. The parameters need not be constants, but if they\ndepend on the argument then not much implemented functionality should be\nexpected.</p>\n\n<p>All the subvectors of parameters are available:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">pprint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span> <span class=\"o\">=</span> <span class=\"n\">meijerg</span><span class=\"p\">([</span><span class=\"mi\">1</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">3</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">4</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">pprint</span><span class=\"p\">(</span><span class=\"n\">g</span><span class=\"p\">,</span> <span class=\"n\">use_unicode</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\"> __1, 1 /1  2 |  \\</span>\n<span class=\"go\">/__     |     | x|</span>\n<span class=\"go\">\\_|2, 2 \\3  4 |  /</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">an</span>\n<span class=\"go\">(1,)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">ap</span>\n<span class=\"go\">(1, 2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">aother</span>\n<span class=\"go\">(2,)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">bm</span>\n<span class=\"go\">(3,)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">bq</span>\n<span class=\"go\">(3, 4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">bother</span>\n<span class=\"go\">(4,)</span>\n</code></pre></div>\n\n<p>The Meijer G-function generalizes the hypergeometric functions.\nIn some cases it can be expressed in terms of hypergeometric functions,\nusing Slater's theorem. For example:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyperexpand</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">([</span><span class=\"n\">a</span><span class=\"p\">],</span> <span class=\"p\">[],</span> <span class=\"p\">[</span><span class=\"n\">c</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"n\">b</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">allow_hyper</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">x**c*gamma(-a + c + 1)*hyper((-a + c + 1,),</span>\n<span class=\"go\">                             (-b + c + 1,), -x)/gamma(-b + c + 1)</span>\n</code></pre></div>\n\n<p>Thus the Meijer G-function also subsumes many named functions as special\ncases. You can use <code>expand_func()</code> or <code>hyperexpand()</code> to (try to)\nrewrite a Meijer G-function in terms of named special functions. For\nexample:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">([[],[]],</span> <span class=\"p\">[[</span><span class=\"mi\">0</span><span class=\"p\">],[]],</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">exp(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">([[],[]],</span> <span class=\"p\">[[</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">],[</span><span class=\"mi\">0</span><span class=\"p\">]],</span> <span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">sin(x)/sqrt(pi)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hyper\nsympy.simplify.hyperexpand</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations,\n       Volume 1\n.. [2] https://en.wikipedia.org/wiki/Meijer_G-function</p>\n", "bases": "sympy.functions.special.hyper.meijerg, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.meijerg.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "meijerg.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.meijerg.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "meijerg.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.meijerg.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "meijerg.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'commutative': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.appellf1", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "appellf1", "type": "class", "doc": "<p>This is the Appell hypergeometric function of two variables as:</p>\n\n<p>.. math ::\n    F_1(a,b_1,b_2,c,x,y) = \\sum_{m=0}^{\\infty} \\sum_{n=0}^{\\infty}\n    \\frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}}\n    \\frac{x^m y^n}{m! n!}.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">appellf1</span><span class=\"p\">,</span> <span class=\"n\">symbols</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b1</span><span class=\"p\">,</span> <span class=\"n\">b2</span><span class=\"p\">,</span> <span class=\"n\">c</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;x y a b1 b2 c&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mf\">2.</span><span class=\"p\">,</span> <span class=\"mf\">1.</span><span class=\"p\">,</span> <span class=\"mf\">6.</span><span class=\"p\">,</span> <span class=\"mf\">4.</span><span class=\"p\">,</span> <span class=\"mf\">5.</span><span class=\"p\">,</span> <span class=\"mf\">6.</span><span class=\"p\">)</span>\n<span class=\"go\">0.0063339426292673</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mf\">12.</span><span class=\"p\">,</span> <span class=\"mf\">12.</span><span class=\"p\">,</span> <span class=\"mf\">6.</span><span class=\"p\">,</span> <span class=\"mf\">4.</span><span class=\"p\">,</span> <span class=\"mf\">0.5</span><span class=\"p\">,</span> <span class=\"mf\">0.12</span><span class=\"p\">)</span>\n<span class=\"go\">172870711.659936</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">6</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">15</span><span class=\"p\">,</span> <span class=\"mi\">60</span><span class=\"p\">)</span>\n<span class=\"go\">appellf1(40, 2, 6, 4, 15, 60)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mf\">20.</span><span class=\"p\">,</span> <span class=\"mf\">12.</span><span class=\"p\">,</span> <span class=\"mf\">10.</span><span class=\"p\">,</span> <span class=\"mf\">3.</span><span class=\"p\">,</span> <span class=\"mf\">0.5</span><span class=\"p\">,</span> <span class=\"mf\">0.12</span><span class=\"p\">)</span>\n<span class=\"go\">15605338197184.4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">6</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">appellf1(40, 2, 6, 4, x, y)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b1</span><span class=\"p\">,</span> <span class=\"n\">b2</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">appellf1(a, b1, b2, c, x, y)</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Appell_series\n.. [2] http://functions.wolfram.com/HypergeometricFunctions/AppellF1/</p>\n", "bases": "sympy.functions.special.hyper.appellf1, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.appellf1.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "appellf1.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.appellf1.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "appellf1.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.legendre", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "legendre", "type": "class", "doc": "<p><code>legendre(n, x)</code> gives the nth Legendre polynomial of x, $P_n(x)$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Legendre polynomials are orthogonal on [-1, 1] with respect to\nthe constant weight 1. They satisfy $P_n(1) = 1$ for all n; further,\n$P_n$ is odd for odd n and even for even n.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">legendre</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">3*x**2/2 - 1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">legendre(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nassoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Legendre_polynomial\n.. [2] http://mathworld.wolfram.com/LegendrePolynomial.html\n.. [3] http://functions.wolfram.com/Polynomials/LegendreP/\n.. [4] http://functions.wolfram.com/Polynomials/LegendreP2/</p>\n", "bases": "sympy.functions.special.polynomials.legendre, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.legendre.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "legendre.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.legendre.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "legendre.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_legendre", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_legendre", "type": "class", "doc": "<p><code>assoc_legendre(n, m, x)</code> gives $P_n^m(x)$, where n and m are\nthe degree and order or an expression which is related to the nth\norder Legendre polynomial, $P_n(x)$ in the following manner:</p>\n\n<p>.. math::\n    P_n^m(x) = (-1)^m (1 - x^2)^{\\frac{m}{2}}\n               \\frac{\\mathrm{d}^m P_n(x)}{\\mathrm{d} x^m}</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Associated Legendre polynomials are orthogonal on [-1, 1] with:</p>\n\n<ul>\n<li>weight = 1            for the same m, and different n.</li>\n<li>weight = 1/(1-x**2)   for the same n, and different m.</li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">assoc_legendre</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_legendre</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_legendre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_legendre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(1 - x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_legendre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">assoc_legendre(n, m, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials\n.. [2] http://mathworld.wolfram.com/LegendrePolynomial.html\n.. [3] http://functions.wolfram.com/Polynomials/LegendreP/\n.. [4] http://functions.wolfram.com/Polynomials/LegendreP2/</p>\n", "bases": "sympy.functions.special.polynomials.assoc_legendre, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_legendre.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_legendre.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_legendre.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_legendre.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hermite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hermite", "type": "class", "doc": "<p><code>hermite(n, x)</code> gives the nth Hermite polynomial in x, $H_n(x)$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Hermite polynomials are orthogonal on $(-\\infty, \\infty)$\nwith respect to the weight $\\exp\\left(-x^2\\right)$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hermite</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">4*x**2 - 2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">hermite(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*n*hermite(n - 1, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*hermite(n, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Hermite_polynomial\n.. [2] http://mathworld.wolfram.com/HermitePolynomial.html\n.. [3] http://functions.wolfram.com/Polynomials/HermiteH/</p>\n", "bases": "sympy.functions.special.polynomials.hermite, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hermite.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hermite.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hermite.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hermite.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt", "type": "class", "doc": "<p>Chebyshev polynomial of the first kind, $T_n(x)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>chebyshevt(n, x)</code> gives the nth Chebyshev polynomial (of the first\nkind) in x, $T_n(x)$.</p>\n\n<p>The Chebyshev polynomials of the first kind are orthogonal on\n$[-1, 1]$ with respect to the weight $\\frac{1}{\\sqrt{1-x^2}}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">chebyshevt</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x**2 - 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">chebyshevt(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*chebyshevt(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">chebyshevt(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">cos(pi*n/2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">n*chebyshevu(n - 1, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial\n.. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html\n.. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html\n.. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/\n.. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/</p>\n", "bases": "sympy.functions.special.polynomials.chebyshevt, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu", "type": "class", "doc": "<p>Chebyshev polynomial of the second kind, $U_n(x)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>chebyshevu(n, x)</code> gives the nth Chebyshev polynomial of the second\nkind in x, $U_n(x)$.</p>\n\n<p>The Chebyshev polynomials of the second kind are orthogonal on\n$[-1, 1]$ with respect to the weight $\\sqrt{1-x^2}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">chebyshevu</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">4*x**2 - 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">chebyshevu(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*chebyshevu(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-chebyshevu(n - 2, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">cos(pi*n/2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">n + 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial\n.. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html\n.. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html\n.. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/\n.. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/</p>\n", "bases": "sympy.functions.special.polynomials.chebyshevu, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu_root", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu_root", "type": "class", "doc": "<p><code>chebyshevu_root(n, k)</code> returns the kth root (indexed from zero) of the\nnth Chebyshev polynomial of the second kind; that is, if 0 &lt;= k &lt; n,\n<code>chebyshevu(n, chebyshevu_root(n, k)) == 0</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">chebyshevu</span><span class=\"p\">,</span> <span class=\"n\">chebyshevu_root</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu_root</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(2)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">chebyshevu_root</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>chebyshevt, chebyshevt_root, chebyshevu,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n", "bases": "sympy.functions.special.polynomials.chebyshevu_root, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu_root.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu_root.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu_root.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu_root.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt_root", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt_root", "type": "class", "doc": "<p><code>chebyshev_root(n, k)</code> returns the kth root (indexed from zero) of\nthe nth Chebyshev polynomial of the first kind; that is, if\n0 &lt;= k &lt; n, <code>chebyshevt(n, chebyshevt_root(n, k)) == 0</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">chebyshevt</span><span class=\"p\">,</span> <span class=\"n\">chebyshevt_root</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt_root</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(3)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">chebyshevt_root</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n", "bases": "sympy.functions.special.polynomials.chebyshevt_root, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt_root.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt_root.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt_root.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt_root.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.laguerre", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "laguerre", "type": "class", "doc": "<p>Returns the nth Laguerre polynomial in x, $L_n(x)$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">laguerre</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1 - x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**2/2 - 2*x + 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-x**3/6 + 3*x**2/2 - 3*x + 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">laguerre(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-assoc_laguerre(n - 1, 1, x)</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>n : int\n    Degree of Laguerre polynomial. Must be <code>n &gt;= 0</code>.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nassoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial\n.. [2] http://mathworld.wolfram.com/LaguerrePolynomial.html\n.. [3] http://functions.wolfram.com/Polynomials/LaguerreL/\n.. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/</p>\n", "bases": "sympy.functions.special.polynomials.laguerre, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.laguerre.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "laguerre.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.laguerre.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "laguerre.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_laguerre", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_laguerre", "type": "class", "doc": "<p>Returns the nth generalized Laguerre polynomial in x, $L_n(x)$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">assoc_laguerre</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a - x + 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +</span>\n<span class=\"go\">    x*(-a**2/2 - 5*a/2 - 3) + 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">binomial(a + n, a)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">assoc_laguerre(n, a, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">laguerre(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-assoc_laguerre(n - 1, a + 1, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>n : int\n    Degree of Laguerre polynomial. Must be <code>n &gt;= 0</code>.</p>\n\n<p>alpha : Expr\n    Arbitrary expression. For <code>alpha=0</code> regular Laguerre\n    polynomials will be generated.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials\n.. [2] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html\n.. [3] http://functions.wolfram.com/Polynomials/LaguerreL/\n.. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/</p>\n", "bases": "sympy.functions.special.polynomials.assoc_laguerre, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_laguerre.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_laguerre.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_laguerre.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_laguerre.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.gegenbauer", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gegenbauer", "type": "class", "doc": "<p>Gegenbauer polynomial $C_n^{\\left(\\alpha\\right)}(x)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>gegenbauer(n, alpha, x)</code> gives the nth Gegenbauer polynomial\nin x, $C_n^{\\left(\\alpha\\right)}(x)$.</p>\n\n<p>The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with\nrespect to the weight $\\left(1-x^2\\right)^{\\alpha-\\frac{1}{2}}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">gegenbauer</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*a*x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-a + x**2*(2*a**2 + 2*a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">gegenbauer(n, a, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*gegenbauer(n, a, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">gegenbauer(n, conjugate(a), conjugate(x))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*a*gegenbauer(n - 1, a + 1, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi,\nchebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials\n.. [2] http://mathworld.wolfram.com/GegenbauerPolynomial.html\n.. [3] http://functions.wolfram.com/Polynomials/GegenbauerC3/</p>\n", "bases": "sympy.functions.special.polynomials.gegenbauer, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.gegenbauer.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gegenbauer.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.gegenbauer.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gegenbauer.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.jacobi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jacobi", "type": "class", "doc": "<p>Jacobi polynomial $P_n^{\\left(\\alpha, \\beta\\right)}(x)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>jacobi(n, alpha, beta, x)</code> gives the nth Jacobi polynomial\nin x, $P_n^{\\left(\\alpha, \\beta\\right)}(x)$.</p>\n\n<p>The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect\nto the weight $\\left(1-x\\right)^\\alpha \\left(1+x\\right)^\\beta$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">jacobi</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a/2 - b/2 + x*(a/2 + b/2 + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">jacobi(n, a, b, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">RisingFactorial(a + 1, n)*gegenbauer(n,</span>\n<span class=\"go\">    a + 1/2, x)/RisingFactorial(2*a + 1, n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">legendre(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*jacobi(n, b, a, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(2**n*factorial(n)*gamma(a + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">RisingFactorial(a + 1, n)/factorial(n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">jacobi(n, conjugate(a), conjugate(b), conjugate(x))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gegenbauer,\nchebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly,\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials\n.. [2] http://mathworld.wolfram.com/JacobiPolynomial.html\n.. [3] http://functions.wolfram.com/Polynomials/JacobiP/</p>\n", "bases": "sympy.functions.special.polynomials.jacobi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.jacobi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jacobi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.jacobi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jacobi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ynm", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ynm", "type": "class", "doc": "<p>Spherical harmonics defined as</p>\n\n<p>.. math::\n    Y_n^m(\\theta, \\varphi) := \\sqrt{\\frac{(2n+1)(n-m)!}{4\\pi(n+m)!}}\n                              \\exp(i m \\varphi)\n                              \\mathrm{P}_n^m\\left(\\cos(\\theta)\\right)</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>Ynm()</code> gives the spherical harmonic function of order $n$ and $m$\nin $\\theta$ and $\\varphi$, $Y_n^m(\\theta, \\varphi)$. The four\nparameters are as follows: $n \\geq 0$ an integer and $m$ an integer\nsuch that $-n \\leq m \\leq n$ holds. The two angles are real-valued\nwith $\\theta \\in [0, \\pi]$ and $\\varphi \\in [0, 2\\pi]$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ynm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">simplify</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>Several symmetries are known, for the order:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>As well as for the angles:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">exp(-2*I*m*phi)*Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>For specific integers $n$ and $m$ we can evaluate the harmonics\nto more useful expressions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">1/(2*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(3)*cos(theta)/(2*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">-sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">-sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))</span>\n</code></pre></div>\n\n<p>We can differentiate the functions with respect\nto both angles:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ynm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">),</span> <span class=\"n\">theta</span><span class=\"p\">)</span>\n<span class=\"go\">m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">),</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">I*m*Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>Further we can compute the complex conjugation:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ynm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">))</span>\n<span class=\"go\">(-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>To get back the well known expressions in spherical\ncoordinates, we use full expansion:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ynm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Ynm_c, Znm</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Spherical_harmonics\n.. [2] http://mathworld.wolfram.com/SphericalHarmonic.html\n.. [3] http://functions.wolfram.com/Polynomials/SphericalHarmonicY/\n.. [4] http://dlmf.nist.gov/14.30</p>\n", "bases": "sympy.functions.special.spherical_harmonics.Ynm, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ynm.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ynm.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ynm.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ynm.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Znm", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Znm", "type": "class", "doc": "<p>Real spherical harmonics defined as</p>\n\n<p>.. math::</p>\n\n<pre><code>Z_n^m(\\theta, \\varphi) :=\n\\begin{cases}\n  \\frac{Y_n^m(\\theta, \\varphi) + \\overline{Y_n^m(\\theta, \\varphi)}}{\\sqrt{2}} &amp;\\quad m &gt; 0 \\\\\n  Y_n^m(\\theta, \\varphi) &amp;\\quad m = 0 \\\\\n  \\frac{Y_n^m(\\theta, \\varphi) - \\overline{Y_n^m(\\theta, \\varphi)}}{i \\sqrt{2}} &amp;\\quad m &lt; 0 \\\\\n\\end{cases}\n</code></pre>\n\n<p>which gives in simplified form</p>\n\n<p>.. math::</p>\n\n<pre><code>Z_n^m(\\theta, \\varphi) =\n\\begin{cases}\n  \\frac{Y_n^m(\\theta, \\varphi) + (-1)^m Y_n^{-m}(\\theta, \\varphi)}{\\sqrt{2}} &amp;\\quad m &gt; 0 \\\\\n  Y_n^m(\\theta, \\varphi) &amp;\\quad m = 0 \\\\\n  \\frac{Y_n^m(\\theta, \\varphi) - (-1)^m Y_n^{-m}(\\theta, \\varphi)}{i \\sqrt{2}} &amp;\\quad m &lt; 0 \\\\\n\\end{cases}\n</code></pre>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Znm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">simplify</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Znm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">Znm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>For specific integers n and m we can evaluate the harmonics\nto more useful expressions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Znm</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">1/(2*sqrt(pi))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Znm</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">-sqrt(3)*sin(theta)*cos(phi)/(2*sqrt(pi))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Znm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">-sqrt(15)*sin(2*theta)*cos(phi)/(4*sqrt(pi))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Ynm, Ynm_c</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Spherical_harmonics\n.. [2] http://mathworld.wolfram.com/SphericalHarmonic.html\n.. [3] http://functions.wolfram.com/Polynomials/SphericalHarmonicY/</p>\n", "bases": "sympy.functions.special.spherical_harmonics.Znm, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Znm.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Znm.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Znm.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Znm.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_k", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_k", "type": "class", "doc": "<p>The complete elliptic integral of the first kind, defined by</p>\n\n<p>.. math:: K(m) = F\\left(\\tfrac{\\pi}{2}\\middle| m\\right)</p>\n\n<p>where $F\\left(z\\middle| m\\right)$ is the Legendre incomplete\nelliptic integral of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The function $K(m)$ is a single-valued function on the complex\nplane with branch cut along the interval $(1, \\infty)$.</p>\n\n<p>Note that our notation defines the incomplete elliptic integral\nin terms of the parameter $m$ instead of the elliptic modulus\n(eccentricity) $k$.\nIn this case, the parameter $m$ is defined as $m=k^2$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">elliptic_k</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_k</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_k</span><span class=\"p\">(</span><span class=\"mf\">1.0</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">1.50923695405127 + 0.625146415202697*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_k</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>elliptic_f</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals\n.. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticK</p>\n", "bases": "sympy.functions.special.elliptic_integrals.elliptic_k, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_k.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_k.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_k.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_k.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_f", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_f", "type": "class", "doc": "<p>The Legendre incomplete elliptic integral of the first\nkind, defined by</p>\n\n<p>.. math:: F\\left(z\\middle| m\\right) =\n          \\int_0^z \\frac{dt}{\\sqrt{1 - m \\sin^2 t}}</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function reduces to a complete elliptic integral of\nthe first kind, $K(m)$, when $z = \\pi/2$.</p>\n\n<p>Note that our notation defines the incomplete elliptic integral\nin terms of the parameter $m$ instead of the elliptic modulus\n(eccentricity) $k$.\nIn this case, the parameter $m$ is defined as $m=k^2$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">elliptic_f</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_f</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_f</span><span class=\"p\">(</span><span class=\"mf\">3.0</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mf\">1.0</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">2.909449841483 + 1.74720545502474*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>elliptic_k</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals\n.. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticF</p>\n", "bases": "sympy.functions.special.elliptic_integrals.elliptic_f, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_f.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_f.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_f.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_f.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_e", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_e", "type": "class", "doc": "<p>Called with two arguments $z$ and $m$, evaluates the\nincomplete elliptic integral of the second kind, defined by</p>\n\n<p>.. math:: E\\left(z\\middle| m\\right) = \\int_0^z \\sqrt{1 - m \\sin^2 t} dt</p>\n\n<p>Called with a single argument $m$, evaluates the Legendre complete\nelliptic integral of the second kind</p>\n\n<p>.. math:: E(m) = E\\left(\\tfrac{\\pi}{2}\\middle| m\\right)</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The function $E(m)$ is a single-valued function on the complex\nplane with branch cut along the interval $(1, \\infty)$.</p>\n\n<p>Note that our notation defines the incomplete elliptic integral\nin terms of the parameter $m$ instead of the elliptic modulus\n(eccentricity) $k$.\nIn this case, the parameter $m$ is defined as $m=k^2$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">elliptic_e</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">=</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">()</span>\n<span class=\"go\">1.55203744279187 + 0.290764986058437*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"mf\">2.0</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">0.991052601328069 + 0.81879421395609*I</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals\n.. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticE2\n.. [3] http://functions.wolfram.com/EllipticIntegrals/EllipticE</p>\n", "bases": "sympy.functions.special.elliptic_integrals.elliptic_e, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_e.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_e.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_e.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_e.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_pi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_pi", "type": "class", "doc": "<p>Called with three arguments $n$, $z$ and $m$, evaluates the\nLegendre incomplete elliptic integral of the third kind, defined by</p>\n\n<p>.. math:: \\Pi\\left(n; z\\middle| m\\right) = \\int_0^z \\frac{dt}\n          {\\left(1 - n \\sin^2 t\\right) \\sqrt{1 - m \\sin^2 t}}</p>\n\n<p>Called with two arguments $n$ and $m$, evaluates the complete\nelliptic integral of the third kind:</p>\n\n<p>.. math:: \\Pi\\left(n\\middle| m\\right) =\n          \\Pi\\left(n; \\tfrac{\\pi}{2}\\middle| m\\right)</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Note that our notation defines the incomplete elliptic integral\nin terms of the parameter $m$ instead of the elliptic modulus\n(eccentricity) $k$.\nIn this case, the parameter $m$ is defined as $m=k^2$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">elliptic_pi</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_pi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"o\">=</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">z + z**3*(m/6 + n/3) + O(z**4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_pi</span><span class=\"p\">(</span><span class=\"mf\">0.5</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"mf\">1.0</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"mf\">1.2</span><span class=\"p\">)</span>\n<span class=\"go\">2.50232379629182 - 0.760939574180767*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_pi</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_pi</span><span class=\"p\">(</span><span class=\"mf\">1.0</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mf\">2.0</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">3.29136443417283 + 0.32555634906645*I</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals\n.. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticPi3\n.. [3] http://functions.wolfram.com/EllipticIntegrals/EllipticPi</p>\n", "bases": "sympy.functions.special.elliptic_integrals.elliptic_pi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_pi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_pi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_pi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_pi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.beta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "beta", "type": "class", "doc": "<p>The beta integral is called the Eulerian integral of the first kind by\nLegendre:</p>\n\n<p>.. math::\n    \\mathrm{B}(x,y)  \\int^{1}_{0} t^{x-1} (1-t)^{y-1} \\mathrm{d}t.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Beta function or Euler's first integral is closely associated\nwith the gamma function. The Beta function is often used in probability\ntheory and mathematical statistics. It satisfies properties like:</p>\n\n<p>.. math::\n    \\mathrm{B}(a,1) = \\frac{1}{a} \\\n    \\mathrm{B}(a,b) = \\mathrm{B}(b,a)  \\\n    \\mathrm{B}(a,b) = \\frac{\\Gamma(a) \\Gamma(b)}{\\Gamma(a+b)}</p>\n\n<p>Therefore for integral values of $a$ and $b$:</p>\n\n<p>.. math::\n    \\mathrm{B} = \\frac{(a-1)! (b-1)!}{(a+b-1)!}</p>\n\n<p>A special case of the Beta function when <code>x = y</code> is the\nCentral Beta function. It satisfies properties like:</p>\n\n<p>.. math::\n    \\mathrm{B}(x) = 2^{1 - 2x}\\mathrm{B}(x, \\frac{1}{2})\n    \\mathrm{B}(x) = 2^{1 - 2x} cos(\\pi x) \\mathrm{B}(\\frac{1}{2} - x, x)\n    \\mathrm{B}(x) = \\int_{0}^{1} \\frac{t^x}{(1 + t)^{2x}} dt\n    \\mathrm{B}(x) = \\frac{2}{x} \\prod_{n = 1}^{\\infty} \\frac{n(n + 2x)}{(n + x)^2}</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n</code></pre></div>\n\n<p>The Beta function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">beta</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">))</span>\n<span class=\"go\">beta(conjugate(x), conjugate(y))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to both $x$ and $y$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">beta</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(polygamma(0, x) - polygamma(0, x + y))*beta(x, y)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">(polygamma(0, y) - polygamma(0, x + y))*beta(x, y)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*(polygamma(0, x) - polygamma(0, 2*x))*beta(x, x)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Beta function to\narbitrary precision for any complex numbers x and y:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">beta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">)</span>\n<span class=\"go\">0.02671848900111377452242355235388489324562</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">-0.2112723729365330143 - 0.7655283165378005676*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nuppergamma: Upper incomplete gamma function.\nlowergamma: Lower incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Beta_function\n.. [2] http://mathworld.wolfram.com/BetaFunction.html\n.. [3] http://dlmf.nist.gov/5.12</p>\n", "bases": "sympy.functions.special.beta_functions.beta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.beta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "beta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.beta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "beta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieus", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieus", "type": "class", "doc": "<p>The Mathieu Sine function $S(a,q,z)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is one solution of the Mathieu differential equation:</p>\n\n<p>.. math ::\n    y(x)^{\\prime\\prime} + (a - 2 q \\cos(2 x)) y(x) = 0</p>\n\n<p>The other solution is the Mathieu Cosine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">mathieus</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieus</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieus(a, q, z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieus</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sin(sqrt(a)*z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">mathieus</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieusprime(a, q, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>mathieuc: Mathieu cosine function.\nmathieusprime: Derivative of Mathieu sine function.\nmathieucprime: Derivative of Mathieu cosine function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Mathieu_function\n.. [2] http://dlmf.nist.gov/28\n.. [3] http://mathworld.wolfram.com/MathieuBase.html\n.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/</p>\n", "bases": "sympy.functions.special.mathieu_functions.mathieus, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieus.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieus.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieus.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieus.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieuc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieuc", "type": "class", "doc": "<p>The Mathieu Cosine function $C(a,q,z)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is one solution of the Mathieu differential equation:</p>\n\n<p>.. math ::\n    y(x)^{\\prime\\prime} + (a - 2 q \\cos(2 x)) y(x) = 0</p>\n\n<p>The other solution is the Mathieu Sine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">mathieuc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieuc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieuc(a, q, z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieuc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">cos(sqrt(a)*z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">mathieuc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieucprime(a, q, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>mathieus: Mathieu sine function\nmathieusprime: Derivative of Mathieu sine function\nmathieucprime: Derivative of Mathieu cosine function</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Mathieu_function\n.. [2] http://dlmf.nist.gov/28\n.. [3] http://mathworld.wolfram.com/MathieuBase.html\n.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/</p>\n", "bases": "sympy.functions.special.mathieu_functions.mathieuc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieuc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieuc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieuc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieuc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieusprime", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieusprime", "type": "class", "doc": "<p>The derivative $S^{\\prime}(a,q,z)$ of the Mathieu Sine function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is one solution of the Mathieu differential equation:</p>\n\n<p>.. math ::\n    y(x)^{\\prime\\prime} + (a - 2 q \\cos(2 x)) y(x) = 0</p>\n\n<p>The other solution is the Mathieu Cosine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">mathieusprime</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieusprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieusprime(a, q, z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieusprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(a)*cos(sqrt(a)*z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">mathieusprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">(-a + 2*q*cos(2*z))*mathieus(a, q, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>mathieus: Mathieu sine function\nmathieuc: Mathieu cosine function\nmathieucprime: Derivative of Mathieu cosine function</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Mathieu_function\n.. [2] http://dlmf.nist.gov/28\n.. [3] http://mathworld.wolfram.com/MathieuBase.html\n.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/</p>\n", "bases": "sympy.functions.special.mathieu_functions.mathieusprime, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieusprime.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieusprime.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieusprime.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieusprime.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieucprime", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieucprime", "type": "class", "doc": "<p>The derivative $C^{\\prime}(a,q,z)$ of the Mathieu Cosine function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is one solution of the Mathieu differential equation:</p>\n\n<p>.. math ::\n    y(x)^{\\prime\\prime} + (a - 2 q \\cos(2 x)) y(x) = 0</p>\n\n<p>The other solution is the Mathieu Sine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">mathieucprime</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieucprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieucprime(a, q, z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieucprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(a)*sin(sqrt(a)*z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">mathieucprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">(-a + 2*q*cos(2*z))*mathieuc(a, q, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>mathieus: Mathieu sine function\nmathieuc: Mathieu cosine function\nmathieusprime: Derivative of Mathieu sine function</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Mathieu_function\n.. [2] http://dlmf.nist.gov/28\n.. [3] http://mathworld.wolfram.com/MathieuBase.html\n.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/</p>\n", "bases": "sympy.functions.special.mathieu_functions.mathieucprime, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieucprime.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieucprime.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieucprime.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieucprime.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.riemann_xi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "riemann_xi", "type": "class", "doc": "<p>Riemann Xi function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>The Riemann Xi function is closely related to the Riemann zeta function.\nThe zeros of Riemann Xi function are precisely the non-trivial zeros\nof the zeta function.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">riemann_xi</span><span class=\"p\">,</span> <span class=\"n\">zeta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">riemann_xi</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">zeta</span><span class=\"p\">)</span>\n<span class=\"go\">s*(s - 1)*gamma(s/2)*zeta(s)/(2*pi**(s/2))</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Riemann_Xi_function</p>\n", "bases": "sympy.functions.special.zeta_functions.riemann_xi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.riemann_xi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "riemann_xi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.riemann_xi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "riemann_xi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc", "type": "class", "doc": "<p>The Generalized Incomplete Beta function is defined as</p>\n\n<p>.. math::\n    \\mathrm{B}_{(x_1, x_2)}(a, b) = \\int_{x_1}^{x_2} t^{a - 1} (1 - t)^{b - 1} dt</p>\n\n<p>The Incomplete Beta function is a special case\nof the Generalized Incomplete Beta function :</p>\n\n<p>.. math:: \\mathrm{B}_z (a, b) = \\mathrm{B}_{(0, z)}(a, b)</p>\n\n<p>The Incomplete Beta function satisfies :</p>\n\n<p>.. math:: \\mathrm{B}_z (a, b) = (-1)^a \\mathrm{B}_{\\frac{z}{z - 1}} (a, 1 - a - b)</p>\n\n<p>The Beta function is a special case of the Incomplete Beta function :</p>\n\n<p>.. math:: \\mathrm{B}(a, b) = \\mathrm{B}_{1}(a, b)</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">betainc</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;a b x x1 x2&#39;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<p>The Generalized Incomplete Beta function is given by:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">)</span>\n<span class=\"go\">betainc(a, b, x1, x2)</span>\n</code></pre></div>\n\n<p>The Incomplete Beta function can be obtained as follows:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">betainc(a, b, 0, x)</span>\n</code></pre></div>\n\n<p>The Incomplete Beta function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">))</span>\n<span class=\"go\">betainc(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Incomplete Beta function to\narbitrary precision for any complex numbers a, b, x1 and x2:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">betainc</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)</span>\n<span class=\"go\">56.08333333</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"mf\">0.75</span><span class=\"p\">,</span> <span class=\"mi\">1</span> <span class=\"o\">-</span> <span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">25</span><span class=\"p\">)</span>\n<span class=\"go\">0.2241657956955709603655887 + 0.3619619242700451992411724*I</span>\n</code></pre></div>\n\n<p>The Generalized Incomplete Beta function can be expressed\nin terms of the Generalized Hypergeometric function.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">(-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/a</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>beta: Beta function\nhyper: Generalized Hypergeometric function</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function\n.. [2] https://dlmf.nist.gov/8.17\n.. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/\n.. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/</p>\n", "bases": "sympy.functions.special.beta_functions.betainc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc_regularized", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc_regularized", "type": "class", "doc": "<p>The Generalized Regularized Incomplete Beta function is given by</p>\n\n<p>.. math::\n    \\mathrm{I}_{(x_1, x_2)}(a, b) = \\frac{\\mathrm{B}_{(x_1, x_2)}(a, b)}{\\mathrm{B}(a, b)}</p>\n\n<p>The Regularized Incomplete Beta function is a special case\nof the Generalized Regularized Incomplete Beta function :</p>\n\n<p>.. math:: \\mathrm{I}_z (a, b) = \\mathrm{I}_{(0, z)}(a, b)</p>\n\n<p>The Regularized Incomplete Beta function is the cumulative distribution\nfunction of the beta distribution.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">betainc_regularized</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;a b x x1 x2&#39;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<p>The Generalized Regularized Incomplete Beta\nfunction is given by:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">)</span>\n<span class=\"go\">betainc_regularized(a, b, x1, x2)</span>\n</code></pre></div>\n\n<p>The Regularized Incomplete Beta function\ncan be obtained as follows:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">betainc_regularized(a, b, 0, x)</span>\n</code></pre></div>\n\n<p>The Regularized Incomplete Beta function\nobeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">))</span>\n<span class=\"go\">betainc_regularized(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Regularized Incomplete Beta function\nto arbitrary precision for any complex numbers a, b, x1 and x2:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">betainc_regularized</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">E</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mf\">0.25</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)</span>\n<span class=\"go\">0.4375000000</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000</span>\n</code></pre></div>\n\n<p>The Generalized Regularized Incomplete Beta function can be\nexpressed in terms of the Generalized Hypergeometric function.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">(-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/(a*beta(a, b))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>beta: Beta function\nhyper: Generalized Hypergeometric function</p>\n\n<h1 id=\"references\">References</h1>\n\n<p>.. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function\n.. [2] https://dlmf.nist.gov/8.17\n.. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/\n.. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/</p>\n", "bases": "sympy.functions.special.beta_functions.betainc_regularized, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc_regularized.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc_regularized.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc_regularized.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc_regularized.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}];
+    /** pdoc search index */const docs = [{"fullname": "algebra_with_sympy", "modulename": "algebra_with_sympy", "type": "module", "doc": "<h2 id=\"algebraic-equations-with-sympy\">Algebraic Equations with SymPy</h2>\n\n<p><a href=\"#introduction\">Introduction</a> | <a href=\"#controlling-the-format-of-interactive-outputs\">Output Formatting</a>\n| <a href=\"#setupinstallation\">Installation</a> |\n<a href=\"#try-in-binder\">Try Live</a> | <a href=\"#issues-or-comments\">Issues or Comments</a> |\n<a href=\"#change-log\">Change Log</a> |\n<a href=\"#this-software-is-distributed-under-the-gnu-v3-licensehttpsgnuorglicenses\">License</a>\n| <a href=\"https://github.com/gutow/Algebra_with_Sympy\">GIT Repository</a>\n| <a href=\"https://pypi.org/project/Algebra-with-SymPy/\">PyPi Link</a></p>\n\n<h4 id=\"websitedocumentation-including-apihttpsgutowgithubioalgebra_with_sympy\"><a href=\"https://gutow.github.io/Algebra_with_Sympy/\">Website/Documentation (including API)</a></h4>\n\n<h4 id=\"introduction\">Introduction</h4>\n\n<p>This tool defines relations that all high school and college students would\nrecognize as mathematical equations. \nThey consist of a left hand side (lhs) and a right hand side (rhs) connected by\nthe relation operator \"=\".</p>\n\n<p>This tool applies operations to both sides of the equation simultaneously, just\nas students are taught to do when \nattempting to isolate (solve for) a variable. Thus the statement <code>Equation/b</code>\nyields a new equation <code>Equation.lhs/b = Equation.rhs/b</code></p>\n\n<p>The intent is to allow using the mathematical tools in SymPy to rearrange\nequations and perform algebra\nin a stepwise fashion using as close to standard mathematical notation as \npossible. In this way more people can successfully perform \nalgebraic rearrangements without stumbling\nover missed details such as a negative sign.</p>\n\n<p>A simple example as it would appear in a <a href=\"https://jupyter.org\">Jupyter</a> \nnotebook is shown immediately below:\n<img src=\"https://gutow.github.io/Algebra_with_Sympy/resources/simple_example.png\" alt=\"screenshot of simple example\" /></p>\n\n<p>Once the algebra is complete it is possible to substitute numbers with \nunits into the solved equation to calculate a numerical solution with \nproper units.</p>\n\n<p>In IPython environments (IPython and Jupyter) there is also a shorthand \nsyntax for entering equations provided through the IPython preparser. An \nequation can be specified as <code>eq1 =@ a/b = c/d</code>. If no Python name is \nspecified for the equation (<code>eq1</code> left out in the previous example), the \nequation will still \nbe defined, but will not be easily accessible for further computation. The \n<code>=@</code> symbol combination was chosen to avoid conflicts with reserved python \nsymbols while minimizing impacts on syntax highlighting and autoformatting. A \nscreenshot of an example in Jupyter is shown immediately below:</p>\n\n<p><img src=\"https://gutow.github.io/Algebra_with_Sympy/resources/short_syntax.png\" alt=\"screenshot of short syntax\" /></p>\n\n<p><a href=\"https://gutow.github.io/Algebra_with_Sympy/Demonstration%20of%20equation%20class.html\">More examples of the capabilities of Algebra with Sympy are \nhere</a>.</p>\n\n<p>Many math packages such as <a href=\"https://www.sagemath.org/\">SageMath</a> \nand <a href=\"http://maxima.sourceforge.net/\">Maxima</a> have similar capabilities, \nbut require more knowledge of command syntax, plus they cannot easily be \ninstalled in a generic python environment.</p>\n\n<h4 id=\"controlling-the-format-of-interactive-outputs\">Controlling the Format of Interactive Outputs</h4>\n\n<ul>\n<li><p><strong>In graphical environments (Jupyter)</strong> you will get rendered Latex such as \n$\\frac{a}{b} = \\frac{c}{d}$. To also see the code representation (what can \nbe copied and pasted for \nadditional computation) set <code>algwsym_config.output.show_code = True</code>. \nThis will print the code version (e.g. <code>Equation(a,b/c)</code>) of the equation as \nwell. This code version can be accessed directly by calling <code>repr()</code> on the \nequation.</p></li>\n<li><p><strong>In interactive text environments (ipython and command line)</strong> the \nrepresentation (code version) is returned by default. Calling <code>print()</code> \nor <code>str()</code> on an equation will return the human readable version with an \nequals sign. To have the human readable version returned by default set \n<code>algwsym_config.output.human_text = True</code>. If combined with \n<code>algwsym_config.output.show_code = True</code>, both code and human readable \nversions will be shown.</p></li>\n<li><p><strong>The equation label</strong> can be turned off by setting\n<code>algwsym_config.output.label = False</code>.</p></li>\n</ul>\n\n<h4 id=\"setupinstallation\">Setup/Installation</h4>\n\n<ol>\n<li>Use pip to install in your python environment: \n<code>pip install -U Algebra-with-SymPy</code></li>\n<li>To use in a running python session issue\nthe following command : <code>from algebra_with_sympy import *</code>. \nThis will also import the SymPy tools. </li>\n<li>If you want to isolate this tool from the global namespace you are \nworking with change the import statement \nto <code>import algebra_with_sympy as spa</code>, where \n<code>spa</code> stands for \"SymPy Algebra\". Then all calls would be made to <code>\nspa.funcname()</code>.</li>\n</ol>\n\n<h4 id=\"try-in-binder\">Try in binder</h4>\n\n<p><a href=\"https://mybinder.org/v2/gh/gutow/Algebra_with_Sympy.git/master/?urlpath=/tree/Demonstration%20of%20equation%20class.ipynb\"><img src=\"https://mybinder.org/badge_logo.svg\" alt=\"Binder\" /></a></p>\n\n<h4 id=\"issues-or-comments\">Issues or Comments</h4>\n\n<ul>\n<li>Issues and bug reports should be <a href=\"https://github.com/gutow/Algebra_with_Sympy/issues\">filed on \ngithub</a>.</li>\n<li>Comments, questions, show and tell, etc. should go in the <a href=\"https://github.com/gutow/Algebra_with_Sympy/discussions\">project \ndiscussions</a>.</li>\n</ul>\n\n<h4 id=\"change-log\">Change Log</h4>\n\n<ul>\n<li>0.9.2\n<ul>\n<li><code>=@</code> shorthand syntax for defining equations in IPython compatible \nenvironments.</li>\n<li>Fixed bug where <code>root()</code> override called <code>sqrt()</code> on bare expressions.</li>\n</ul></li>\n<li>0.9.1\n<ul>\n<li>Equations labeled with their python name, if they have one.</li>\n<li>Added flags to adjust human readable output and equation labeling.</li>\n<li>Accept equation as function argument in any position.</li>\n<li>First pass at <code>solve()</code> accepting equations.</li>\n<li>Added override of <code>root()</code> to avoid warning messages.</li>\n<li>More unit tests.</li>\n<li>First pass at documentation.</li>\n</ul></li>\n<li>0.9.0 functionality equivalent to extension of SymPy in\n<a href=\"https://github.com/sympy/sympy/pull/21333\">PR#21333</a>.</li>\n</ul>\n\n<h5 id=\"this-software-is-distributed-under-the-gnu-v3-licensehttpsgnuorglicenses\"><a href=\"https://gnu.org/licenses\">This software is distributed under the GNU V3 license</a></h5>\n\n<p>This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU General Public License as published by\n    the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n    GNU General Public License for more details.</p>\n\n<p>Copyright - Jonathan Gutow 2021, 2022</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation", "modulename": "algebra_with_sympy.algebraic_equation", "type": "module", "doc": "<h1 id=\"algebraic-equations-with-sympy\">Algebraic Equations with SymPy</h1>\n\n<p>These tools define relations that all high school and college students would\nrecognize as mathematical equations. They consist of a left hand side (lhs)\nand a right hand side (rhs) connected by a relation operator such as \"=\". At\npresent the \"=\" relation operator is the only option. The relation operator may\nnot be set.</p>\n\n<p>This class should not be confused with the Boolean class <code>Equality</code>\n(abbreviated <code>Eq</code>) which specifies that the equality of two objects is\n<code>True</code>.</p>\n\n<p>This tool applies operations to both sides of the equation simultaneously, just\nas students are taught to do when attempting to isolate (solve for) a\nvariable. Thus the statement <code>Equation/b</code> yields a new equation\n<code>Equation.lhs/b = Equation.rhs/b</code></p>\n\n<p>The intent is to allow using the mathematical tools in SymPy to rearrange\nequations and perform algebra in a stepwise fashion. In this way more people\ncan successfully perform algebraic rearrangements without stumbling over\nmissed details such as a negative sign. This mimics the capabilities available\nin <a href=\"https://www.sagemath.org/\">SageMath</a> and\n<a href=\"http://maxima.sourceforge.net/\">Maxima</a>.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config", "type": "class", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.__init__", "type": "function", "doc": "<p>This is a class to hold parameters that control behavior of\nthe algebra_with_sympy package.</p>\n\n<h1 id=\"settings\">Settings</h1>\n\n<h2 id=\"printing\">Printing</h2>\n\n<p>In interactive environments the default output of an equation is a\nhuman readable string with the two sides connected by an equals\nsign or a typeset equation with the two sides connected by an equals sign.\n<code>print(Eqn)</code> or <code>str(Eqn)</code> will return this human readable text version of\nthe equation as well. This is consistent with python standards, but not\nsympy, where <code>str()</code> is supposed to return something that can be\ncopy-pasted into code. If the equation has a declared name as in <code>eq1 =\nEqn(a,b/c)</code> the name will be displayed to the right of the equation in\nparentheses (eg. <code>a = b/c    (eq1)</code>). Use <code>print(repr(Eqn))</code> instead of\n<code>print(Eqn)</code> or <code>repr(Eqn)</code> instead of <code>str(Eqn)</code> to get a code\ncompatible version of the equation.</p>\n\n<p>You can adjust this behvior using some flags that impact output:</p>\n\n<ul>\n<li><code>algwsym_config.output.show_code</code> default is <code>False</code>.</li>\n<li><code>algwsym_config.output.human_text</code> default is <code>False</code>.</li>\n<li><code>algwsym_config.output.label</code> default is <code>True</code>.</li>\n</ul>\n\n<p>In interactive environments you can get both types of output by setting\nthe <code>algwsym_config.output.show_code</code> flag. If this flag is true\ncalls to <code>latex</code> and <code>str</code> will also print an additional line \"code\nversion: <code>repr(Eqn)</code>\". Thus in Jupyter you will get a line of typeset\nmathematics output preceded by the code version that can be copy-pasted.\nDefault is <code>False</code>.</p>\n\n<p>A second flag <code>algwsym_config.output.human_text</code> is useful in\ntext-based interactive environments such as command line python or\nipython. If this flag is true <code>repr</code> will return <code>str</code>. Thus the human\nreadable text will be printed as the output of a line that is an\nexpression containing an equation.\nDefault is <code>False</code>.</p>\n\n<p>Setting both of these flags to true in a command line or ipython\nenvironment will show both the code version and the human readable text.\nThese flags impact the behavior of the <code>print(Eqn)</code> statement.</p>\n\n<p>The third flag <code>algwsym_config.output.label</code> has a default value of\n<code>True</code>. Setting this to <code>False</code> suppresses the labeling of an equation\nwith its python name off to the right of the equation.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output", "type": "class", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.__init__", "type": "function", "doc": "<p>This holds settings that impact output.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.show_code", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.show_code", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.human_text", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.human_text", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.label", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.label", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation", "type": "class", "doc": "<p>This class defines an equation with a left-hand-side (lhs) and a right-\nhand-side (rhs) connected by the \"=\" operator (e.g. <code>p*V = n*R*T</code>).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This class defines relations that all high school and college students\nwould recognize as mathematical equations. At present only the \"=\" relation\noperator is recognized.</p>\n\n<p>This class is intended to allow using the mathematical tools in SymPy to\nrearrange equations and perform algebra in a stepwise fashion. In this\nway more people can successfully perform algebraic rearrangements without\nstumbling over missed details such as a negative sign.</p>\n\n<p>Create an equation with the call <code>Equation(lhs,rhs)</code>, where <code>lhs</code> and\n<code>rhs</code> are any valid Sympy expression. <code>Eqn(...)</code> is a synonym for\n<code>Equation(...)</code>.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>lhs: sympy expression, <code>class Expr</code>.\nrhs: sympy expression, <code>class Expr</code>.\nkwargs:</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">algebra_with_sympy</span> <span class=\"kn\">import</span> <span class=\"o\">*</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;a b c x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Equation</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">*</span><span class=\"n\">c</span>\n<span class=\"go\">Equation(a*c, b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a*c, b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(exp(a), exp(b/c))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(a, b/c)</span>\n</code></pre></div>\n\n<p>Simplification and Expansion</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span>\n<span class=\"go\">Equation(x**2 - 1, c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">simplify</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(x - 1, c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(x - 1, c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factor</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"p\">)</span>\n<span class=\"go\">Equation((x - 1)*(x + 1), c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span><span class=\"o\">.</span><span class=\"n\">factor</span><span class=\"p\">()</span>\n<span class=\"go\">Equation((x - 1)*(x + 1), c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f2</span> <span class=\"o\">=</span> <span class=\"n\">f</span><span class=\"o\">+</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">+</span><span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span><span class=\"n\">c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f2</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">f2</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>\n</code></pre></div>\n\n<p>Apply operation to only one side</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applyrhs</span><span class=\"p\">(</span><span class=\"n\">factor</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">factor</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">collect</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>\n</code></pre></div>\n\n<p><code>.apply...</code> also works with user defined python functions</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">def</span> <span class=\"nf\">addsquare</span><span class=\"p\">(</span><span class=\"n\">eqn</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span>    <span class=\"k\">return</span> <span class=\"n\">eqn</span><span class=\"o\">+</span><span class=\"n\">eqn</span><span class=\"o\">**</span><span class=\"mi\">2</span>\n<span class=\"gp\">...</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">apply</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">applyrhs</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">apply</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">,</span> <span class=\"n\">side</span> <span class=\"o\">=</span> <span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">addsquare</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b**2/c**2 + b/c)</span>\n</code></pre></div>\n\n<p>Inaddition to <code>.apply...</code> there is also the less general <code>.do</code>,\n<code>.dolhs</code>, <code>.dorhs</code>, which only works for operations defined on the\n<code>Expr</code> class (e.g.<code>.collect(), .factor(), .expand()</code>, etc...).</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dolhs</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">do</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">factor</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>\n</code></pre></div>\n\n<p><code>poly.do.exp()</code> or other sympy math functions will raise an error.</p>\n\n<p>Rearranging an equation (simple example made complicated as illustration)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span><span class=\"p\">,</span> <span class=\"n\">V</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">R</span><span class=\"p\">,</span> <span class=\"n\">T</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;p V n R T&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq1</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">p</span><span class=\"o\">*</span><span class=\"n\">V</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"o\">*</span><span class=\"n\">R</span><span class=\"o\">*</span><span class=\"n\">T</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq1</span>\n<span class=\"go\">Equation(V*p, R*T*n)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span> <span class=\"o\">=</span><span class=\"n\">eq1</span><span class=\"o\">/</span><span class=\"n\">V</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span>\n<span class=\"go\">Equation(p, R*T*n/V)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq3</span> <span class=\"o\">=</span> <span class=\"n\">eq2</span><span class=\"o\">/</span><span class=\"n\">R</span><span class=\"o\">/</span><span class=\"n\">T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq3</span>\n<span class=\"go\">Equation(p/(R*T), n/V)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq4</span> <span class=\"o\">=</span> <span class=\"n\">eq3</span><span class=\"o\">*</span><span class=\"n\">R</span><span class=\"o\">/</span><span class=\"n\">p</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq4</span>\n<span class=\"go\">Equation(1/T, R*n/(V*p))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">eq4</span>\n<span class=\"go\">Equation(T, V*p/(R*n))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq5</span> <span class=\"o\">=</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">eq4</span> <span class=\"o\">-</span> <span class=\"n\">T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq5</span>\n<span class=\"go\">Equation(0, -T + V*p/(R*n))</span>\n</code></pre></div>\n\n<p>Substitution (#'s and units)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span><span class=\"p\">,</span> <span class=\"n\">atm</span><span class=\"p\">,</span> <span class=\"n\">mol</span><span class=\"p\">,</span> <span class=\"n\">K</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;L atm mol K&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span> <span class=\"c1\"># units</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">R</span><span class=\"p\">:</span><span class=\"mf\">0.08206</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"o\">*</span><span class=\"n\">atm</span><span class=\"o\">/</span><span class=\"n\">mol</span><span class=\"o\">/</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">T</span><span class=\"p\">:</span><span class=\"mi\">273</span><span class=\"o\">*</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"p\">:</span><span class=\"mf\">1.00</span><span class=\"o\">*</span><span class=\"n\">mol</span><span class=\"p\">,</span><span class=\"n\">V</span><span class=\"p\">:</span><span class=\"mf\">24.0</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"p\">})</span>\n<span class=\"go\">Equation(p, 0.9334325*atm)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">R</span><span class=\"p\">:</span><span class=\"mf\">0.08206</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"o\">*</span><span class=\"n\">atm</span><span class=\"o\">/</span><span class=\"n\">mol</span><span class=\"o\">/</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">T</span><span class=\"p\">:</span><span class=\"mi\">273</span><span class=\"o\">*</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"p\">:</span><span class=\"mf\">1.00</span><span class=\"o\">*</span><span class=\"n\">mol</span><span class=\"p\">,</span><span class=\"n\">V</span><span class=\"p\">:</span><span class=\"mf\">24.0</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"p\">})</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(p, 0.9334*atm)</span>\n</code></pre></div>\n\n<p>Combining equations (Math with equations: lhs with lhs and rhs with rhs)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a*c, b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">+</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a*c + a, b/c + b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">/</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(c, 1/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">**</span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a**(a*c), (b/c)**(b/c**2))</span>\n</code></pre></div>\n\n<p>Utility operations</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">reversed</span>\n<span class=\"go\">Equation(b/c, a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">swap</span>\n<span class=\"go\">Equation(b/c, a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">lhs</span>\n<span class=\"go\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">rhs</span>\n<span class=\"go\">b/c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">as_Boolean</span><span class=\"p\">()</span>\n<span class=\"go\">Eq(a, b/c)</span>\n</code></pre></div>\n\n<p><code>.check()</code> convenience method for <code>.as_Boolean().simplify()</code></p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Equation</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">+</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">+</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">check</span><span class=\"p\">()</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">a</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">check</span><span class=\"p\">()</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<p>Differentiation\nDifferentiation is applied to both sides if the wrt variable appears on\nboth sides.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a*c, b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b), c**(-2))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, -2*b/c**3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">),</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(log(a*c), b), 1/b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a, c), 6*b/c**4)</span>\n</code></pre></div>\n\n<p>If you specify multiple differentiation all at once the assumption\nis order of differentiation matters and the lhs will not be\nevaluated.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b, c), -2/c**3)</span>\n</code></pre></div>\n\n<p>To overcome this specify the order of operations.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">),</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a, b), -2/c**3)</span>\n</code></pre></div>\n\n<p>But the reverse order returns an unevaulated lhs (a may depend on b).</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">),</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b, c), -2/c**3)</span>\n</code></pre></div>\n\n<p>Integration can only be performed on one side at a time.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">b**2/(2*c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;lhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">a*b*c</span>\n</code></pre></div>\n\n<p>Make a pretty statement of integration from an equation</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">Integral</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"o\">.</span><span class=\"n\">lhs</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">),</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(Integral(a*c, b), b**2/(2*c))</span>\n</code></pre></div>\n\n<p>Integration of each side with respect to different variables</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">b</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">dolhs</span><span class=\"o\">.</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2*c/2, b**2/(2*c))</span>\n</code></pre></div>\n\n<p>SymPy's solvers do not understand these equations. They expect an\nexpression that the solver assumes = 0. Thus to use the solver the\nequation must be rearranged so that all non-zero symbols are on one side.\nThen just the non-zero symbolic side is passed to <code>solve()</code>.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t2</span> <span class=\"o\">=</span> <span class=\"n\">t</span><span class=\"o\">-</span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">rhs</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t2</span>\n<span class=\"go\">Equation(a - b/c, 0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">solve</span><span class=\"p\">(</span><span class=\"n\">t2</span><span class=\"o\">.</span><span class=\"n\">lhs</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">[b/a]</span>\n</code></pre></div>\n", "bases": "sympy.core.basic.Basic, sympy.core.evalf.EvalfMixin"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.lhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.lhs", "type": "variable", "doc": "<p>Returns the lhs of the equation.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.rhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.rhs", "type": "variable", "doc": "<p>Returns the rhs of the equation.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.as_Boolean", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.as_Boolean", "type": "function", "doc": "<p>Converts the equation to an Equality.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.check", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.check", "type": "function", "doc": "<p>Forces simplification and casts as <code>Equality</code> to check validity.</p>\n\n<h2 id=\"parameters\">Parameters</h2>\n\n<p>kwargs any appropriate for <code>Equality</code>.</p>\n\n<h2 id=\"returns\">Returns</h2>\n\n<p>True, False or an unevaluated <code>Equality</code> if truth cannot be determined.</p>\n", "signature": "(self, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.reversed", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.reversed", "type": "variable", "doc": "<p>Swaps the lhs and the rhs.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.swap", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.swap", "type": "variable", "doc": "<p>Synonym for <code>.reversed</code></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.apply", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.apply", "type": "function", "doc": "<p>Apply an operation/function/method to the equation returning the\nresulting equation.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>func: object\n    object to apply usually a function</p>\n\n<p>args: as necessary for the function</p>\n\n<p>side: 'both', 'lhs', 'rhs', optional\n    Specifies which side of the equation the operation will be applied\n    to. Default is 'both'.</p>\n\n<p>kwargs: as necessary for the function</p>\n", "signature": "(self, func, *args, side='both', **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.applylhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.applylhs", "type": "function", "doc": "<p>If lhs side of the equation has a defined subfunction (attribute) of\nname <code>func</code>, that will be applied instead of the global function.\nThe operation is applied to only the lhs.</p>\n", "signature": "(self, func, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.applyrhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.applyrhs", "type": "function", "doc": "<p>If rhs side of the equation has a defined subfunction (attribute) of\nname <code>func</code>, that will be applied instead of the global function.\nThe operation is applied to only the rhs.</p>\n", "signature": "(self, func, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.do", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.do", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.dolhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.dolhs", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.dorhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.dorhs", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.expand", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.expand", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.simplify", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.simplify", "type": "function", "doc": "<p>See the simplify function in sympy.simplify</p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.factor", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.factor", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.collect", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.collect", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.evalf", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.evalf", "type": "function", "doc": "<p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>subs : dict, optional\n    Substitute numerical values for symbols, e.g.\n    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a\n    dictionary.</p>\n\n<p>maxn : int, optional\n    Allow a maximum temporary working precision of maxn digits.</p>\n\n<p>chop : bool or number, optional\n    Specifies how to replace tiny real or imaginary parts in\n    subresults by exact zeros.</p>\n\n<pre><code>When ``True`` the chop value defaults to standard precision.\n\nOtherwise the chop value is used to determine the\nmagnitude of \"small\" for purposes of chopping.\n\n&gt;&gt;&gt; from sympy import N\n&gt;&gt;&gt; x = 1e-4\n&gt;&gt;&gt; N(x, chop=True)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-5)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-4)\n0\n</code></pre>\n\n<p>strict : bool, optional\n    Raise <code>PrecisionExhausted</code> if any subresult fails to\n    evaluate to full accuracy, given the available maxprec.</p>\n\n<p>quad : str, optional\n    Choose algorithm for numerical quadrature. By default,\n    tanh-sinh quadrature is used. For oscillatory\n    integrals on an infinite interval, try <code>quad='osc'</code>.</p>\n\n<p>verbose : bool, optional\n    Print debug information.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>When Floats are naively substituted into an expression,\nprecision errors may adversely affect the result. For example,\nadding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is\nthen subtracted, the result will be 0.\nThat is exactly what happens in the following:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">values</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">}</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Using the subs argument for evalf is the accurate way to\nevaluate such an expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"n\">subs</span><span class=\"o\">=</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000000000000</span>\n</code></pre></div>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.n", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.n", "type": "function", "doc": "<p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>subs : dict, optional\n    Substitute numerical values for symbols, e.g.\n    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a\n    dictionary.</p>\n\n<p>maxn : int, optional\n    Allow a maximum temporary working precision of maxn digits.</p>\n\n<p>chop : bool or number, optional\n    Specifies how to replace tiny real or imaginary parts in\n    subresults by exact zeros.</p>\n\n<pre><code>When ``True`` the chop value defaults to standard precision.\n\nOtherwise the chop value is used to determine the\nmagnitude of \"small\" for purposes of chopping.\n\n&gt;&gt;&gt; from sympy import N\n&gt;&gt;&gt; x = 1e-4\n&gt;&gt;&gt; N(x, chop=True)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-5)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-4)\n0\n</code></pre>\n\n<p>strict : bool, optional\n    Raise <code>PrecisionExhausted</code> if any subresult fails to\n    evaluate to full accuracy, given the available maxprec.</p>\n\n<p>quad : str, optional\n    Choose algorithm for numerical quadrature. By default,\n    tanh-sinh quadrature is used. For oscillatory\n    integrals on an infinite interval, try <code>quad='osc'</code>.</p>\n\n<p>verbose : bool, optional\n    Print debug information.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>When Floats are naively substituted into an expression,\nprecision errors may adversely affect the result. For example,\nadding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is\nthen subtracted, the result will be 0.\nThat is exactly what happens in the following:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">values</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">}</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Using the subs argument for evalf is the accurate way to\nevaluate such an expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"n\">subs</span><span class=\"o\">=</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000000000000</span>\n</code></pre></div>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Equation.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equation.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn", "type": "class", "doc": "<p>This class defines an equation with a left-hand-side (lhs) and a right-\nhand-side (rhs) connected by the \"=\" operator (e.g. <code>p*V = n*R*T</code>).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This class defines relations that all high school and college students\nwould recognize as mathematical equations. At present only the \"=\" relation\noperator is recognized.</p>\n\n<p>This class is intended to allow using the mathematical tools in SymPy to\nrearrange equations and perform algebra in a stepwise fashion. In this\nway more people can successfully perform algebraic rearrangements without\nstumbling over missed details such as a negative sign.</p>\n\n<p>Create an equation with the call <code>Equation(lhs,rhs)</code>, where <code>lhs</code> and\n<code>rhs</code> are any valid Sympy expression. <code>Eqn(...)</code> is a synonym for\n<code>Equation(...)</code>.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>lhs: sympy expression, <code>class Expr</code>.\nrhs: sympy expression, <code>class Expr</code>.\nkwargs:</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">algebra_with_sympy</span> <span class=\"kn\">import</span> <span class=\"o\">*</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;a b c x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Equation</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">*</span><span class=\"n\">c</span>\n<span class=\"go\">Equation(a*c, b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a*c, b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(exp(a), exp(b/c))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(a, b/c)</span>\n</code></pre></div>\n\n<p>Simplification and Expansion</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span>\n<span class=\"go\">Equation(x**2 - 1, c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">Equation((x**2 - 1)/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">simplify</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(x - 1, c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(x - 1, c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"o\">/</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factor</span><span class=\"p\">(</span><span class=\"n\">f</span><span class=\"p\">)</span>\n<span class=\"go\">Equation((x - 1)*(x + 1), c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span><span class=\"o\">.</span><span class=\"n\">factor</span><span class=\"p\">()</span>\n<span class=\"go\">Equation((x - 1)*(x + 1), c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f2</span> <span class=\"o\">=</span> <span class=\"n\">f</span><span class=\"o\">+</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">+</span><span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span><span class=\"n\">c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f2</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">f2</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)</span>\n</code></pre></div>\n\n<p>Apply operation to only one side</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">b</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">c</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applyrhs</span><span class=\"p\">(</span><span class=\"n\">factor</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">factor</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">collect</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>\n</code></pre></div>\n\n<p><code>.apply...</code> also works with user defined python functions</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">def</span> <span class=\"nf\">addsquare</span><span class=\"p\">(</span><span class=\"n\">eqn</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span>    <span class=\"k\">return</span> <span class=\"n\">eqn</span><span class=\"o\">+</span><span class=\"n\">eqn</span><span class=\"o\">**</span><span class=\"mi\">2</span>\n<span class=\"gp\">...</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">apply</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">applyrhs</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">apply</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">,</span> <span class=\"n\">side</span> <span class=\"o\">=</span> <span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, b**2/c**2 + b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">applylhs</span><span class=\"p\">(</span><span class=\"n\">addsquare</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">addsquare</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2 + a, b**2/c**2 + b/c)</span>\n</code></pre></div>\n\n<p>Inaddition to <code>.apply...</code> there is also the less general <code>.do</code>,\n<code>.dolhs</code>, <code>.dorhs</code>, which only works for operations defined on the\n<code>Expr</code> class (e.g.<code>.collect(), .factor(), .expand()</code>, etc...).</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dolhs</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">do</span><span class=\"o\">.</span><span class=\"n\">collect</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(b*x + x**2*(a + c), c*x + x**3*(a + b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">poly</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">factor</span><span class=\"p\">()</span>\n<span class=\"go\">Equation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))</span>\n</code></pre></div>\n\n<p><code>poly.do.exp()</code> or other sympy math functions will raise an error.</p>\n\n<p>Rearranging an equation (simple example made complicated as illustration)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span><span class=\"p\">,</span> <span class=\"n\">V</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">R</span><span class=\"p\">,</span> <span class=\"n\">T</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;p V n R T&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq1</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">p</span><span class=\"o\">*</span><span class=\"n\">V</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"o\">*</span><span class=\"n\">R</span><span class=\"o\">*</span><span class=\"n\">T</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq1</span>\n<span class=\"go\">Equation(V*p, R*T*n)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span> <span class=\"o\">=</span><span class=\"n\">eq1</span><span class=\"o\">/</span><span class=\"n\">V</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span>\n<span class=\"go\">Equation(p, R*T*n/V)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq3</span> <span class=\"o\">=</span> <span class=\"n\">eq2</span><span class=\"o\">/</span><span class=\"n\">R</span><span class=\"o\">/</span><span class=\"n\">T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq3</span>\n<span class=\"go\">Equation(p/(R*T), n/V)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq4</span> <span class=\"o\">=</span> <span class=\"n\">eq3</span><span class=\"o\">*</span><span class=\"n\">R</span><span class=\"o\">/</span><span class=\"n\">p</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq4</span>\n<span class=\"go\">Equation(1/T, R*n/(V*p))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">eq4</span>\n<span class=\"go\">Equation(T, V*p/(R*n))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq5</span> <span class=\"o\">=</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">eq4</span> <span class=\"o\">-</span> <span class=\"n\">T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq5</span>\n<span class=\"go\">Equation(0, -T + V*p/(R*n))</span>\n</code></pre></div>\n\n<p>Substitution (#'s and units)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span><span class=\"p\">,</span> <span class=\"n\">atm</span><span class=\"p\">,</span> <span class=\"n\">mol</span><span class=\"p\">,</span> <span class=\"n\">K</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;L atm mol K&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span> <span class=\"c1\"># units</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">R</span><span class=\"p\">:</span><span class=\"mf\">0.08206</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"o\">*</span><span class=\"n\">atm</span><span class=\"o\">/</span><span class=\"n\">mol</span><span class=\"o\">/</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">T</span><span class=\"p\">:</span><span class=\"mi\">273</span><span class=\"o\">*</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"p\">:</span><span class=\"mf\">1.00</span><span class=\"o\">*</span><span class=\"n\">mol</span><span class=\"p\">,</span><span class=\"n\">V</span><span class=\"p\">:</span><span class=\"mf\">24.0</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"p\">})</span>\n<span class=\"go\">Equation(p, 0.9334325*atm)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">eq2</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">R</span><span class=\"p\">:</span><span class=\"mf\">0.08206</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"o\">*</span><span class=\"n\">atm</span><span class=\"o\">/</span><span class=\"n\">mol</span><span class=\"o\">/</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">T</span><span class=\"p\">:</span><span class=\"mi\">273</span><span class=\"o\">*</span><span class=\"n\">K</span><span class=\"p\">,</span><span class=\"n\">n</span><span class=\"p\">:</span><span class=\"mf\">1.00</span><span class=\"o\">*</span><span class=\"n\">mol</span><span class=\"p\">,</span><span class=\"n\">V</span><span class=\"p\">:</span><span class=\"mf\">24.0</span><span class=\"o\">*</span><span class=\"n\">L</span><span class=\"p\">})</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(p, 0.9334*atm)</span>\n</code></pre></div>\n\n<p>Combining equations (Math with equations: lhs with lhs and rhs with rhs)</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span> <span class=\"o\">=</span> <span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a*c, b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a, b/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">+</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(a*c + a, b/c + b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">/</span><span class=\"n\">t</span>\n<span class=\"go\">Equation(c, 1/c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">**</span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a**(a*c), (b/c)**(b/c**2))</span>\n</code></pre></div>\n\n<p>Utility operations</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">reversed</span>\n<span class=\"go\">Equation(b/c, a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">swap</span>\n<span class=\"go\">Equation(b/c, a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">lhs</span>\n<span class=\"go\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">rhs</span>\n<span class=\"go\">b/c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">as_Boolean</span><span class=\"p\">()</span>\n<span class=\"go\">Eq(a, b/c)</span>\n</code></pre></div>\n\n<p><code>.check()</code> convenience method for <code>.as_Boolean().simplify()</code></p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Equation</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">+</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">+</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">check</span><span class=\"p\">()</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">a</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">check</span><span class=\"p\">()</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<p>Differentiation\nDifferentiation is applied to both sides if the wrt variable appears on\nboth sides.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span>\n<span class=\"go\">Equation(a*c, b/c**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b), c**(-2))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a, -2*b/c**3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">),</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(log(a*c), b), 1/b)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a, c), 6*b/c**4)</span>\n</code></pre></div>\n\n<p>If you specify multiple differentiation all at once the assumption\nis order of differentiation matters and the lhs will not be\nevaluated.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b, c), -2/c**3)</span>\n</code></pre></div>\n\n<p>To overcome this specify the order of operations.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">),</span><span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a, b), -2/c**3)</span>\n</code></pre></div>\n\n<p>But the reverse order returns an unevaulated lhs (a may depend on b).</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">),</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(Derivative(a*c, b, c), -2/c**3)</span>\n</code></pre></div>\n\n<p>Integration can only be performed on one side at a time.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">=</span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">*</span><span class=\"n\">c</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"o\">/</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">b**2/(2*c)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;lhs&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">a*b*c</span>\n</code></pre></div>\n\n<p>Make a pretty statement of integration from an equation</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eqn</span><span class=\"p\">(</span><span class=\"n\">Integral</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"o\">.</span><span class=\"n\">lhs</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">),</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">q</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">side</span><span class=\"o\">=</span><span class=\"s1\">&#39;rhs&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">Equation(Integral(a*c, b), b**2/(2*c))</span>\n</code></pre></div>\n\n<p>Integration of each side with respect to different variables</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">q</span><span class=\"o\">.</span><span class=\"n\">dorhs</span><span class=\"o\">.</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">b</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">dolhs</span><span class=\"o\">.</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">Equation(a**2*c/2, b**2/(2*c))</span>\n</code></pre></div>\n\n<p>SymPy's solvers do not understand these equations. They expect an\nexpression that the solver assumes = 0. Thus to use the solver the\nequation must be rearranged so that all non-zero symbols are on one side.\nThen just the non-zero symbolic side is passed to <code>solve()</code>.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t2</span> <span class=\"o\">=</span> <span class=\"n\">t</span><span class=\"o\">-</span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">rhs</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t2</span>\n<span class=\"go\">Equation(a - b/c, 0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">solve</span><span class=\"p\">(</span><span class=\"n\">t2</span><span class=\"o\">.</span><span class=\"n\">lhs</span><span class=\"p\">,</span><span class=\"n\">c</span><span class=\"p\">)</span>\n<span class=\"go\">[b/a]</span>\n</code></pre></div>\n", "bases": "sympy.core.basic.Basic, sympy.core.evalf.EvalfMixin"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.lhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.lhs", "type": "variable", "doc": "<p>Returns the lhs of the equation.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.rhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.rhs", "type": "variable", "doc": "<p>Returns the rhs of the equation.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.as_Boolean", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.as_Boolean", "type": "function", "doc": "<p>Converts the equation to an Equality.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.check", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.check", "type": "function", "doc": "<p>Forces simplification and casts as <code>Equality</code> to check validity.</p>\n\n<h2 id=\"parameters\">Parameters</h2>\n\n<p>kwargs any appropriate for <code>Equality</code>.</p>\n\n<h2 id=\"returns\">Returns</h2>\n\n<p>True, False or an unevaluated <code>Equality</code> if truth cannot be determined.</p>\n", "signature": "(self, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.reversed", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.reversed", "type": "variable", "doc": "<p>Swaps the lhs and the rhs.</p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.swap", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.swap", "type": "variable", "doc": "<p>Synonym for <code>.reversed</code></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.apply", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.apply", "type": "function", "doc": "<p>Apply an operation/function/method to the equation returning the\nresulting equation.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>func: object\n    object to apply usually a function</p>\n\n<p>args: as necessary for the function</p>\n\n<p>side: 'both', 'lhs', 'rhs', optional\n    Specifies which side of the equation the operation will be applied\n    to. Default is 'both'.</p>\n\n<p>kwargs: as necessary for the function</p>\n", "signature": "(self, func, *args, side='both', **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.applylhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.applylhs", "type": "function", "doc": "<p>If lhs side of the equation has a defined subfunction (attribute) of\nname <code>func</code>, that will be applied instead of the global function.\nThe operation is applied to only the lhs.</p>\n", "signature": "(self, func, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.applyrhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.applyrhs", "type": "function", "doc": "<p>If rhs side of the equation has a defined subfunction (attribute) of\nname <code>func</code>, that will be applied instead of the global function.\nThe operation is applied to only the rhs.</p>\n", "signature": "(self, func, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.do", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.do", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.dolhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.dolhs", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.dorhs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.dorhs", "type": "variable", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.expand", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.expand", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.simplify", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.simplify", "type": "function", "doc": "<p>See the simplify function in sympy.simplify</p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.factor", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.factor", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.collect", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.collect", "type": "function", "doc": "<p></p>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.evalf", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.evalf", "type": "function", "doc": "<p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>subs : dict, optional\n    Substitute numerical values for symbols, e.g.\n    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a\n    dictionary.</p>\n\n<p>maxn : int, optional\n    Allow a maximum temporary working precision of maxn digits.</p>\n\n<p>chop : bool or number, optional\n    Specifies how to replace tiny real or imaginary parts in\n    subresults by exact zeros.</p>\n\n<pre><code>When ``True`` the chop value defaults to standard precision.\n\nOtherwise the chop value is used to determine the\nmagnitude of \"small\" for purposes of chopping.\n\n&gt;&gt;&gt; from sympy import N\n&gt;&gt;&gt; x = 1e-4\n&gt;&gt;&gt; N(x, chop=True)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-5)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-4)\n0\n</code></pre>\n\n<p>strict : bool, optional\n    Raise <code>PrecisionExhausted</code> if any subresult fails to\n    evaluate to full accuracy, given the available maxprec.</p>\n\n<p>quad : str, optional\n    Choose algorithm for numerical quadrature. By default,\n    tanh-sinh quadrature is used. For oscillatory\n    integrals on an infinite interval, try <code>quad='osc'</code>.</p>\n\n<p>verbose : bool, optional\n    Print debug information.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>When Floats are naively substituted into an expression,\nprecision errors may adversely affect the result. For example,\nadding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is\nthen subtracted, the result will be 0.\nThat is exactly what happens in the following:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">values</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">}</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Using the subs argument for evalf is the accurate way to\nevaluate such an expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"n\">subs</span><span class=\"o\">=</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000000000000</span>\n</code></pre></div>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.n", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.n", "type": "function", "doc": "<p>Evaluate the given formula to an accuracy of <em>n</em> digits.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>subs : dict, optional\n    Substitute numerical values for symbols, e.g.\n    <code>subs={x:3, y:1+pi}</code>. The substitutions must be given as a\n    dictionary.</p>\n\n<p>maxn : int, optional\n    Allow a maximum temporary working precision of maxn digits.</p>\n\n<p>chop : bool or number, optional\n    Specifies how to replace tiny real or imaginary parts in\n    subresults by exact zeros.</p>\n\n<pre><code>When ``True`` the chop value defaults to standard precision.\n\nOtherwise the chop value is used to determine the\nmagnitude of \"small\" for purposes of chopping.\n\n&gt;&gt;&gt; from sympy import N\n&gt;&gt;&gt; x = 1e-4\n&gt;&gt;&gt; N(x, chop=True)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-5)\n0.000100000000000000\n&gt;&gt;&gt; N(x, chop=1e-4)\n0\n</code></pre>\n\n<p>strict : bool, optional\n    Raise <code>PrecisionExhausted</code> if any subresult fails to\n    evaluate to full accuracy, given the available maxprec.</p>\n\n<p>quad : str, optional\n    Choose algorithm for numerical quadrature. By default,\n    tanh-sinh quadrature is used. For oscillatory\n    integrals on an infinite interval, try <code>quad='osc'</code>.</p>\n\n<p>verbose : bool, optional\n    Print debug information.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>When Floats are naively substituted into an expression,\nprecision errors may adversely affect the result. For example,\nadding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is\nthen subtracted, the result will be 0.\nThat is exactly what happens in the following:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">values</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">:</span> <span class=\"mf\">1e16</span><span class=\"p\">}</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Using the subs argument for evalf is the accurate way to\nevaluate such an expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"n\">y</span> <span class=\"o\">-</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"n\">subs</span><span class=\"o\">=</span><span class=\"n\">values</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000000000000</span>\n</code></pre></div>\n", "signature": "(self, *args, **kwargs)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eqn.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.solve", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "solve", "type": "function", "doc": "<p>Override of sympy <code>solve()</code>. If it is passed an equation it will\noutput the solutions as typeset equations and return\nthe answer as a list of equations that can be accessed for additional\nmanipulations.</p>\n", "signature": "(f, *symbols, **flags)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sqrt", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sqrt", "type": "function", "doc": "<p>Override of sympy convenience function <code>sqrt</code>. Simply divides equations\ninto two sides if <code>arg</code> is an instance of <code>Equation</code>. This avoids an\nissue with the way sympy is delaying specialized applications of _Pow_ on\nobjects that are not basic sympy expressions.\nReturns the principal square root.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>evaluate : bool, optional\n    The parameter determines if the expression should be evaluated.\n    If <code>None</code>, its value is taken from\n    <code>global_parameters.evaluate</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">**</span><span class=\"mi\">2</span>\n<span class=\"go\">x</span>\n</code></pre></div>\n\n<p>Note that sqrt(x**2) does not simplify to x.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(x**2)</span>\n</code></pre></div>\n\n<p>This is because the two are not equal to each other in general.\nFor example, consider x == -1:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Eq</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Eq</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<p>This is because sqrt computes the principal square root, so the square may\nput the argument in a different branch.  This identity does hold if x is\npositive:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;y&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">y</span>\n</code></pre></div>\n\n<p>You can force this simplification by using the powdenest() function with\nthe force option set to True:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">powdenest</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">powdenest</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">force</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">x</span>\n</code></pre></div>\n\n<p>To get both branches of the square root you can use the rootof function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">rootof</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">rootof</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">-</span><span class=\"mi\">3</span><span class=\"p\">,</span><span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span><span class=\"mi\">1</span><span class=\"p\">)]</span>\n<span class=\"go\">[-sqrt(3), sqrt(3)]</span>\n</code></pre></div>\n\n<p>Although <code>sqrt</code> is printed, there is no <code>sqrt</code> function so looking for\n<code>sqrt</code> in an expression will fail:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.utilities.misc</span> <span class=\"kn\">import</span> <span class=\"n\">func_name</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">func_name</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">&#39;Pow&#39;</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">has</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<p>To find <code>sqrt</code> look for <code>Pow</code> with an exponent of <code>1/2</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">find</span><span class=\"p\">(</span><span class=\"k\">lambda</span> <span class=\"n\">i</span><span class=\"p\">:</span> <span class=\"n\">i</span><span class=\"o\">.</span><span class=\"n\">is_Pow</span> <span class=\"ow\">and</span> <span class=\"nb\">abs</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"o\">.</span><span class=\"n\">exp</span><span class=\"p\">)</span> <span class=\"ow\">is</span> <span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">Half</span><span class=\"p\">)</span>\n<span class=\"go\">{1/sqrt(x)}</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.polys.rootoftools.rootof, root, real_root</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "signature": "(arg, evaluate=None)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.root", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "root", "type": "function", "doc": "<p>Override of sympy convenience function <code>root</code>. Simply divides equations\ninto two sides if <code>arg</code> is an instance of <code>Equation</code>. This avoids an\nissue with the way sympy is delaying specialized applications of _Pow_ on\nobjects that are not basic sympy expressions.\nReturns the <em>k</em>-th <em>n</em>-th root of <code>arg</code>.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>k : int, optional\n    Should be an integer in ${0, 1, ..., n-1}$.\n    Defaults to the principal root if $0$.</p>\n\n<p>evaluate : bool, optional\n    The parameter determines if the expression should be evaluated.\n    If <code>None</code>, its value is taken from\n    <code>global_parameters.evaluate</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">root</span><span class=\"p\">,</span> <span class=\"n\">Rational</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">x**(1/3)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">x**(1/n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">x**(-3/2)</span>\n</code></pre></div>\n\n<p>To get the k-th n-th root, specify k:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-(-1)**(2/3)*2**(1/3)</span>\n</code></pre></div>\n\n<p>To get all n n-th roots you can use the rootof function.\nThe following examples show the roots of unity for n\nequal 2, 3 and 4:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">rootof</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">rootof</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)]</span>\n<span class=\"go\">[-1, 1]</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">rootof</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">rootof</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">4</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)]</span>\n<span class=\"go\">[-1, 1, -I, I]</span>\n</code></pre></div>\n\n<p>SymPy, like other symbolic algebra systems, returns the\ncomplex root of negative numbers. This is the principal\nroot and differs from the text-book result that one might\nbe expecting. For example, the cube root of -8 does not\ncome back as -2:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">8</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">2*(-1)**(1/3)</span>\n</code></pre></div>\n\n<p>The real_root function can be used to either make the principal\nresult real (or simply to return the real root directly):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">real_root</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">real_root</span><span class=\"p\">(</span><span class=\"n\">_</span><span class=\"p\">)</span>\n<span class=\"go\">-2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">real_root</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">32</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">-2</span>\n</code></pre></div>\n\n<p>Alternatively, the n//2-th n-th root of a negative number can be\ncomputed with root:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">root</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">32</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"o\">//</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.polys.rootoftools.rootof\nsympy.core.power.integer_nthroot\nsqrt, real_root</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "signature": "(arg, n, k=0, evaluate=None)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.collect", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "collect", "type": "function", "doc": "<p>Override of sympy <code>collect()</code>.</p>\n", "signature": "(\n    expr,\n    syms,\n    func=None,\n    evaluate=None,\n    exact=False,\n    distribute_order_term=True\n)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eq", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eq", "type": "class", "doc": "<p>Extension of Equality class to include the ability to convert it to an\nEquation.</p>\n", "bases": "sympy.core.relational.Equality"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eq.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eq.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eq.to_Equation", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eq.to_Equation", "type": "function", "doc": "<p>Return: recasts the Equality as an Equation.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eq.to_Eqn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eq.to_Eqn", "type": "function", "doc": "<p>Synonym for to_Equation.\nReturn: recasts the Equality as an Equation.</p>\n", "signature": "(self)", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Eq.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eq.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.EqnFunction", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "EqnFunction", "type": "class", "doc": "<p>Extension of the sympy Function class to understand equations. Each\nsympy function impacted by this extension is listed in the documentation\nthat follows.</p>\n", "bases": "sympy.core.function.Function"}, {"fullname": "algebra_with_sympy.algebraic_equation.EqnFunction.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "EqnFunction.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.EqnFunction.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "EqnFunction.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial", "type": "class", "doc": "<p>Implementation of factorial function over nonnegative integers.\nBy convention (consistent with the gamma function and the binomial\ncoefficients), factorial of a negative integer is complex infinity.</p>\n\n<p>The factorial is very important in combinatorics where it gives\nthe number of ways in which <code>n</code> objects can be permuted. It also\narises in calculus, probability, number theory, etc.</p>\n\n<p>There is strict relation of factorial with gamma function. In\nfact <code>n! = gamma(n+1)</code> for nonnegative integers. Rewrite of this\nkind is very useful in case of combinatorial simplification.</p>\n\n<p>Computation of the factorial is done using two algorithms. For\nsmall arguments a precomputed look up table is used. However for bigger\ninput algorithm Prime-Swing is used. It is the fastest algorithm\nknown and computes <code>n!</code> via prime factorization of special class\nof numbers, called here the 'Swing Numbers'.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"mi\">7</span><span class=\"p\">)</span>\n<span class=\"go\">5040</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(2*n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(1/2)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial2, RisingFactorial, FallingFactorial</p>\n", "bases": "sympy.functions.combinatorial.factorials.factorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial2", "type": "class", "doc": "<p>The double factorial <code>n!!</code>, not to be confused with <code>(n!)!</code></p>\n\n<p>The double factorial is defined for nonnegative integers and for odd\nnegative integers as:</p>\n\n<p>$$n!! = \\begin{cases} 1 &amp; n = 0 \\n(n-2)(n-4) \\cdots 1 &amp; n\\ \\text{positive odd} \\\nn(n-2)(n-4) \\cdots 2 &amp; n\\ \\text{positive even} \\\n(n+2)!!/(n+2) &amp; n\\ \\text{negative odd} \\end{cases}$$</p>\n\n<h1 id=\"references\">References</h1>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">factorial2</span><span class=\"p\">,</span> <span class=\"n\">var</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">var</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span>\n<span class=\"go\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial2</span><span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">factorial2(n + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial2</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">15</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">factorial2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">1/3</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, RisingFactorial, FallingFactorial</p>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.factorials.factorial2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.factorial2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "factorial2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.rf", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "rf", "type": "class", "doc": "<p>Rising factorial (also called Pochhammer symbol <sup class=\"footnote-ref\" id=\"fnref-1\"><a href=\"#fn-1\">1</a></sup>) is a double valued\nfunction arising in concrete mathematics, hypergeometric functions\nand series expansions. It is defined by:</p>\n\n<p>$$rf(x,k) = x \\cdot (x+1) \\cdots (x+k-1)$$</p>\n\n<p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For\nmore information check \"Concrete mathematics\" by Graham, pp. 66\nor visit http://mathworld.wolfram.com/RisingFactorial.html page.</p>\n\n<p>When <code>x</code> is a Poly instance of degree $\\ge 1$ with a single variable,\n<code>rf(x,k) = x(y) \\cdot x(y+1) \\cdots x(y+k-1)</code>, where <code>y</code> is the\nvariable of <code>x</code>. This is as described in <sup class=\"footnote-ref\" id=\"fnref-2\"><a href=\"#fn-2\">2</a></sup>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">Poly</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">120</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">4</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">Poly</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain=&#39;ZZ&#39;)</span>\n</code></pre></div>\n\n<p>Rewriting is complicated unless the relationship between\nthe arguments is known, but rising factorial can\nbe rewritten in terms of gamma, factorial, binomial,\nand falling factorial.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">R</span> <span class=\"o\">=</span> <span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span> <span class=\"n\">R</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">RisingFactorial(n, n + 2)</span>\n<span class=\"go\">FallingFactorial(2*n + 1, n + 2)</span>\n<span class=\"go\">factorial(2*n + 1)/factorial(n - 1)</span>\n<span class=\"go\">binomial(2*n + 1, n + 2)*factorial(n + 2)</span>\n<span class=\"go\">gamma(2*n + 2)/gamma(n)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, factorial2, FallingFactorial</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-1\">\n<p>https://en.wikipedia.org/wiki/Pochhammer_symbol&#160;<a href=\"#fnref-1\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n\n<li id=\"fn-2\">\n<p>Peter Paule, \"Greatest Factorial Factorization and Symbolic\nSummation\", Journal of Symbolic Computation, vol. 20, pp. 235-268,\n1995.&#160;<a href=\"#fnref-2\" class=\"footnoteBackLink\" title=\"Jump back to footnote 2 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.factorials.RisingFactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.rf.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "rf.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.rf.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "rf.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.ff", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ff", "type": "class", "doc": "<p>Falling factorial (related to rising factorial) is a double valued\nfunction arising in concrete mathematics, hypergeometric functions\nand series expansions. It is defined by</p>\n\n<p>$$ff(x,k) = x \\cdot (x-1) \\cdots (x-k+1)$$</p>\n\n<p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For\nmore information check \"Concrete mathematics\" by Graham, pp. 66\nor visit http://mathworld.wolfram.com/FallingFactorial.html page.</p>\n\n<p>When <code>x</code> is a Poly instance of degree &gt;= 1 with single variable,\n<code>ff(x,k) = x(y) \\cdot x(y-1) \\cdots x(y-k+1)</code>, where <code>y</code> is the\nvariable of <code>x</code>. This is as described in Peter Paule, \"Greatest\nFactorial Factorization and Symbolic Summation\", Journal of\nSymbolic Computation, vol. 20, pp. 235-268, 1995.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">Poly</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">120</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">Poly</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Poly(x**4 - 2*x**3 + x**2, x, domain=&#39;ZZ&#39;)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(n)</span>\n</code></pre></div>\n\n<p>Rewriting is complicated unless the relationship between\nthe arguments is known, but falling factorial can\nbe rewritten in terms of gamma, factorial and binomial\nand rising factorial.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">F</span> <span class=\"o\">=</span> <span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span> <span class=\"n\">F</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">RisingFactorial(3, n - 2)</span>\n<span class=\"go\">FallingFactorial(n, n - 2)</span>\n<span class=\"go\">factorial(n)/2</span>\n<span class=\"go\">binomial(n, n - 2)*factorial(n - 2)</span>\n<span class=\"go\">gamma(n + 1)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, factorial2, RisingFactorial</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.factorials.FallingFactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.ff.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ff.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.ff.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ff.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.binomial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "binomial", "type": "class", "doc": "<p>Implementation of the binomial coefficient. It can be defined\nin two ways depending on its desired interpretation:</p>\n\n<p>$$\\binom{n}{k} = \\frac{n!}{k!(n-k)!}\\ \\text{or}\\binom{n}{k} = \\frac{ff(n, k)}{k!}$$</p>\n\n<p>First, in a strict combinatorial sense it defines the\nnumber of ways we can choose <code>k</code> elements from a set of\n<code>n</code> elements. In this case both arguments are nonnegative\nintegers and binomial is computed using an efficient\nalgorithm based on prime factorization.</p>\n\n<p>The other definition is generalization for arbitrary <code>n</code>,\nhowever <code>k</code> must also be nonnegative. This case is very\nuseful when evaluating summations.</p>\n\n<p>For the sake of convenience for negative integer <code>k</code> this function\nwill return zero no matter what valued is the other argument.</p>\n\n<p>To expand the binomial when <code>n</code> is a symbol, use either\n<code>expand_func()</code> or <code>expand(func=True)</code>. The former will keep\nthe polynomial in factored form while the latter will expand the\npolynomial itself. See examples for details.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">Rational</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"mi\">15</span><span class=\"p\">,</span> <span class=\"mi\">8</span><span class=\"p\">)</span>\n<span class=\"go\">6435</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Rows of Pascal's triangle can be generated with the binomial function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">N</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span>    <span class=\"nb\">print</span><span class=\"p\">([</span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">N</span><span class=\"p\">,</span> <span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">N</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)])</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">[1]</span>\n<span class=\"go\">[1, 1]</span>\n<span class=\"go\">[1, 2, 1]</span>\n<span class=\"go\">[1, 3, 3, 1]</span>\n<span class=\"go\">[1, 4, 6, 4, 1]</span>\n<span class=\"go\">[1, 5, 10, 10, 5, 1]</span>\n<span class=\"go\">[1, 6, 15, 20, 15, 6, 1]</span>\n<span class=\"go\">[1, 7, 21, 35, 35, 21, 7, 1]</span>\n</code></pre></div>\n\n<p>As can a given diagonal, e.g. the 4th diagonal:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">N</span> <span class=\"o\">=</span> <span class=\"o\">-</span><span class=\"mi\">4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">N</span><span class=\"p\">,</span> <span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">-</span> <span class=\"n\">N</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, -4, 10, -20, 35]</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">),</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">-5/128</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">),</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">-195/128</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">binomial(n, 3)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">n**3/6 - n**2/2 + n/3</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">binomial</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">n*(n - 2)*(n - 1)/6</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.factorials.binomial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.binomial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "binomial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.binomial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "binomial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.RisingFactorial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "RisingFactorial", "type": "class", "doc": "<p>Rising factorial (also called Pochhammer symbol <sup class=\"footnote-ref\" id=\"fnref-1\"><a href=\"#fn-1\">1</a></sup>) is a double valued\nfunction arising in concrete mathematics, hypergeometric functions\nand series expansions. It is defined by:</p>\n\n<p>$$rf(x,k) = x \\cdot (x+1) \\cdots (x+k-1)$$</p>\n\n<p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For\nmore information check \"Concrete mathematics\" by Graham, pp. 66\nor visit http://mathworld.wolfram.com/RisingFactorial.html page.</p>\n\n<p>When <code>x</code> is a Poly instance of degree $\\ge 1$ with a single variable,\n<code>rf(x,k) = x(y) \\cdot x(y+1) \\cdots x(y+k-1)</code>, where <code>y</code> is the\nvariable of <code>x</code>. This is as described in <sup class=\"footnote-ref\" id=\"fnref-2\"><a href=\"#fn-2\">2</a></sup>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">Poly</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">120</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">4</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">Poly</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain=&#39;ZZ&#39;)</span>\n</code></pre></div>\n\n<p>Rewriting is complicated unless the relationship between\nthe arguments is known, but rising factorial can\nbe rewritten in terms of gamma, factorial, binomial,\nand falling factorial.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">R</span> <span class=\"o\">=</span> <span class=\"n\">rf</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span> <span class=\"n\">R</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">RisingFactorial(n, n + 2)</span>\n<span class=\"go\">FallingFactorial(2*n + 1, n + 2)</span>\n<span class=\"go\">factorial(2*n + 1)/factorial(n - 1)</span>\n<span class=\"go\">binomial(2*n + 1, n + 2)*factorial(n + 2)</span>\n<span class=\"go\">gamma(2*n + 2)/gamma(n)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, factorial2, FallingFactorial</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-1\">\n<p>https://en.wikipedia.org/wiki/Pochhammer_symbol&#160;<a href=\"#fnref-1\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n\n<li id=\"fn-2\">\n<p>Peter Paule, \"Greatest Factorial Factorization and Symbolic\nSummation\", Journal of Symbolic Computation, vol. 20, pp. 235-268,\n1995.&#160;<a href=\"#fnref-2\" class=\"footnoteBackLink\" title=\"Jump back to footnote 2 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.factorials.RisingFactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.RisingFactorial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "RisingFactorial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.RisingFactorial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "RisingFactorial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.FallingFactorial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "FallingFactorial", "type": "class", "doc": "<p>Falling factorial (related to rising factorial) is a double valued\nfunction arising in concrete mathematics, hypergeometric functions\nand series expansions. It is defined by</p>\n\n<p>$$ff(x,k) = x \\cdot (x-1) \\cdots (x-k+1)$$</p>\n\n<p>where <code>x</code> can be arbitrary expression and <code>k</code> is an integer. For\nmore information check \"Concrete mathematics\" by Graham, pp. 66\nor visit http://mathworld.wolfram.com/FallingFactorial.html page.</p>\n\n<p>When <code>x</code> is a Poly instance of degree &gt;= 1 with single variable,\n<code>ff(x,k) = x(y) \\cdot x(y-1) \\cdots x(y-k+1)</code>, where <code>y</code> is the\nvariable of <code>x</code>. This is as described in Peter Paule, \"Greatest\nFactorial Factorization and Symbolic Summation\", Journal of\nSymbolic Computation, vol. 20, pp. 235-268, 1995.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">Poly</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">120</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">Poly</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">Poly(x**4 - 2*x**3 + x**2, x, domain=&#39;ZZ&#39;)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">factorial(n)</span>\n</code></pre></div>\n\n<p>Rewriting is complicated unless the relationship between\nthe arguments is known, but falling factorial can\nbe rewritten in terms of gamma, factorial and binomial\nand rising factorial.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">F</span> <span class=\"o\">=</span> <span class=\"n\">ff</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">n</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"n\">rf</span><span class=\"p\">,</span> <span class=\"n\">ff</span><span class=\"p\">,</span> <span class=\"n\">factorial</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">):</span>\n<span class=\"gp\">... </span> <span class=\"n\">F</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"gp\">...</span>\n<span class=\"go\">RisingFactorial(3, n - 2)</span>\n<span class=\"go\">FallingFactorial(n, n - 2)</span>\n<span class=\"go\">factorial(n)/2</span>\n<span class=\"go\">binomial(n, n - 2)*factorial(n - 2)</span>\n<span class=\"go\">gamma(n + 1)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>factorial, factorial2, RisingFactorial</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.factorials.FallingFactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.FallingFactorial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "FallingFactorial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.FallingFactorial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "FallingFactorial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.subfactorial", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "subfactorial", "type": "class", "doc": "<p>The subfactorial counts the derangements of n items and is\ndefined for non-negative integers as:</p>\n\n<p>$$!n = \\begin{cases} 1 &amp; n = 0 \\ 0 &amp; n = 1 \\(n-1)(!(n-1) + !(n-2)) &amp; n &gt; 1 \\end{cases}$$</p>\n\n<p>It can also be written as <code>int(round(n!/exp(1)))</code> but the\nrecursive definition with caching is implemented for this function.</p>\n\n<p>An interesting analytic expression is the following <sup class=\"footnote-ref\" id=\"fnref-2\"><a href=\"#fn-2\">1</a></sup></p>\n\n<p>$$!x = \\Gamma(x + 1, -1)/e$$</p>\n\n<p>which is valid for non-negative integers <code>x</code>. The above formula\nis not very useful incase of non-integers. \\( \\Gamma(x + 1, -1) \\) is\nsingle-valued only for integral arguments <code>x</code>, elsewhere on the positive\nreal axis it has an infinite number of branches none of which are real.</p>\n\n<h1 id=\"references\">References</h1>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">subfactorial</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">subfactorial</span><span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">subfactorial(n + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">subfactorial</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">44</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.combinatorial.factorials.factorial,\nsympy.utilities.iterables.generate_derangements,\nsympy.functions.special.gamma_functions.uppergamma</p>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-2\">\n<p>http://mathworld.wolfram.com/Subfactorial.html&#160;<a href=\"#fnref-2\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.factorials.subfactorial, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.subfactorial.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "subfactorial.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.subfactorial.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "subfactorial.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.carmichael", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "carmichael", "type": "class", "doc": "<p>Carmichael Numbers:</p>\n\n<p>Certain cryptographic algorithms make use of big prime numbers.\nHowever, checking whether a big number is prime is not so easy.\nRandomized prime number checking tests exist that offer a high degree of confidence of\naccurate determination at low cost, such as the Fermat test.</p>\n\n<p>Let 'a' be a random number between 2 and n - 1, where n is the number whose primality we are testing.\nThen, n is probably prime if it satisfies the modular arithmetic congruence relation :</p>\n\n<p>a^(n-1) = 1(mod n).\n(where mod refers to the modulo operation)</p>\n\n<p>If a number passes the Fermat test several times, then it is prime with a\nhigh probability.</p>\n\n<p>Unfortunately, certain composite numbers (non-primes) still pass the Fermat test\nwith every number smaller than themselves.\nThese numbers are called Carmichael numbers.</p>\n\n<p>A Carmichael number will pass a Fermat primality test to every base b relatively prime to the number,\neven though it is not actually prime. This makes tests based on Fermat's Little Theorem less effective than\nstrong probable prime tests such as the Baillie-PSW primality test and the Miller-Rabin primality test.\nmr functions given in sympy/sympy/ntheory/primetest.py will produce wrong results for each and every\ncarmichael number.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">carmichael</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">find_first_n_carmichaels</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">[561, 1105, 1729, 2465, 2821]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">is_prime</span><span class=\"p\">(</span><span class=\"mi\">2465</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">is_prime</span><span class=\"p\">(</span><span class=\"mi\">1729</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">find_carmichael_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">562</span><span class=\"p\">)</span>\n<span class=\"go\">[561]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">find_carmichael_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span><span class=\"mi\">1000</span><span class=\"p\">)</span>\n<span class=\"go\">[561]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">carmichael</span><span class=\"o\">.</span><span class=\"n\">find_carmichael_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span><span class=\"mi\">2000</span><span class=\"p\">)</span>\n<span class=\"go\">[561, 1105, 1729]</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.carmichael, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.carmichael.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "carmichael.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.carmichael.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "carmichael.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.fibonacci", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fibonacci", "type": "class", "doc": "<p>Fibonacci numbers / Fibonacci polynomials</p>\n\n<p>The Fibonacci numbers are the integer sequence defined by the\ninitial terms <code>F_0 = 0</code>, <code>F_1 = 1</code> and the two-term recurrence\nrelation <code>F_n = F_{n-1} + F_{n-2}</code>.  This definition\nextended to arbitrary real and complex arguments using\nthe formula</p>\n\n<p>.. math :: F_z = \\frac{\\phi^z - \\cos(\\pi z) \\phi^{-z}}{\\sqrt 5}</p>\n\n<p>The Fibonacci polynomials are defined by <code>F_1(x) = 1</code>,\n<code>F_2(x) = x</code>, and <code>F_n(x) = x*F_{n-1}(x) + F_{n-2}(x)</code> for <code>n &gt; 2</code>.\nFor all positive integers <code>n</code>, <code>F_n(1) = F_n</code>.</p>\n\n<ul>\n<li><code>fibonacci(n)</code> gives the <code>n^{th}</code> Fibonacci number, <code>F_n</code></li>\n<li><code>fibonacci(n, x)</code> gives the <code>n^{th}</code> Fibonacci polynomial in <code>x</code>, <code>F_n(x)</code></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">fibonacci</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">fibonacci</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fibonacci</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;t&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">t**4 + 3*t**2 + 1</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, harmonic, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.fibonacci, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.fibonacci.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fibonacci.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.fibonacci.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fibonacci.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.lucas", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lucas", "type": "class", "doc": "<p>Lucas numbers</p>\n\n<p>Lucas numbers satisfy a recurrence relation similar to that of\nthe Fibonacci sequence, in which each term is the sum of the\npreceding two. They are generated by choosing the initial\nvalues <code>L_0 = 2</code> and <code>L_1 = 1</code>.</p>\n\n<ul>\n<li><code>lucas(n)</code> gives the <code>n^{th}</code> Lucas number</li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lucas</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">lucas</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, harmonic, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.lucas, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.lucas.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lucas.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.lucas.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lucas.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.motzkin", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "motzkin", "type": "class", "doc": "<p>The nth Motzkin number is the number\nof ways of drawing non-intersecting chords\nbetween n points on a circle (not necessarily touching\nevery point by a chord). The Motzkin numbers are named\nafter Theodore Motzkin and have diverse applications\nin geometry, combinatorics and number theory.</p>\n\n<p>Motzkin numbers are the integer sequence defined by the\ninitial terms <code>M_0 = 1</code>, <code>M_1 = 1</code> and the two-term recurrence relation\n<code>M_n = \frac{2*n + 1}{n + 2} * M_{n-1} + \frac{3n - 3}{n + 2} * M_{n-2}</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">motzkin</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">motzkin</span><span class=\"o\">.</span><span class=\"n\">is_motzkin</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">motzkin</span><span class=\"o\">.</span><span class=\"n\">find_motzkin_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span><span class=\"mi\">300</span><span class=\"p\">)</span>\n<span class=\"go\">[2, 4, 9, 21, 51, 127]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">motzkin</span><span class=\"o\">.</span><span class=\"n\">find_motzkin_numbers_in_range</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span><span class=\"mi\">900</span><span class=\"p\">)</span>\n<span class=\"go\">[2, 4, 9, 21, 51, 127, 323, 835]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">motzkin</span><span class=\"o\">.</span><span class=\"n\">find_first_n_motzkins</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)</span>\n<span class=\"go\">[1, 1, 2, 4, 9, 21, 51, 127, 323, 835]</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.motzkin, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.motzkin.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "motzkin.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.motzkin.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "motzkin.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.tribonacci", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tribonacci", "type": "class", "doc": "<p>Tribonacci numbers / Tribonacci polynomials</p>\n\n<p>The Tribonacci numbers are the integer sequence defined by the\ninitial terms <code>T_0 = 0</code>, <code>T_1 = 1</code>, <code>T_2 = 1</code> and the three-term\nrecurrence relation <code>T_n = T_{n-1} + T_{n-2} + T_{n-3}</code>.</p>\n\n<p>The Tribonacci polynomials are defined by <code>T_0(x) = 0</code>, <code>T_1(x) = 1</code>,\n<code>T_2(x) = x^2</code>, and <code>T_n(x) = x^2 T_{n-1}(x) + x T_{n-2}(x) + T_{n-3}(x)</code>\nfor <code>n &gt; 2</code>.  For all positive integers <code>n</code>, <code>T_n(1) = T_n</code>.</p>\n\n<ul>\n<li><code>tribonacci(n)</code> gives the <code>n^{th}</code> Tribonacci number, <code>T_n</code></li>\n<li><code>tribonacci(n, x)</code> gives the <code>n^{th}</code> Tribonacci polynomial in <code>x</code>, <code>T_n(x)</code></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">tribonacci</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">tribonacci</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tribonacci</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;t&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">t**8 + 3*t**5 + 3*t**2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.tribonacci, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.tribonacci.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tribonacci.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.tribonacci.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tribonacci.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.harmonic", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "harmonic", "type": "class", "doc": "<p>Harmonic numbers</p>\n\n<p>The nth harmonic number is given by <code>\\operatorname{H}_{n} =\n1 + \\frac{1}{2} + \\frac{1}{3} + \\ldots + \\frac{1}{n}</code>.</p>\n\n<p>More generally:</p>\n\n<p>$$\\operatorname{H}_{n,m} = \\sum_{k=1}^{n} \\frac{1}{k^m}$$</p>\n\n<p>As <code>n \\rightarrow \\infty</code>, <code>\\operatorname{H}_{n,m} \\rightarrow \\zeta(m)</code>,\nthe Riemann zeta function.</p>\n\n<ul>\n<li><p><code>harmonic(n)</code> gives the nth harmonic number, <code>\\operatorname{H}_n</code></p></li>\n<li><p><code>harmonic(n, m)</code> gives the nth generalized harmonic number\nof order <code>m</code>, <code>\\operatorname{H}_{n,m}</code>, where\n<code>harmonic(n) == harmonic(n, 1)</code></p></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">harmonic</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">)]</span>\n<span class=\"go\">[0, 1, 3/2, 11/6, 25/12, 137/60]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">)]</span>\n<span class=\"go\">[0, 1, 5/4, 49/36, 205/144, 5269/3600]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">pi**2/6</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">Sum</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Sum</span><span class=\"p\">)</span>\n<span class=\"go\">Sum(1/_k, (_k, 1, n))</span>\n</code></pre></div>\n\n<p>We can evaluate harmonic numbers for all integral and positive\nrational arguments:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">simplify</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">)</span>\n<span class=\"go\">761/280</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)</span>\n<span class=\"go\">83711/27720</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">H</span> <span class=\"o\">=</span> <span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">H</span>\n<span class=\"go\">harmonic(1/3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span> <span class=\"o\">=</span> <span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">H</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span>\n<span class=\"go\">-log(6) - sqrt(3)*pi/6 + 2*Sum(log(sin(_k*pi/3))*cos(2*_k*pi/3), (_k, 1, 1))</span>\n<span class=\"go\">                       + 3*Sum(1/(3*_k + 1), (_k, 0, 0))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">()</span>\n<span class=\"go\">-log(6) - sqrt(3)*pi/6 - log(sqrt(3)/2) + 3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">H</span> <span class=\"o\">=</span> <span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">25</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">7</span><span class=\"p\">))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span> <span class=\"o\">=</span> <span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">H</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">())</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span>\n<span class=\"go\">log(sin(2*pi/7)**(2*cos(16*pi/7))/(14*sin(pi/7)**(2*cos(pi/7))*cos(pi/14)**(2*sin(pi/14)))) + pi*tan(pi/14)/2 + 30247/9900</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">He</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">)</span>\n<span class=\"go\">1.983697455232980674869851942390639915940</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"mi\">25</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">7</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">)</span>\n<span class=\"go\">1.983697455232980674869851942390639915940</span>\n</code></pre></div>\n\n<p>We can rewrite harmonic numbers in terms of polygamma functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">digamma</span><span class=\"p\">,</span> <span class=\"n\">polygamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">m</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;m&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">digamma</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(0, n + 1) + EulerGamma</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(0, n + 1) + EulerGamma</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(2, n + 1)/2 - polygamma(2, 1)/2</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)</span>\n</code></pre></div>\n\n<p>Integer offsets in the argument can be pulled out:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">+</span><span class=\"mi\">4</span><span class=\"p\">))</span>\n<span class=\"go\">harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">-</span><span class=\"mi\">4</span><span class=\"p\">))</span>\n<span class=\"go\">harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n</span>\n</code></pre></div>\n\n<p>Some limits can be computed as well:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">limit</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">limit</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">limit</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">pi**2/6</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">limit</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-polygamma(2, 1)/2</span>\n</code></pre></div>\n\n<p>However we cannot compute the general relation yet:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">limit</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">harmonic(oo, m)</span>\n</code></pre></div>\n\n<p>which equals <code>zeta(m)</code> for <code>m &gt; 1</code>.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.harmonic, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.harmonic.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "harmonic.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.harmonic.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "harmonic.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.bernoulli", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bernoulli", "type": "class", "doc": "<p>Bernoulli numbers / Bernoulli polynomials</p>\n\n<p>The Bernoulli numbers are a sequence of rational numbers\ndefined by <code>B_0 = 1</code> and the recursive relation (<code>n &gt; 0</code>):</p>\n\n<p>.. math :: 0 = \\sum_{k=0}^n \\binom{n+1}{k} B_k</p>\n\n<p>They are also commonly defined by their exponential generating\nfunction, which is <code>\\frac{x}{e^x - 1}</code>. For odd indices &gt; 1, the\nBernoulli numbers are zero.</p>\n\n<p>The Bernoulli polynomials satisfy the analogous formula:</p>\n\n<p>.. math :: B_n(x) = \\sum_{k=0}^n \\binom{n}{k} B_k x^{n-k}</p>\n\n<p>Bernoulli numbers and Bernoulli polynomials are related as\n<code>B_n(0) = B_n</code>.</p>\n\n<p>We compute Bernoulli numbers using Ramanujan's formula:</p>\n\n<p>.. math :: B_n = \\frac{A(n) - S(n)}{\\binom{n+3}{n}}</p>\n\n<p>where:</p>\n\n<p>.. math :: A(n) = \\begin{cases} \\frac{n+3}{3} &amp;\n    n \\equiv 0\\ \\text{or}\\ 2 \\pmod{6} \\\n    -\\frac{n+3}{6} &amp; n \\equiv 4 \\pmod{6} \\end{cases}</p>\n\n<p>and:</p>\n\n<p>.. math :: S(n) = \\sum_{k=1}^{[n/6]} \\binom{n+3}{n-6k} B_{n-6k}</p>\n\n<p>This formula is similar to the sum given in the definition, but\ncuts 2/3 of the terms. For Bernoulli polynomials, we use the\nformula in the definition.</p>\n\n<ul>\n<li><code>bernoulli(n)</code> gives the nth Bernoulli number, <code>B_n</code></li>\n<li><code>bernoulli(n, x)</code> gives the nth Bernoulli polynomial in <code>x</code>, <code>B_n(x)</code></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">bernoulli</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">bernoulli</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">bernoulli</span><span class=\"p\">(</span><span class=\"mi\">1000001</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.bernoulli, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.bernoulli.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bernoulli.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.bernoulli.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bernoulli.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.bell", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bell", "type": "class", "doc": "<p>Bell numbers / Bell polynomials</p>\n\n<p>The Bell numbers satisfy <code>B_0 = 1</code> and</p>\n\n<p>$$B_n = \\sum_{k=0}^{n-1} \\binom{n-1}{k} B_k.$$</p>\n\n<p>They are also given by:</p>\n\n<p>$$B_n = \\frac{1}{e} \\sum_{k=0}^{\\infty} \\frac{k^n}{k!}.$$</p>\n\n<p>The Bell polynomials are given by <code>B_0(x) = 1</code> and</p>\n\n<p>$$B_n(x) = x \\sum_{k=1}^{n-1} \\binom{n-1}{k-1} B_{k-1}(x).$$</p>\n\n<p>The second kind of Bell polynomials (are sometimes called \"partial\" Bell\npolynomials or incomplete Bell polynomials) are defined as</p>\n\n<p>$$B_{n,k}(x_1, x_2,\\dotsc x_{n-k+1}) =\\sum_{j_1+j_2+j_2+\\dotsb=k \\atop j_1+2j_2+3j_2+\\dotsb=n}\n    \\frac{n!}{j_1!j_2!\\dotsb j_{n-k+1}!}\n    \\left(\\frac{x_1}{1!} \\right)^{j_1}\n    \\left(\\frac{x_2}{2!} \\right)^{j_2} \\dotsb\n    \\left(\\frac{x_{n-k+1}}{(n-k+1)!} \\right) ^{j_{n-k+1}}.$$</p>\n\n<ul>\n<li><code>bell(n)</code> gives the <code>n^{th}</code> Bell number, <code>B_n</code>.</li>\n<li><code>bell(n, x)</code> gives the <code>n^{th}</code> Bell polynomial, <code>B_n(x)</code>.</li>\n<li><code>bell(n, k, (x1, x2, ...))</code> gives Bell polynomials of the second kind,\n<code>B_{n,k}(x_1, x_2, \\dotsc, x_{n-k+1})</code>.</li>\n</ul>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p>Not to be confused with Bernoulli numbers and Bernoulli polynomials,\nwhich use the same notation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">bell</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">symbols</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">bell</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">11</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">bell</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">846749014511809332450147</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">bell</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;t&#39;</span><span class=\"p\">))</span>\n<span class=\"go\">t**4 + 6*t**3 + 7*t**2 + t</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">bell</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;x:6&#39;</span><span class=\"p\">)[</span><span class=\"mi\">1</span><span class=\"p\">:])</span>\n<span class=\"go\">6*x1*x5 + 15*x2*x4 + 10*x3**2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.bell, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.bell.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bell.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.bell.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bell.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.euler", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "euler", "type": "class", "doc": "<p>Euler numbers / Euler polynomials</p>\n\n<p>The Euler numbers are given by:</p>\n\n<p>$$E_{2n} = I \\sum_{k=1}^{2n+1} \\sum_{j=0}^k \\binom{k}{j}\\frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k}$$</p>\n\n<p>$$E_{2n+1} = 0$$</p>\n\n<p>Euler numbers and Euler polynomials are related by</p>\n\n<p>$$E_n = 2^n E_n\\left(\\frac{1}{2}\\right).$$</p>\n\n<p>We compute symbolic Euler polynomials using <sup class=\"footnote-ref\" id=\"fnref-5\"><a href=\"#fn-5\">1</a></sup></p>\n\n<p>$$E_n(x) = \\sum_{k=0}^n \\binom{n}{k} \\frac{E_k}{2^k}\\left(x - \\frac{1}{2}\\right)^{n-k}.$$</p>\n\n<p>However, numerical evaluation of the Euler polynomial is computed\nmore efficiently (and more accurately) using the mpmath library.</p>\n\n<ul>\n<li><code>euler(n)</code> gives the <code>n^{th}</code> Euler number, <code>E_n</code>.</li>\n<li><code>euler(n, x)</code> gives the <code>n^{th}</code> Euler polynomial, <code>E_n(x)</code>.</li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">euler</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">euler(3*n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;x&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">euler(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x - 1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**2 - x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**3 - 3*x**2/2 + 1/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**4 - 2*x**3 + x</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">12</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">Half</span><span class=\"p\">)</span>\n<span class=\"go\">2702765/4096</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">euler</span><span class=\"p\">(</span><span class=\"mi\">12</span><span class=\"p\">)</span>\n<span class=\"go\">2702765</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, fibonacci, harmonic, lucas, genocchi, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-5\">\n<p>http://dlmf.nist.gov/24.2#ii&#160;<a href=\"#fnref-5\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.euler, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.euler.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "euler.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.euler.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "euler.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.catalan", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "catalan", "type": "class", "doc": "<p>Catalan numbers</p>\n\n<p>The <code>n^{th}</code> catalan number is given by:</p>\n\n<p>.. math :: C_n = \\frac{1}{n+1} \\binom{2n}{n}</p>\n\n<ul>\n<li><code>catalan(n)</code> gives the <code>n^{th}</code> Catalan number, <code>C_n</code></li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"p\">(</span><span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">binomial</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"p\">,</span> <span class=\"n\">hyper</span><span class=\"p\">,</span>\n<span class=\"gp\">... </span>    <span class=\"n\">catalan</span><span class=\"p\">,</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">combsimp</span><span class=\"p\">,</span> <span class=\"n\">Rational</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"mi\">10</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, 2, 5, 14, 42, 132, 429, 1430, 4862]</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">catalan(n)</span>\n</code></pre></div>\n\n<p>Catalan numbers can be transformed into several other, identical\nexpressions involving other mathematical functions</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">binomial</span><span class=\"p\">)</span>\n<span class=\"go\">binomial(2*n, n)/(n + 1)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">)</span>\n<span class=\"go\">4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">hyper((1 - n, -n), (2,), 1)</span>\n</code></pre></div>\n\n<p>For some non-integer values of n we can get closed form\nexpressions by rewriting in terms of gamma functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">)</span>\n<span class=\"go\">8/(3*pi)</span>\n</code></pre></div>\n\n<p>We can differentiate the Catalan numbers C(n) interpreted as a\ncontinuous real function in n:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">),</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">(polygamma(0, n + 1/2) - polygamma(0, n + 2) + log(4))*catalan(n)</span>\n</code></pre></div>\n\n<p>As a more advanced example consider the following ratio\nbetween consecutive numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">combsimp</span><span class=\"p\">((</span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">binomial</span><span class=\"p\">))</span>\n<span class=\"go\">2*(2*n + 1)/(n + 2)</span>\n</code></pre></div>\n\n<p>The Catalan numbers can be generalized to complex numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">)</span>\n<span class=\"go\">4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))</span>\n</code></pre></div>\n\n<p>and evaluated with arbitrary precision:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">catalan</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">0.39764993382373624267 - 0.020884341620842555705*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci\nsympy.functions.combinatorial.factorials.binomial</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.catalan, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.catalan.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "catalan.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.catalan.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "catalan.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.genocchi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "genocchi", "type": "class", "doc": "<p>Genocchi numbers</p>\n\n<p>The Genocchi numbers are a sequence of integers <code>G_n</code> that satisfy the\nrelation:</p>\n\n<p>$$\\frac{2t}{e^t + 1} = \\sum_{n=1}^\\infty \\frac{G_n t^n}{n!}$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">genocchi</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">genocchi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">9</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, -1, 0, 1, 0, -3, 0, 17]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">genocchi</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">n</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, partition, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.genocchi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.genocchi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "genocchi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.genocchi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "genocchi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.partition", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "partition", "type": "class", "doc": "<p>Partition numbers</p>\n\n<p>The Partition numbers are a sequence of integers <code>p_n</code> that represent the\nnumber of distinct ways of representing <code>n</code> as a sum of natural numbers\n(with order irrelevant). The generating function for <code>p_n</code> is given by:</p>\n\n<p>$$\\sum_{n=0}^\\infty p_n x^n = \\prod_{k=1}^\\infty (1 - x^k)^{-1}$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">partition</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">[</span><span class=\"n\">partition</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">n</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">9</span><span class=\"p\">)]</span>\n<span class=\"go\">[1, 1, 2, 3, 5, 7, 11, 15, 22]</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">negative</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">partition</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, tribonacci</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.combinatorial.numbers.partition, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.partition.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "partition.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.partition.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "partition.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Rem", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Rem", "type": "class", "doc": "<p>Returns the remainder when <code>p</code> is divided by <code>q</code> where <code>p</code> is finite\nand <code>q</code> is not equal to zero. The result, <code>p - int(p/q)*q</code>, has the same sign\nas the divisor.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>p : Expr\n    Dividend.</p>\n\n<p>q : Expr\n    Divisor.</p>\n\n<h1 id=\"notes\">Notes</h1>\n\n<p><code>Rem</code> corresponds to the <code>%</code> operator in C.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Rem</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Rem</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">Rem(x**3, y)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Rem</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"o\">-</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">:</span> <span class=\"mi\">3</span><span class=\"p\">})</span>\n<span class=\"go\">-2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Mod</p>\n", "bases": "sympy.functions.elementary.miscellaneous.Rem, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Rem.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Rem.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Rem.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Rem.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.re", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re", "type": "class", "doc": "<p>Returns real part of expression. This function performs only\nelementary analysis and so it will fail to decompose properly\nmore complicated expressions. If completely simplified result\nis needed then use <code>Basic.as_real_imag()</code> or perform complex\nexpansion on instance of this function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">re</span><span class=\"p\">,</span> <span class=\"n\">im</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"n\">symbols</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;x y&#39;</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">E</span><span class=\"p\">)</span>\n<span class=\"go\">2*E</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">17</span><span class=\"p\">)</span>\n<span class=\"go\">17</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"mi\">5</span> <span class=\"o\">+</span> <span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">7</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>expr : Expr\n    Real part of expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>im</p>\n", "bases": "sympy.functions.elementary.complexes.re, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'extended_real': True, 'imaginary': False, 'commutative': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.re.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "re.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.im", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im", "type": "class", "doc": "<p>Returns imaginary part of expression. This function performs only\nelementary analysis and so it will fail to decompose properly more\ncomplicated expressions. If completely simplified result is needed then\nuse <code>Basic.as_real_imag()</code> or perform complex expansion on instance of\nthis function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">re</span><span class=\"p\">,</span> <span class=\"n\">im</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">E</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">17</span><span class=\"p\">)</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">re(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"n\">re</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">im(y)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">im</span><span class=\"p\">(</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">3</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>expr : Expr\n    Imaginary part of expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>re</p>\n", "bases": "sympy.functions.elementary.complexes.im, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'extended_real': True, 'imaginary': False, 'commutative': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.im.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "im.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign", "type": "class", "doc": "<p>Returns the complex sign of an expression:</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>If the expression is real the sign will be:</p>\n\n<pre><code>* $1$ if expression is positive\n* $0$ if expression is equal to zero\n* $-1$ if expression is negative\n</code></pre>\n\n<p>If the expression is imaginary the sign will be:</p>\n\n<pre><code>* $I$ if im(expression) is positive\n* $-I$ if im(expression) is negative\n</code></pre>\n\n<p>Otherwise an unevaluated expression will be returned. When evaluated, the\nresult (in general) will be <code>cos(arg(expr)) + I*sin(arg(expr))</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sign</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sign</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sign</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sign</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">-I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sign</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">sign(1 + I)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">_</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">()</span>\n<span class=\"go\">0.707106781186548 + 0.707106781186548*I</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or imaginary expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>expr : Expr\n    Complex sign of expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Abs, conjugate</p>\n", "bases": "sympy.functions.elementary.complexes.sign, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'complex': True, 'commutative': True, 'infinite': False, 'finite': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.sign.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sign.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs", "type": "class", "doc": "<p>Return the absolute value of the argument.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This is an extension of the built-in function <code>abs()</code> to accept symbolic\nvalues.  If you pass a SymPy expression to the built-in <code>abs()</code>, it will\npass it automatically to <code>Abs()</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Abs</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">Abs(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">x**2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">abs</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"c1\"># The Python built-in</span>\n<span class=\"go\">Abs(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">x</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(9*x**2 + 4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Abs</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">8</span>\n</code></pre></div>\n\n<p>Note that the Python built-in will return either an Expr or int depending on\nthe argument::</p>\n\n<pre><code>&gt;&gt;&gt; type(abs(-1))\n&lt;... 'int'&gt;\n&gt;&gt;&gt; type(abs(S.NegativeOne))\n&lt;class 'sympy.core.numbers.One'&gt;\n</code></pre>\n\n<p>Abs will always return a SymPy object.</p>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>expr : Expr\n    Absolute value returned can be an expression or integer depending on\n    input arg.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sign, conjugate</p>\n", "bases": "sympy.functions.elementary.complexes.Abs, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_extended_negative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_extended_negative", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_extended_nonnegative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_extended_nonnegative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'extended_negative': False, 'negative': False, 'extended_real': True, 'imaginary': False, 'commutative': True, 'extended_nonnegative': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_negative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_negative", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Abs.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Abs.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.conjugate", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "conjugate", "type": "class", "doc": "<p>Returns the <em>complex conjugate</em> <sup class=\"footnote-ref\" id=\"fnref-1\"><a href=\"#fn-1\">1</a></sup> of an argument.\nIn mathematics, the complex conjugate of a complex number\nis given by changing the sign of the imaginary part.</p>\n\n<p>Thus, the conjugate of the complex number\n\\( a + ib \\) (where $a$ and $b$ are real numbers) is \\( a - ib \\)</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">-I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"mi\">3</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">3 - 2*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"mi\">5</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">5 + I</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>arg : Expr\n    Complex conjugate of arg as real, imaginary or mixed expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sign, Abs</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-1\">\n<p>https://en.wikipedia.org/wiki/Complex_conjugation&#160;<a href=\"#fnref-1\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.complexes.conjugate, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.conjugate.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "conjugate.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.conjugate.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "conjugate.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg", "type": "class", "doc": "<p>Returns the argument (in radians) of a complex number. The argument is\nevaluated in consistent convention with <code>atan2</code> where the branch-cut is\ntaken along the negative real axis and <code>arg(z)</code> is in the interval\n$(-\\pi,\\pi]$. For a positive number, the argument is always 0; the\nargument of a negative number is $\\pi$; and the argument of 0\nis undefined and returns <code>nan</code>. So the <code>arg</code> function will never nest\ngreater than 3 levels since at the 4th application, the result must be\nnan; for a real number, nan is returned on the 3rd application.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">arg</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">Dummy</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"mf\">2.0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">pi/6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"mi\">4</span> <span class=\"o\">+</span> <span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">atan(3/4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"mf\">0.8</span> <span class=\"o\">+</span> <span class=\"mf\">0.6</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">0.643501108793284</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))))</span>\n<span class=\"go\">nan</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">real</span> <span class=\"o\">=</span> <span class=\"n\">Dummy</span><span class=\"p\">(</span><span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">arg</span><span class=\"p\">(</span><span class=\"n\">real</span><span class=\"p\">)))</span>\n<span class=\"go\">nan</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>value : Expr\n    Returns arc tangent of arg measured in radians.</p>\n", "bases": "sympy.functions.elementary.complexes.arg, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'finite': True, 'infinite': False, 'extended_real': True, 'imaginary': False, 'commutative': True, 'real': True, 'hermitian': True, 'complex': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.arg.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "arg.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.polar_lift", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polar_lift", "type": "class", "doc": "<p>Lift argument to the Riemann surface of the logarithm, using the\nstandard branch.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">polar_lift</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;p&#39;</span><span class=\"p\">,</span> <span class=\"n\">polar</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">4*exp_polar(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">4*exp_polar(I*pi)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">exp_polar(-I*pi/2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"n\">I</span> <span class=\"o\">+</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polar_lift(2 + I)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">4*polar_lift(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">p</span><span class=\"p\">)</span>\n<span class=\"go\">4*p</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr\n    Real or complex expression.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.exponential.exp_polar\nperiodic_argument</p>\n", "bases": "sympy.functions.elementary.complexes.polar_lift, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.polar_lift.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polar_lift.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.polar_lift.is_polar", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polar_lift.is_polar", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.polar_lift.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polar_lift.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'polar': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.periodic_argument", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "periodic_argument", "type": "class", "doc": "<p>Represent the argument on a quotient of the Riemann surface of the\nlogarithm. That is, given a period $P$, always return a value in\n$(-P/2, P/2]$, by using $\\exp(PI) = 1$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">periodic_argument</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">periodic_argument</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">-pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">periodic_argument</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">),</span> <span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>ar : Expr\n    A polar number.</p>\n\n<p>period : Expr\n    The period $P$.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.exponential.exp_polar\npolar_lift : Lift argument to the Riemann surface of the logarithm\nprincipal_branch</p>\n", "bases": "sympy.functions.elementary.complexes.periodic_argument, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.periodic_argument.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "periodic_argument.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.periodic_argument.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "periodic_argument.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.principal_branch", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "principal_branch", "type": "class", "doc": "<p>Represent a polar number reduced to its principal branch on a quotient\nof the Riemann surface of the logarithm.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This is a function of two arguments. The first argument is a polar\nnumber <code>z</code>, and the second one a positive real number or infinity, <code>p</code>.\nThe result is <code>z mod exp_polar(I*p)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">principal_branch</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">principal_branch</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">principal_branch</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">3*exp_polar(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">principal_branch</span><span class=\"p\">(</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">3*principal_branch(z, 2*pi)</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>x : Expr\n    A polar number.</p>\n\n<p>period : Expr\n    Positive real number or infinity.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.exponential.exp_polar\npolar_lift : Lift argument to the Riemann surface of the logarithm\nperiodic_argument</p>\n", "bases": "sympy.functions.elementary.complexes.principal_branch, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.principal_branch.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "principal_branch.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.principal_branch.is_polar", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "principal_branch.is_polar", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.principal_branch.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "principal_branch.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'polar': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.transpose", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "transpose", "type": "class", "doc": "<p>Linear map transposition.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">transpose</span><span class=\"p\">,</span> <span class=\"n\">Matrix</span><span class=\"p\">,</span> <span class=\"n\">MatrixSymbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">A</span> <span class=\"o\">=</span> <span class=\"n\">MatrixSymbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;A&#39;</span><span class=\"p\">,</span> <span class=\"mi\">25</span><span class=\"p\">,</span> <span class=\"mi\">9</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">transpose</span><span class=\"p\">(</span><span class=\"n\">A</span><span class=\"p\">)</span>\n<span class=\"go\">A.T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">B</span> <span class=\"o\">=</span> <span class=\"n\">MatrixSymbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;B&#39;</span><span class=\"p\">,</span> <span class=\"mi\">9</span><span class=\"p\">,</span> <span class=\"mi\">22</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">transpose</span><span class=\"p\">(</span><span class=\"n\">B</span><span class=\"p\">)</span>\n<span class=\"go\">B.T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">transpose</span><span class=\"p\">(</span><span class=\"n\">A</span><span class=\"o\">*</span><span class=\"n\">B</span><span class=\"p\">)</span>\n<span class=\"go\">B.T*A.T</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">M</span> <span class=\"o\">=</span> <span class=\"n\">Matrix</span><span class=\"p\">([[</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">90</span><span class=\"p\">,</span> <span class=\"mi\">12</span><span class=\"p\">]])</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">M</span>\n<span class=\"go\">Matrix([</span>\n<span class=\"go\">[ 4,  5],</span>\n<span class=\"go\">[ 2,  1],</span>\n<span class=\"go\">[90, 12]])</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">transpose</span><span class=\"p\">(</span><span class=\"n\">M</span><span class=\"p\">)</span>\n<span class=\"go\">Matrix([</span>\n<span class=\"go\">[4, 2, 90],</span>\n<span class=\"go\">[5, 1, 12]])</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Matrix\n     Matrix or matrix expression to take the transpose of.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>value : Matrix\n    Transpose of arg.</p>\n", "bases": "sympy.functions.elementary.complexes.transpose, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.transpose.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "transpose.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.transpose.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "transpose.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.adjoint", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "adjoint", "type": "class", "doc": "<p>Conjugate transpose or Hermite conjugation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">adjoint</span><span class=\"p\">,</span> <span class=\"n\">MatrixSymbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">A</span> <span class=\"o\">=</span> <span class=\"n\">MatrixSymbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;A&#39;</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">adjoint</span><span class=\"p\">(</span><span class=\"n\">A</span><span class=\"p\">)</span>\n<span class=\"go\">Adjoint(A)</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Matrix\n    Matrix or matrix expression to take the adjoint of.</p>\n\n<h1 id=\"returns\">Returns</h1>\n\n<p>value : Matrix\n    Represents the conjugate transpose or Hermite\n    conjugation of arg.</p>\n", "bases": "sympy.functions.elementary.complexes.adjoint, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.adjoint.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "adjoint.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.adjoint.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "adjoint.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sin", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sin", "type": "class", "doc": "<p>The sine function.</p>\n\n<p>Returns the sine of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function will evaluate automatically in the\ncase $x/\\pi$ is some rational number <sup class=\"footnote-ref\" id=\"fnref-4\"><a href=\"#fn-4\">1</a></sup>.  For example,\nif $x$ is a multiple of $\\pi$, $\\pi/2$, $\\pi/3$, $\\pi/4$, and $\\pi/6$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sin</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x*cos(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">6</span><span class=\"p\">)</span>\n<span class=\"go\">1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">12</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(2)/4 + sqrt(6)/4</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>csc, cos, sec, tan, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-4\">\n<p>http://mathworld.wolfram.com/TrigonometryAngles.html&#160;<a href=\"#fnref-4\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.sin, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sin.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sin.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sin.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sin.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.cos", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cos", "type": "class", "doc": "<p>The cosine function.</p>\n\n<p>Returns the cosine of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See <code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">cos</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*x*sin(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">-1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">12</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)/4 + sqrt(6)/4</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, sec, tan, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.cos, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.cos.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cos.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.cos.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cos.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.tan", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tan", "type": "class", "doc": "<p>The tangent function.</p>\n\n<p>Returns the tangent of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See <code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">tan</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tan</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x*(tan(x**2)**2 + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tan</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tan</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">8</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">()</span>\n<span class=\"go\">-1 + sqrt(2)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.tan, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.tan.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tan.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.tan.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tan.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sec", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sec", "type": "class", "doc": "<p>The secant function.</p>\n\n<p>Returns the secant of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See <code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sec</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sec</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x*tan(x**2)*sec(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sec</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, tan, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.sec, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sec.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sec.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sec.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sec.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.csc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csc", "type": "class", "doc": "<p>The cosecant function.</p>\n\n<p>Returns the cosecant of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See <code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">csc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">csc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*x*cot(x**2)*csc(x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">csc</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, cos, sec, tan, cot\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.csc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.csc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.csc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.cot", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cot", "type": "class", "doc": "<p>The cotangent function.</p>\n\n<p>Returns the cotangent of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>See <code>sin</code> for notes about automatic evaluation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">cot</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cot</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x*(-cot(x**2)**2 - 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cot</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cot</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">/</span><span class=\"mi\">12</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(3) + 2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan\nasin, acsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.cot, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.cot.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cot.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.cot.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cot.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinc", "type": "class", "doc": "<p>Represents an unnormalized sinc function:</p>\n\n<p>$$\\operatorname{sinc}(x) =\n\\begin{cases}\n  \\frac{\\sin x}{x} &amp; \\qquad x \\neq 0 \\\n  1 &amp; \\qquad x = 0\n\\end{cases}$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sinc</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">jn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sinc(x)</span>\n</code></pre></div>\n\n<ul>\n<li>Automated Evaluation</li>\n</ul>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<ul>\n<li>Differentiation</li>\n</ul>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">()</span>\n<span class=\"go\">cos(x)/x - sin(x)/x**2</span>\n</code></pre></div>\n\n<ul>\n<li>Series Expansion</li>\n</ul>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">()</span>\n<span class=\"go\">1 - x**2/6 + x**4/120 + O(x**6)</span>\n</code></pre></div>\n\n<ul>\n<li>As zero'th order spherical Bessel Function</li>\n</ul>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinc</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">)</span>\n<span class=\"go\">jn(0, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See also</h1>\n\n<p>sin</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.sinc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.asin", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asin", "type": "class", "doc": "<p>The inverse sine function.</p>\n\n<p>Returns the arcsine of x in radians.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>asin(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$ and for some instances when the\nresult is a rational multiple of $\\pi$ (see the <code>eval</code> class method).</p>\n\n<p>A purely imaginary argument will lead to an asinh expression.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">asin</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asin</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asin</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asin</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asin</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-oo*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nacsc, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.asin, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.asin.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asin.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.asin.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asin.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acos", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acos", "type": "class", "doc": "<p>The inverse cosine function.</p>\n\n<p>Returns the arc cosine of x (measured in radians).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p><code>acos(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$ and for some instances when\nthe result is a rational multiple of $\\pi$ (see the eval class method).</p>\n\n<p><code>acos(zoo)</code> evaluates to <code>zoo</code>\n(see note in <code>sympy.functions.elementary.trigonometric.asec</code>)</p>\n\n<p>A purely imaginary argument will be rewritten to asinh.</p>\n\n<h1 id=\"examples-2\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acos</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acos</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acos</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acos</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.acos, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acos.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acos.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acos.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acos.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan", "type": "class", "doc": "<p>The inverse tangent function.</p>\n\n<p>Returns the arc tangent of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>atan(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$ and for some instances when the\nresult is a rational multiple of $\\pi$ (see the eval class method).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atan</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, acos, asec, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.atan, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.asec", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asec", "type": "class", "doc": "<p>The inverse secant function.</p>\n\n<p>Returns the arc secant of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>asec(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$ and for some instances when the\nresult is a rational multiple of $\\pi$ (see the eval class method).</p>\n\n<p><code>asec(x)</code> has branch cut in the interval $[-1, 1]$. For complex arguments,\nit can be defined <sup class=\"footnote-ref\" id=\"fnref-4\"><a href=\"#fn-4\">1</a></sup> as</p>\n\n<p>$$\\operatorname{sec^{-1}}(z) = -i\\frac{\\log\\left(\\sqrt{1 - z^2} + 1\\right)}{z}$$</p>\n\n<p>At <code>x = 0</code>, for positive branch cut, the limit evaluates to <code>zoo</code>. For\nnegative branch cut, the limit</p>\n\n<p>$$\\lim_{z \\to 0}-i\\frac{\\log\\left(-\\sqrt{1 - z^2} + 1\\right)}{z}$$</p>\n\n<p>simplifies to \\( -i\\log\\left(z/2 + O\\left(z^3\\right)\\right) \\) which\nultimately evaluates to <code>zoo</code>.</p>\n\n<p>As <code>acos(x) = asec(1/x)</code>, a similar argument can be given for\n<code>acos(x)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">asec</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asec</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asec</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asec</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asec</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, acos, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-4\">\n<p>http://reference.wolfram.com/language/ref/ArcSec.html&#160;<a href=\"#fnref-4\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.asec, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.asec.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asec.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.asec.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asec.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsc", "type": "class", "doc": "<p>The inverse cosecant function.</p>\n\n<p>Returns the arc cosecant of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>acsc(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, 0, 1, -1}$` and for some instances when the\nresult is a rational multiple of $\\pi$ (see the <code>eval</code> class method).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acsc</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsc</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acos, asec, atan, acot, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.acsc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acot", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acot", "type": "class", "doc": "<p>The inverse cotangent function.</p>\n\n<p>Returns the arc cotangent of x (measured in radians).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>acot(x)</code> will evaluate automatically in the cases\n$x \\in {\\infty, -\\infty, \\tilde{\\infty}, 0, 1, -1}$\nand for some instances when the result is a rational multiple of $\\pi$\n(see the eval class method).</p>\n\n<p>A purely imaginary argument will lead to an <code>acoth</code> expression.</p>\n\n<p><code>acot(x)</code> has a branch cut along $(-i, i)$, hence it is discontinuous\nat 0. Its range for real $x$ is $(-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acot</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acot</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acot</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acot</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-5*pi/12</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, acos, asec, atan, atan2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.acot, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acot.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acot.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acot.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acot.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan2", "type": "class", "doc": "<p>The function <code>atan2(y, x)</code> computes <code>\\operatorname{atan}(y/x)</code> taking\ntwo arguments <code>y</code> and <code>x</code>.  Signs of both <code>y</code> and <code>x</code> are considered to\ndetermine the appropriate quadrant of <code>\\operatorname{atan}(y/x)</code>.\nThe range is <code>(-\\pi, \\pi]</code>. The complete definition reads as follows:</p>\n\n<p>$$\\operatorname{atan2}(y, x) =\n\\begin{cases}\n  \\arctan\\left(\\frac y x\\right) &amp; \\qquad x &gt; 0 \\\n  \\arctan\\left(\\frac y x\\right) + \\pi&amp; \\qquad y \\ge 0, x &lt; 0 \\\n  \\arctan\\left(\\frac y x\\right) - \\pi&amp; \\qquad y &lt; 0, x &lt; 0 \\\n  +\\frac{\\pi}{2} &amp; \\qquad y &gt; 0, x = 0 \\\n  -\\frac{\\pi}{2} &amp; \\qquad y &lt; 0, x = 0 \\\n  \\text{undefined} &amp; \\qquad y = 0, x = 0\n\\end{cases}$$</p>\n\n<p>Attention: Note the role reversal of both arguments. The <code>y</code>-coordinate\nis the first argument and the <code>x</code>-coordinate the second.</p>\n\n<p>If either <code>x</code> or <code>y</code> is complex:</p>\n\n<p>$$\\operatorname{atan2}(y, x) =\n    -i\\log\\left(\\frac{x + iy}{\\sqrt{x^2 + y^2}}\\right)$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Going counter-clock wise around the origin we find the\nfollowing angles:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atan2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">3*pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-3*pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/4</span>\n</code></pre></div>\n\n<p>which are all correct. Compare this to the results of the ordinary\n<code>\\operatorname{atan}</code> function for the point <code>(x, y) = (-1, 1)</code></p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atan</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">3*pi/4</span>\n</code></pre></div>\n\n<p>where only the <code>\\operatorname{atan2}</code> function reurns what we expect.\nWe can differentiate the function with respect to both arguments:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-y/(x**2 + y**2)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">x/(x**2 + y**2)</span>\n</code></pre></div>\n\n<p>We can express the <code>\\operatorname{atan2}</code> function in terms of\ncomplex logarithms:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">log</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">)</span>\n<span class=\"go\">-I*log((x + I*y)/sqrt(x**2 + y**2))</span>\n</code></pre></div>\n\n<p>and in terms of <code>\\operatorname(atan)</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atan</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atan2</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">atan</span><span class=\"p\">)</span>\n<span class=\"go\">Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) &lt; 0), (0, Ne(x, 0)), (nan, True))</span>\n</code></pre></div>\n\n<p>but note that this form is undefined on the negative real axis.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sin, csc, cos, sec, tan, cot\nasin, acsc, acos, asec, atan, acot</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.trigonometric.atan2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.atan2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atan2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp_polar", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp_polar", "type": "class", "doc": "<p>Represent a <em>polar number</em> (see g-function Sphinx documentation).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>exp_polar</code> represents the function\n<code>Exp: \\mathbb{C} \\rightarrow \\mathcal{S}</code>, sending the complex number\n<code>z = a + bi</code> to the polar number <code>r = exp(a), \\theta = b</code>. It is one of\nthe main functions to construct polar numbers.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">exp</span>\n</code></pre></div>\n\n<p>The main difference is that polar numbers don't \"wrap around\" at <code>2 \\pi</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">exp_polar(2*I*pi)</span>\n</code></pre></div>\n\n<p>apart from that they behave mostly like classical complex numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">exp_polar(5)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.simplify.powsimp.powsimp\npolar_lift\nperiodic_argument\nprincipal_branch</p>\n", "bases": "sympy.functions.elementary.exponential.exp_polar, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp_polar.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp_polar.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp_polar.is_polar", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp_polar.is_polar", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp_polar.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp_polar.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'polar': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp", "type": "class", "doc": "<p>The exponential function, \\( e^x \\).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">exp(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">exp(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>arg : Expr</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>log</p>\n", "bases": "sympy.functions.elementary.exponential.exp, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.exp.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "exp.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.ln", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ln", "type": "class", "doc": "<p>The natural logarithm function <code>\\ln(x)</code> or <code>\\log(x)</code>.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Logarithms are taken with the natural base, <code>e</code>. To get\na logarithm of a different base <code>b</code>, use <code>log(x, b)</code>,\nwhich is essentially short-hand for <code>log(x)/log(b)</code>.</p>\n\n<p><code>log</code> represents the principal branch of the natural\nlogarithm. As such it has a branch cut along the negative\nreal axis and returns values having a complex argument in\n<code>(-\\pi, \\pi]</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">log</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-log(3)/log(2) + 3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">log(2) + 2*I*pi/3</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>exp</p>\n", "bases": "sympy.functions.elementary.exponential.log, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.ln.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ln.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.ln.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ln.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.log", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "log", "type": "class", "doc": "<p>The natural logarithm function <code>\\ln(x)</code> or <code>\\log(x)</code>.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Logarithms are taken with the natural base, <code>e</code>. To get\na logarithm of a different base <code>b</code>, use <code>log(x, b)</code>,\nwhich is essentially short-hand for <code>log(x)/log(b)</code>.</p>\n\n<p><code>log</code> represents the principal branch of the natural\nlogarithm. As such it has a branch cut along the negative\nreal axis and returns values having a complex argument in\n<code>(-\\pi, \\pi]</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">log</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">8</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-log(3)/log(2) + 3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">log(2) + 2*I*pi/3</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>exp</p>\n", "bases": "sympy.functions.elementary.exponential.log, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.log.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "log.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.log.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "log.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.LambertW", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LambertW", "type": "class", "doc": "<p>The Lambert W function $W(z)$ is defined as the inverse\nfunction of $w \\exp(w)$ <sup class=\"footnote-ref\" id=\"fnref-1\"><a href=\"#fn-1\">1</a></sup>.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>In other words, the value of $W(z)$ is such that $z = W(z) \\exp(W(z))$\nfor any complex number $z$.  The Lambert W function is a multivalued\nfunction with infinitely many branches $W_k(z)$, indexed by\n$k \\in \\mathbb{Z}$.  Each branch gives a different solution $w$\nof the equation $z = w \\exp(w)$.</p>\n\n<p>The Lambert W function has two partially real branches: the\nprincipal branch ($k = 0$) is real for real $z &gt; -1/e$, and the\n$k = -1$ branch is real for $-1/e &lt; z &lt; 0$. All branches except\n$k = 0$ have a logarithmic singularity at $z = 0$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">LambertW</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LambertW</span><span class=\"p\">(</span><span class=\"mf\">1.2</span><span class=\"p\">)</span>\n<span class=\"go\">0.635564016364870</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LambertW</span><span class=\"p\">(</span><span class=\"mf\">1.2</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">()</span>\n<span class=\"go\">-1.34747534407696 - 4.41624341514535*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LambertW</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">is_real</span>\n<span class=\"go\">False</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-1\">\n<p>https://en.wikipedia.org/wiki/Lambert_W_function&#160;<a href=\"#fnref-1\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.exponential.LambertW, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.LambertW.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LambertW.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.LambertW.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LambertW.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinh", "type": "class", "doc": "<p><code>sinh(x)</code> is the hyperbolic sine of <code>x</code>.</p>\n\n<p>The hyperbolic sine function is $\\frac{e^x - e^{-x}}{2}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sinh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sinh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sinh(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>cosh, tanh, asinh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.sinh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sinh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sinh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.cosh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cosh", "type": "class", "doc": "<p><code>cosh(x)</code> is the hyperbolic cosine of <code>x</code>.</p>\n\n<p>The hyperbolic cosine function is $\\frac{e^x + e^{-x}}{2}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">cosh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cosh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">cosh(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, tanh, acosh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.cosh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.cosh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cosh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.cosh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "cosh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.tanh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tanh", "type": "class", "doc": "<p><code>tanh(x)</code> is the hyperbolic tangent of <code>x</code>.</p>\n\n<p>The hyperbolic tangent function is $\\frac{\\sinh(x)}{\\cosh(x)}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">tanh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">tanh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">tanh(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, cosh, atanh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.tanh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.tanh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tanh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.tanh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "tanh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.coth", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "coth", "type": "class", "doc": "<p><code>coth(x)</code> is the hyperbolic cotangent of <code>x</code>.</p>\n\n<p>The hyperbolic cotangent function is $\\frac{\\cosh(x)}{\\sinh(x)}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">coth</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">coth</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">coth(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, cosh, acoth</p>\n", "bases": "sympy.functions.elementary.hyperbolic.coth, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.coth.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "coth.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.coth.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "coth.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.sech", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sech", "type": "class", "doc": "<p><code>sech(x)</code> is the hyperbolic secant of <code>x</code>.</p>\n\n<p>The hyperbolic secant function is $\\frac{2}{e^x + e^{-x}}$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sech</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sech</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sech(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, cosh, tanh, coth, csch, asinh, acosh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.sech, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.sech.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sech.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.sech.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "sech.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.csch", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csch", "type": "class", "doc": "<p><code>csch(x)</code> is the hyperbolic cosecant of <code>x</code>.</p>\n\n<p>The hyperbolic cosecant function is $\\frac{2}{e^x - e^{-x}}$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">csch</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">csch</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">csch(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sinh, cosh, tanh, sech, asinh, acosh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.csch, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.csch.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csch.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.csch.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "csch.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.asinh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asinh", "type": "class", "doc": "<p><code>asinh(x)</code> is the inverse hyperbolic sine of <code>x</code>.</p>\n\n<p>The inverse hyperbolic sine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">asinh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asinh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1/sqrt(x**2 + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asinh</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">log(1 + sqrt(2))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>acosh, atanh, sinh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.asinh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.asinh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asinh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.asinh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asinh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acosh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acosh", "type": "class", "doc": "<p><code>acosh(x)</code> is the inverse hyperbolic cosine of <code>x</code>.</p>\n\n<p>The inverse hyperbolic cosine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acosh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acosh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1/sqrt(x**2 - 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acosh</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh, atanh, cosh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.acosh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acosh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acosh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acosh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acosh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.atanh", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atanh", "type": "class", "doc": "<p><code>atanh(x)</code> is the inverse hyperbolic tangent of <code>x</code>.</p>\n\n<p>The inverse hyperbolic tangent function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">atanh</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">atanh</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1/(1 - x**2)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh, acosh, tanh</p>\n", "bases": "sympy.functions.elementary.hyperbolic.atanh, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.atanh.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atanh.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.atanh.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "atanh.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acoth", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acoth", "type": "class", "doc": "<p><code>acoth(x)</code> is the inverse hyperbolic cotangent of <code>x</code>.</p>\n\n<p>The inverse hyperbolic cotangent function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acoth</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acoth</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1/(1 - x**2)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh, acosh, coth</p>\n", "bases": "sympy.functions.elementary.hyperbolic.acoth, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acoth.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acoth.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acoth.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acoth.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.asech", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asech", "type": "class", "doc": "<p><code>asech(x)</code> is the inverse hyperbolic secant of <code>x</code>.</p>\n\n<p>The inverse hyperbolic secant function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">asech</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-1/(x*sqrt(1 - x**2))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">I*pi/3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">3*I*pi/4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">asech</span><span class=\"p\">((</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)))</span>\n<span class=\"go\">I*pi/12</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh, atanh, cosh, acoth</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.hyperbolic.asech, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.asech.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asech.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.asech.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "asech.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsch", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsch", "type": "class", "doc": "<p><code>acsch(x)</code> is the inverse hyperbolic cosecant of <code>x</code>.</p>\n\n<p>The inverse hyperbolic cosecant function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">acsch</span><span class=\"p\">,</span> <span class=\"n\">sqrt</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-1/(x**2*sqrt(1 + x**(-2)))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">log(1 + sqrt(2))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">ImaginaryUnit</span><span class=\"p\">)</span>\n<span class=\"go\">-I*pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">ImaginaryUnit</span><span class=\"p\">)</span>\n<span class=\"go\">I*pi/6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">acsch</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"o\">.</span><span class=\"n\">ImaginaryUnit</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">6</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)))</span>\n<span class=\"go\">-5*I*pi/12</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>asinh</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.hyperbolic.acsch, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsch.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsch.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.acsch.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "acsch.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.floor", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "floor", "type": "class", "doc": "<p>Floor is a univariate function which returns the largest integer\nvalue not greater than its argument. This implementation\ngeneralizes floor to complex numbers by taking the floor of the\nreal and imaginary parts separately.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">floor</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">Float</span><span class=\"p\">,</span> <span class=\"n\">Rational</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"mi\">17</span><span class=\"p\">)</span>\n<span class=\"go\">17</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">23</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">))</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">E</span><span class=\"p\">)</span>\n<span class=\"go\">5</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">Float</span><span class=\"p\">(</span><span class=\"mf\">0.567</span><span class=\"p\">))</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">floor</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">2 + 2*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.integers.ceiling</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.integers.floor, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.floor.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "floor.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.floor.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "floor.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.ceiling", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ceiling", "type": "class", "doc": "<p>Ceiling is a univariate function which returns the smallest integer\nvalue not less than its argument. This implementation\ngeneralizes ceiling to complex numbers by taking the ceiling of the\nreal and imaginary parts separately.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">ceiling</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">Float</span><span class=\"p\">,</span> <span class=\"n\">Rational</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"mi\">17</span><span class=\"p\">)</span>\n<span class=\"go\">17</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">23</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">))</span>\n<span class=\"go\">3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">E</span><span class=\"p\">)</span>\n<span class=\"go\">6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">Float</span><span class=\"p\">(</span><span class=\"mf\">0.567</span><span class=\"p\">))</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ceiling</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"mi\">5</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">3 + 3*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.integers.floor</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.integers.ceiling, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.ceiling.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ceiling.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.ceiling.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ceiling.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.frac", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "frac", "type": "class", "doc": "<p>Represents the fractional part of x</p>\n\n<p>For real numbers it is defined <sup class=\"footnote-ref\" id=\"fnref-1\"><a href=\"#fn-1\">1</a></sup> as</p>\n\n<p>$$x - \\left\\lfloor{x}\\right\\rfloor$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">frac</span><span class=\"p\">,</span> <span class=\"n\">Rational</span><span class=\"p\">,</span> <span class=\"n\">floor</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">1/3</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">Rational</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">2/3</span>\n</code></pre></div>\n\n<p>returns zero for integer arguments</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>rewrite as floor</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">floor</span><span class=\"p\">)</span>\n<span class=\"go\">x - floor(x)</span>\n</code></pre></div>\n\n<p>for complex arguments</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">r</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;r&#39;</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">t</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;t&#39;</span><span class=\"p\">,</span> <span class=\"n\">real</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">frac</span><span class=\"p\">(</span><span class=\"n\">t</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">r</span><span class=\"p\">)</span>\n<span class=\"go\">I*frac(r) + frac(t)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.functions.elementary.integers.floor\nsympy.functions.elementary.integers.ceiling</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n<li id=\"fn-1\">\n<p>https://en.wikipedia.org/wiki/Fractional_part&#160;<a href=\"#fnref-1\" class=\"footnoteBackLink\" title=\"Jump back to footnote 1 in the text.\">&#8617;</a></p>\n</li>\n</ol>\n</div>\n", "bases": "sympy.functions.elementary.integers.frac, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.frac.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "frac.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.frac.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "frac.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Piecewise", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Piecewise", "type": "class", "doc": "<p>Represents a piecewise function.</p>\n\n<p>Usage:</p>\n\n<p>Piecewise( (expr,cond), (expr,cond), ... )\n    - Each argument is a 2-tuple defining an expression and condition\n    - The conds are evaluated in turn returning the first that is True.\n      If any of the evaluated conds are not explicitly False,\n      e.g. <code>x &lt; 1</code>, the function is returned in symbolic form.\n    - If the function is evaluated at a place where all conditions are False,\n      nan will be returned.\n    - Pairs where the cond is explicitly False, will be removed and no pair\n      appearing after a True condition will ever be retained. If a single\n      pair with a True condition remains, it will be returned, even when\n      evaluation is False.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Piecewise</span><span class=\"p\">,</span> <span class=\"n\">log</span><span class=\"p\">,</span> <span class=\"n\">piecewise_fold</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">f</span> <span class=\"o\">=</span> <span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span> <span class=\"o\">=</span> <span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span> <span class=\"o\">=</span> <span class=\"n\">Piecewise</span><span class=\"p\">((</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">&lt;</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"n\">f</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">&lt;=</span> <span class=\"mi\">1</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"n\">g</span><span class=\"p\">,</span> <span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">log(5)</span>\n</code></pre></div>\n\n<p>Booleans can contain Piecewise elements:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">cond</span> <span class=\"o\">=</span> <span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">&lt;</span> <span class=\"n\">y</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">Piecewise</span><span class=\"p\">((</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span> <span class=\"o\">&lt;</span> <span class=\"mi\">0</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"kc\">True</span><span class=\"p\">)));</span> <span class=\"n\">cond</span>\n<span class=\"go\">Piecewise((2, x &lt; 0), (3, True)) &lt; y</span>\n</code></pre></div>\n\n<p>The folded version of this results in a Piecewise whose\nexpressions are Booleans:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">folded_cond</span> <span class=\"o\">=</span> <span class=\"n\">piecewise_fold</span><span class=\"p\">(</span><span class=\"n\">cond</span><span class=\"p\">);</span> <span class=\"n\">folded_cond</span>\n<span class=\"go\">Piecewise((2 &lt; y, x &lt; 0), (3 &lt; y, True))</span>\n</code></pre></div>\n\n<p>When a Boolean containing Piecewise (like cond) or a Piecewise\nwith Boolean expressions (like folded_cond) is used as a condition,\nit is converted to an equivalent <code>~.ITE</code> object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Piecewise</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">folded_cond</span><span class=\"p\">))</span>\n<span class=\"go\">Piecewise((1, ITE(x &lt; 0, y &gt; 2, y &gt; 3)))</span>\n</code></pre></div>\n\n<p>When a condition is an <code>ITE</code>, it will be converted to a simplified\nBoolean expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">piecewise_fold</span><span class=\"p\">(</span><span class=\"n\">_</span><span class=\"p\">)</span>\n<span class=\"go\">Piecewise((1, ((x &gt;= 0) | (y &gt; 2)) &amp; ((y &gt; 3) | (x &lt; 0))))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>piecewise_fold, ITE</p>\n", "bases": "sympy.functions.elementary.piecewise.Piecewise, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Piecewise.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Piecewise.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Piecewise.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Piecewise.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf", "type": "class", "doc": "<p>The Gauss error function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erf}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2} \\mathrm{d}t.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">erf</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-oo*I</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1 and $I$ from the argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-erf(z)</span>\n</code></pre></div>\n\n<p>The error function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">erf(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">2*exp(-z**2)/sqrt(pi)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the error function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.999999984582742099719981147840</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">-1296959.73071763923152794095062*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erfc: Complementary error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.erf, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfc", "type": "class", "doc": "<p>Complementary Error Function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erfc}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_x^\\infty e^{-t^2} \\mathrm{d}t</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">erfc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-oo*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo*I</span>\n</code></pre></div>\n\n<p>The error function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">erfc(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-2*exp(-z**2)/sqrt(pi)</span>\n</code></pre></div>\n\n<p>It also follows</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">2 - erfc(z)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the complementary error function to arbitrary\nprecision on the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.0000000154172579002800188521596734869</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfc</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">1.0 - 1296959.73071763923152794095062*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.erfc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfi", "type": "class", "doc": "<p>Imaginary error function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The function erfi is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erfi}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{t^2} \\mathrm{d}t</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">erfi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-I</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1 and $I$ from the argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-erfi(z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">erfi(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">2*exp(z**2)/sqrt(pi)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the imaginary error function to arbitrary\nprecision on the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">18.5648024145755525987042919132</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">-0.995322265018952734162069256367*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.erfi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2", "type": "class", "doc": "<p>Two-argument error function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erf2}(x, y) = \\frac{2}{\\sqrt{\\pi}} \\int_x^y e^{-t^2} \\mathrm{d}t</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">erf2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">1 - erf(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-erf(x) - 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">erf(y) - 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">erf(y) + 1</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">-erf2(x, y)</span>\n</code></pre></div>\n\n<p>The error function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">))</span>\n<span class=\"go\">erf2(conjugate(x), conjugate(y))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$, $y$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*exp(-x**2)/sqrt(pi)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf2</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">2*exp(-y**2)/sqrt(pi)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerfi: Imaginary error function.\nerfinv: Inverse error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.erf2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfinv", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfinv", "type": "class", "doc": "<p>Inverse Error Function. The erfinv function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erf}(x) = y \\quad \\Rightarrow \\quad \\mathrm{erfinv}(y) = x</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">erfinv</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfinv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfinv</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erfinv</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)*exp(erfinv(x)**2)/2</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the inverse error function to arbitrary\nprecision on [-1, 1]:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfinv</span><span class=\"p\">(</span><span class=\"mf\">0.2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.179143454621291692285822705344</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfcinv: Inverse Complementary error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.erfinv, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfinv.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfinv.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfinv.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfinv.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfcinv", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfcinv", "type": "class", "doc": "<p>Inverse Complementary Error Function. The erfcinv function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erfc}(x) = y \\quad \\Rightarrow \\quad \\mathrm{erfcinv}(y) = x</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">erfcinv</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfcinv</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erfcinv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erfcinv</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(pi)*exp(erfcinv(x)**2)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerf2inv: Inverse two-argument error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.erfcinv, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfcinv.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfcinv.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erfcinv.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erfcinv.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2inv", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2inv", "type": "class", "doc": "<p>Two-argument Inverse error function. The erf2inv function is defined as:</p>\n\n<p>.. math ::\n    \\mathrm{erf2}(x, w) = y \\quad \\Rightarrow \\quad \\mathrm{erf2inv}(x, y) = w</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">erf2inv</span><span class=\"p\">,</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">erfinv(y)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">erfcinv(-y)</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ and $y$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">exp(-x**2 + erf2inv(x, y)**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">erf2inv</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)*exp(erf2inv(x, y)**2)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>erf: Gaussian error function.\nerfc: Complementary error function.\nerfi: Imaginary error function.\nerf2: Two-argument error function.\nerfinv: Inverse error function.\nerfcinv: Inverse complementary error function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.erf2inv, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2inv.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2inv.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.erf2inv.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "erf2inv.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ei", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ei", "type": "class", "doc": "<p>The classical exponential integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For use in SymPy, this function is defined as</p>\n\n<p>$$\\operatorname{Ei}(x) = \\sum_{n=1}^\\infty \\frac{x^n}{n\\, n!}+ \\log(x) + \\gamma,$$</p>\n\n<p>where $\\gamma$ is the Euler-Mascheroni constant.</p>\n\n<p>If $x$ is a polar number, this defines an analytic function on the\nRiemann surface of the logarithm. Otherwise this defines an analytic\nfunction in the cut plane $\\mathbb{C} \\setminus (-\\infty, 0]$.</p>\n\n<p><strong>Background</strong></p>\n\n<p>The name exponential integral comes from the following statement:</p>\n\n<p>$$\\operatorname{Ei}(x) = \\int_{-\\infty}^x \\frac{e^t}{t} \\mathrm{d}t$$</p>\n\n<p>If the integral is interpreted as a Cauchy principal value, this statement\nholds for $x &gt; 0$ and $\\operatorname{Ei}(x)$ as defined above.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ei</span><span class=\"p\">,</span> <span class=\"n\">polar_lift</span><span class=\"p\">,</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">Ei(-1)</span>\n</code></pre></div>\n\n<p>This yields a real value:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">(</span><span class=\"n\">chop</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">-0.219383934395520</span>\n</code></pre></div>\n\n<p>On the other hand the analytic continuation is not real:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">))</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">(</span><span class=\"n\">chop</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">-0.21938393439552 + 3.14159265358979*I</span>\n</code></pre></div>\n\n<p>The exponential integral has a logarithmic branch point at the origin:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Ei(x) + 2*I*pi</span>\n</code></pre></div>\n\n<p>Differentiation is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">exp(x)/x</span>\n</code></pre></div>\n\n<p>The exponential integral is related to many other special functions.\nFor example:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span><span class=\"p\">,</span> <span class=\"n\">Shi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">-expint(1, x*exp_polar(I*pi)) - I*pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ei</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Shi</span><span class=\"p\">)</span>\n<span class=\"go\">Chi(x) + Shi(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>expint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.\nSi: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.\nuppergamma: Upper incomplete gamma function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.Ei, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ei.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ei.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ei.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ei.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.expint", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "expint", "type": "class", "doc": "<p>Generalized exponential integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>$$\\operatorname{E}_\\nu(z) = z^{\\nu - 1} \\Gamma(1 - \\nu, z),$$</p>\n\n<p>where $\\Gamma(1 - \\nu, z)$ is the upper incomplete gamma function\n(<code>uppergamma</code>).</p>\n\n<p>Hence for $z$ with positive real part we have</p>\n\n<p>$$\\operatorname{E}_\\nu(z)=   \\int_1^\\infty \\frac{e^{-zt}}{t^\\nu} \\mathrm{d}t,$$</p>\n\n<p>which explains the name.</p>\n\n<p>The representation as an incomplete gamma function provides an analytic\ncontinuation for $\\operatorname{E}_\\nu(z)$. If $\\nu$ is a\nnon-positive integer, the exponential integral is thus an unbranched\nfunction of $z$, otherwise there is a branch point at the origin.\nRefer to the incomplete gamma function documentation for details of the\nbranching behavior.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Differentiation is supported. Differentiation with respect to $z$ further\nexplains the name: for integral orders, the exponential integral is an\niterated integral of the exponential function.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-expint(nu - 1, z)</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $\\nu$ has no classical expression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">)</span>\n<span class=\"go\">-z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z)</span>\n</code></pre></div>\n\n<p>At non-postive integer orders, the exponential integral reduces to the\nexponential function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">exp(-z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">exp(-z)/z + exp(-z)/z**2</span>\n</code></pre></div>\n\n<p>At half-integers it reduces to error functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)*erfc(sqrt(z))/sqrt(z)</span>\n</code></pre></div>\n\n<p>At positive integer orders it can be rewritten in terms of exponentials\nand <code>expint(1, z)</code>. Use <code>expand_func()</code> to do this:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24</span>\n</code></pre></div>\n\n<p>The generalised exponential integral is essentially equivalent to the\nincomplete gamma function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">uppergamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">uppergamma</span><span class=\"p\">)</span>\n<span class=\"go\">z**(nu - 1)*uppergamma(1 - nu, z)</span>\n</code></pre></div>\n\n<p>As such it is branched at the origin:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">))</span>\n<span class=\"go\">I*pi*z**3/3 + expint(4, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expint</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">))</span>\n<span class=\"go\">z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Ei: Another related function called exponential integral.\nE1: The classical case, returns expint(1, z).\nli: Logarithmic integral.\nLi: Offset logarithmic integral.\nSi: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.\nuppergamma</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.expint, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.expint.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "expint.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.expint.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "expint.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.li", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "li", "type": "class", "doc": "<p>The classical logarithmic integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For use in SymPy, this function is defined as</p>\n\n<p>$$\\operatorname{li}(x) = \\int_0^x \\frac{1}{\\log(t)} \\mathrm{d}t \\,.$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">li</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">1/log(z)</span>\n</code></pre></div>\n\n<p>Defining the <code>li</code> function via an integral:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">integrate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">z*li(z) - Ei(2*log(z))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z*li(z) - Ei(2*log(z))</span>\n</code></pre></div>\n\n<p>The logarithmic integral can also be defined in terms of <code>Ei</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ei</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Ei</span><span class=\"p\">)</span>\n<span class=\"go\">Ei(log(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Ei</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">1/log(z)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the logarithmic integral to arbitrary precision\non the whole complex plane (except the singular points):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">1.04516378011749278484458888919</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">1.0652795784357498247001125598 + 3.08346052231061726610939702133*I</span>\n</code></pre></div>\n\n<p>We can even compute Soldner's constant by the help of mpmath:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">mpmath</span> <span class=\"kn\">import</span> <span class=\"n\">findroot</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">findroot</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">1.45136923488338</span>\n</code></pre></div>\n\n<p>Further transformations include rewriting <code>li</code> in terms of\nthe trigonometric integrals <code>Si</code>, <code>Ci</code>, <code>Shi</code> and <code>Chi</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Si</span><span class=\"p\">,</span> <span class=\"n\">Ci</span><span class=\"p\">,</span> <span class=\"n\">Shi</span><span class=\"p\">,</span> <span class=\"n\">Chi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Si</span><span class=\"p\">)</span>\n<span class=\"go\">-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Ci</span><span class=\"p\">)</span>\n<span class=\"go\">-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Shi</span><span class=\"p\">)</span>\n<span class=\"go\">-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Chi</span><span class=\"p\">)</span>\n<span class=\"go\">-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Li: Offset logarithmic integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nSi: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.li, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.li.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "li.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.li.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "li.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Li", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Li", "type": "class", "doc": "<p>The offset logarithmic integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For use in SymPy, this function is defined as</p>\n\n<p>$$\\operatorname{Li}(x) = \\operatorname{li}(x) - \\operatorname{li}(2)$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Li</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The following special value is known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">1/log(z)</span>\n</code></pre></div>\n\n<p>The shifted logarithmic integral can be written in terms of $li(z)$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">li</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">li</span><span class=\"p\">)</span>\n<span class=\"go\">li(z) - li(2)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the logarithmic integral to arbitrary precision\non the whole complex plane (except the singular points):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Li</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">1.92242131492155809316615998938</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>li: Logarithmic integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nSi: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.Li, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Li.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Li.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Li.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Li.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Si", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Si", "type": "class", "doc": "<p>Sine integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined by</p>\n\n<p>$$\\operatorname{Si}(z) = \\int_0^z \\frac{\\sin{t}}{t} \\mathrm{d}t.$$</p>\n\n<p>It is an entire function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Si</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The sine integral is an antiderivative of $sin(z)/z$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sin(z)/z</span>\n</code></pre></div>\n\n<p>It is unbranched:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Si(z)</span>\n</code></pre></div>\n\n<p>Sine integral behaves much like ordinary sine under multiplication by <code>I</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">I*Shi(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-Si(z)</span>\n</code></pre></div>\n\n<p>It can also be expressed in terms of exponential integrals, but beware\nthat the latter is branched:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">-I*(-expint(1, z*exp_polar(-I*pi/2))/2 +</span>\n<span class=\"go\">     expint(1, z*exp_polar(I*pi/2))/2) + pi/2</span>\n</code></pre></div>\n\n<p>It can be rewritten in the form of sinc function (by definition):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">sinc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Si</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">sinc</span><span class=\"p\">)</span>\n<span class=\"go\">Integral(sinc(t), (t, 0, z))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Ci: Cosine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nsinc: unnormalized sinc function\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.Si, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Si.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Si.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Si.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Si.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ci", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ci", "type": "class", "doc": "<p>Cosine integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined for positive $x$ by</p>\n\n<p>$$\\operatorname{Ci}(x) = \\gamma + \\log{x}+ \\int_0^x \\frac{\\cos{t} - 1}{t} \\mathrm{d}t\n= -\\int_x^\\infty \\frac{\\cos{t}}{t} \\mathrm{d}t,$$</p>\n\n<p>where $\\gamma$ is the Euler-Mascheroni constant.</p>\n\n<p>We have</p>\n\n<p>$$\\operatorname{Ci}(z) =-\\frac{\\operatorname{E}_1\\left(e^{i\\pi/2} z\\right)\n       + \\operatorname{E}_1\\left(e^{-i \\pi/2} z\\right)}{2}$$</p>\n\n<p>which holds for all polar $z$ and thus provides an analytic\ncontinuation to the Riemann surface of the logarithm.</p>\n\n<p>The formula also holds as stated\nfor $z \\in \\mathbb{C}$ with $\\Re(z) &gt; 0$.\nBy lifting to the principal branch, we obtain an analytic function on the\ncut complex plane.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ci</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The cosine integral is a primitive of $\\cos(z)/z$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">cos(z)/z</span>\n</code></pre></div>\n\n<p>It has a logarithmic branch point at the origin:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Ci(z) + 2*I*pi</span>\n</code></pre></div>\n\n<p>The cosine integral behaves somewhat like ordinary $\\cos$ under\nmultiplication by $i$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">polar_lift</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">Chi(z) + I*pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">Ci(z) + I*pi</span>\n</code></pre></div>\n\n<p>It can also be expressed in terms of exponential integrals:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ci</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">-expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Si: Sine integral.\nShi: Hyperbolic sine integral.\nChi: Hyperbolic cosine integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.Ci, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ci.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ci.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ci.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ci.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Shi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Shi", "type": "class", "doc": "<p>Sinh integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined by</p>\n\n<p>$$\\operatorname{Shi}(z) = \\int_0^z \\frac{\\sinh{t}}{t} \\mathrm{d}t.$$</p>\n\n<p>It is an entire function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Shi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The Sinh integral is a primitive of $\\sinh(z)/z$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sinh(z)/z</span>\n</code></pre></div>\n\n<p>It is unbranched:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Shi(z)</span>\n</code></pre></div>\n\n<p>The $\\sinh$ integral behaves much like ordinary $\\sinh$ under\nmultiplication by $i$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">I*Si(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-Shi(z)</span>\n</code></pre></div>\n\n<p>It can also be expressed in terms of exponential integrals, but beware\nthat the latter is branched:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Shi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Si: Sine integral.\nCi: Cosine integral.\nChi: Hyperbolic cosine integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.Shi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Shi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Shi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Shi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Shi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Chi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Chi", "type": "class", "doc": "<p>Cosh integral.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined for positive $x$ by</p>\n\n<p>$$\\operatorname{Chi}(x) = \\gamma + \\log{x}+ \\int_0^x \\frac{\\cosh{t} - 1}{t} \\mathrm{d}t,$$</p>\n\n<p>where $\\gamma$ is the Euler-Mascheroni constant.</p>\n\n<p>We have</p>\n\n<p>$$\\operatorname{Chi}(z) = \\operatorname{Ci}\\left(e^{i \\pi/2}z\\right)- i\\frac{\\pi}{2},$$</p>\n\n<p>which holds for all polar $z$ and thus provides an analytic\ncontinuation to the Riemann surface of the logarithm.\nBy lifting to the principal branch we obtain an analytic function on the\ncut complex plane.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Chi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>The $\\cosh$ integral is a primitive of $\\cosh(z)/z$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">cosh(z)/z</span>\n</code></pre></div>\n\n<p>It has a logarithmic branch point at the origin:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">exp_polar</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"o\">*</span><span class=\"n\">exp_polar</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">pi</span><span class=\"p\">))</span>\n<span class=\"go\">Chi(z) + 2*I*pi</span>\n</code></pre></div>\n\n<p>The $\\cosh$ integral behaves somewhat like ordinary $\\cosh$ under\nmultiplication by $i$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">polar_lift</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">Ci(z) + I*pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">polar_lift</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">Chi(z) + I*pi</span>\n</code></pre></div>\n\n<p>It can also be expressed in terms of exponential integrals:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Chi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">expint</span><span class=\"p\">)</span>\n<span class=\"go\">-expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Si: Sine integral.\nCi: Cosine integral.\nShi: Hyperbolic sine integral.\nEi: Exponential integral.\nexpint: Generalised exponential integral.\nE1: Special case of the generalised exponential integral.\nli: Logarithmic integral.\nLi: Offset logarithmic integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.Chi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Chi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Chi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Chi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Chi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnels", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnels", "type": "class", "doc": "<p>Fresnel integral S.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined by</p>\n\n<p>$$\\operatorname{S}(z) = \\int_0^z \\sin{\\frac{\\pi}{2} t^2} \\mathrm{d}t.$$</p>\n\n<p>It is an entire function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">fresnels</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-I/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">I/2</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1 and $i$ from the argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-fresnels(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-I*fresnels(z)</span>\n</code></pre></div>\n\n<p>The Fresnel S integral obeys the mirror symmetry\n$\\overline{S(z)} = S(\\bar{z})$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">fresnels(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sin(pi*z**2/2)</span>\n</code></pre></div>\n\n<p>Defining the Fresnel functions via an integral:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">integrate</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">sin</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">fresnels(z)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Fresnel integral to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.343415678363698242195300815958</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnels</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.343415678363698242195300815958*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>fresnelc: Fresnel cosine integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.fresnels, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnels.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnels.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnels.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnels.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnelc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnelc", "type": "class", "doc": "<p>Fresnel integral C.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined by</p>\n\n<p>$$\\operatorname{C}(z) = \\int_0^z \\cos{\\frac{\\pi}{2} t^2} \\mathrm{d}t.$$</p>\n\n<p>It is an entire function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">fresnelc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">I/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">-I/2</span>\n</code></pre></div>\n\n<p>In general one can pull out factors of -1 and $i$ from the argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-fresnelc(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">I*fresnelc(z)</span>\n</code></pre></div>\n\n<p>The Fresnel C integral obeys the mirror symmetry\n$\\overline{C(z)} = C(\\bar{z})$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">fresnelc(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">cos(pi*z**2/2)</span>\n</code></pre></div>\n\n<p>Defining the Fresnel functions via an integral:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">integrate</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">cos</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">fresnelc(z)*gamma(1/4)/(4*gamma(5/4))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">integrate</span><span class=\"p\">(</span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"o\">*</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">fresnelc(z)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Fresnel integral to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">0.488253406075340754500223503357</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fresnelc</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">-0.488253406075340754500223503357*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>fresnels: Fresnel sine integral.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.error_functions.fresnelc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnelc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnelc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.fresnelc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "fresnelc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.gamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gamma", "type": "class", "doc": "<p>The gamma function</p>\n\n<p>$$\\Gamma(x) := \\int^{\\infty}_{0} t^{x-1} e^{-t} \\mathrm{d}t.$$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The <code>gamma</code> function implements the function which passes through the\nvalues of the factorial function (i.e., $\\Gamma(n) = (n - 1)!$ when n is\nan integer). More generally, $\\Gamma(z)$ is defined in the whole complex\nplane except at the negative integers where there are simple poles.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">gamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)/2</span>\n</code></pre></div>\n\n<p>The <code>gamma</code> function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">gamma(conjugate(x))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">gamma(x)*polygamma(0, x)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the <code>gamma</code> function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">)</span>\n<span class=\"go\">2.288037795340032417959588909060233922890</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"o\">+</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">0.49801566811835604271 - 0.15494982830181068512*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>lowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.gamma_functions.gamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.gamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.gamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.lowergamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lowergamma", "type": "class", "doc": "<p>The lower incomplete gamma function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>It can be defined as the meromorphic continuation of</p>\n\n<p>$$\\gamma(s, x) := \\int_0^x t^{s-1} e^{-t} \\mathrm{d}t = \\Gamma(s) - \\Gamma(s, x).$$</p>\n\n<p>This can be shown to be the same as</p>\n\n<p>$$\\gamma(s, x) = \\frac{x^s}{s} {}_1F_1\\left({s \\atop s+1} \\middle| -x\\right),$$</p>\n\n<p>where ${}_1F_1$ is the (confluent) hypergeometric function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lowergamma</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lowergamma</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">lowergamma(s, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lowergamma</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*(x**2/2 + x + 1)*exp(-x) + 2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lowergamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.gamma_functions.lowergamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.lowergamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lowergamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.lowergamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lowergamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.uppergamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "uppergamma", "type": "class", "doc": "<p>The upper incomplete gamma function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>It can be defined as the meromorphic continuation of</p>\n\n<p>$$\\Gamma(s, x) := \\int_x^\\infty t^{s-1} e^{-t} \\mathrm{d}t = \\Gamma(s) - \\gamma(s, x).$$</p>\n\n<p>where $\\gamma(s, x)$ is the lower incomplete gamma function,\n<code>lowergamma</code>. This can be shown to be the same as</p>\n\n<p>$$\\Gamma(s, x) = \\Gamma(s) - \\frac{x^s}{s} {}_1F_1\\left({s \\atop s+1} \\middle| -x\\right),$$</p>\n\n<p>where ${}_1F_1$ is the (confluent) hypergeometric function.</p>\n\n<p>The upper incomplete gamma function is also essentially equivalent to the\ngeneralized exponential integral:</p>\n\n<p>$$\\operatorname{E}_{n}(x) = \\int_{1}^{\\infty}{\\frac{e^{-xt}}{t^n} \\, dt} = x^{n-1}\\Gamma(1-n,x).$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">uppergamma</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uppergamma</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">uppergamma(s, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uppergamma</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*(x**2/2 + x + 1)*exp(-x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uppergamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uppergamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">expint(3, x)/x**2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.gamma_functions.uppergamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.uppergamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "uppergamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.uppergamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "uppergamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.polygamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polygamma", "type": "class", "doc": "<p>The function <code>polygamma(n, z)</code> returns <code>log(gamma(z)).diff(n + 1)</code>.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>It is a meromorphic function on $\\mathbb{C}$ and defined as the $(n+1)$-th\nderivative of the logarithm of the gamma function:</p>\n\n<p>$$\\psi^{(n)} (z) := \\frac{\\mathrm{d}^{n+1}}{\\mathrm{d} z^{n+1}} \\log\\Gamma(z).$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">polygamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">-2*log(2) - EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">))</span>\n<span class=\"go\">-log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"o\">/</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">))</span>\n<span class=\"go\">-pi/2 - log(4) - log(2) - EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">1 - EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">23</span><span class=\"p\">)</span>\n<span class=\"go\">19093197/5173168 - EulerGamma</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">I</span><span class=\"o\">*</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;x&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(1, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(2, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(3, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(2, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(3, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(3, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(4, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(n + 1, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(n + 2, x)</span>\n</code></pre></div>\n\n<p>We can rewrite <code>polygamma</code> functions in terms of harmonic numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">harmonic</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">)</span>\n<span class=\"go\">harmonic(x - 1) - EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">)</span>\n<span class=\"go\">2*harmonic(x - 1, 3) - 2*zeta(3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">ni</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polygamma</span><span class=\"p\">(</span><span class=\"n\">ni</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">harmonic</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.gamma_functions.polygamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.polygamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polygamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.polygamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polygamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.loggamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "loggamma", "type": "class", "doc": "<p>The <code>loggamma</code> function implements the logarithm of the\ngamma function (i.e., $\\log\\Gamma(x)$).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Several special values are known. For numerical integral\narguments we have:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">loggamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">log(2)</span>\n</code></pre></div>\n\n<p>And for symbolic values:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;n&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">log(gamma(n))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n</code></pre></div>\n\n<p>For half-integral values:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">log(3*sqrt(pi)/4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">log(2**(1 - n)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2))</span>\n</code></pre></div>\n\n<p>And general rational arguments:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span> <span class=\"o\">=</span> <span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">16</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">L</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">()</span>\n<span class=\"go\">-5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span> <span class=\"o\">=</span> <span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">19</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">L</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">()</span>\n<span class=\"go\">-4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">L</span> <span class=\"o\">=</span> <span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">23</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">7</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">L</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">doit</span><span class=\"p\">()</span>\n<span class=\"go\">-3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16)</span>\n</code></pre></div>\n\n<p>The <code>loggamma</code> function has the following limits towards infinity:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n</code></pre></div>\n\n<p>The <code>loggamma</code> function obeys the mirror symmetry\nif $x \\in \\mathbb{C} \\setminus {-\\infty, 0}$:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">loggamma(conjugate(x))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $x$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(0, x)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">cancel</span><span class=\"p\">()</span>\n<span class=\"go\">-log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the <code>gamma</code> function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n<span class=\"go\">3.17805383034794561964694160130</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">loggamma</span><span class=\"p\">(</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">-0.65092319930185633889 - 1.8724366472624298171*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.gamma_functions.loggamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.loggamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "loggamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.loggamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "loggamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.digamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "digamma", "type": "class", "doc": "<p>The <code>digamma</code> function is the first derivative of the <code>loggamma</code>\nfunction</p>\n\n<p>$$\\psi(x) := \\frac{\\mathrm{d}}{\\mathrm{d} z} \\log\\Gamma(z)\n        = \\frac{\\Gamma'(z)}{\\Gamma(z) }.$$</p>\n\n<p>In this case, <code>digamma(z) = polygamma(0, z)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">digamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">digamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;z&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">digamma</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(0, z)</span>\n</code></pre></div>\n\n<p>To retain <code>digamma</code> as it is:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">digamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">evaluate</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">digamma(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">digamma</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">evaluate</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">digamma(z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ntrigamma: Trigamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.gamma_functions.digamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.digamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "digamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.digamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "digamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.trigamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "trigamma", "type": "class", "doc": "<p>The <code>trigamma</code> function is the second derivative of the <code>loggamma</code>\nfunction</p>\n\n<p>$$\\psi^{(1)}(z) := \\frac{\\mathrm{d}^{2}}{\\mathrm{d} z^{2}} \\log\\Gamma(z).$$</p>\n\n<p>In this case, <code>trigamma(z) = polygamma(1, z)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">trigamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">trigamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;z&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">trigamma</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">polygamma(1, z)</span>\n</code></pre></div>\n\n<p>To retain <code>trigamma</code> as it is:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">trigamma</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">evaluate</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">trigamma(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">trigamma</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">evaluate</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">trigamma(z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nlowergamma: Lower incomplete gamma function.\nuppergamma: Upper incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\nbeta: Euler Beta function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.gamma_functions.trigamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.trigamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "trigamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.trigamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "trigamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.multigamma", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "multigamma", "type": "class", "doc": "<p>The multivariate gamma function is a generalization of the gamma function</p>\n\n<p>$$\\Gamma_p(z) = \\pi^{p(p-1)/4}\\prod_{k=1}^p \\Gamma[z + (1 - k)/2].$$</p>\n\n<p>In a special case, <code>multigamma(x, 1) = gamma(x)</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">multigamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;x&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">p</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;p&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">p</span><span class=\"p\">)</span>\n<span class=\"go\">pi**(p*(p - 1)/4)*Product(gamma(-_k/2 + x + 1/2), (_k, 1, p))</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)/2</span>\n</code></pre></div>\n\n<p>Writing <code>multigamma</code> in terms of the <code>gamma</code> function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">gamma(x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(pi)*gamma(x)*gamma(x - 1/2)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">multigamma</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">pi**(3/2)*gamma(x)*gamma(x - 1)*gamma(x - 1/2)</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>p : order or dimension of the multivariate gamma function</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma, lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma,\nbeta</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.gamma_functions.multigamma, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.multigamma.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "multigamma.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.multigamma.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "multigamma.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.dirichlet_eta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "dirichlet_eta", "type": "class", "doc": "<p>Dirichlet eta function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For $\\operatorname{Re}(s) &gt; 0$, this function is defined as</p>\n\n<p>$$\\eta(s) = \\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n^s}.$$</p>\n\n<p>It admits a unique analytic continuation to all of $\\mathbb{C}$.\nIt is an entire, unbranched function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>The Dirichlet eta function is closely related to the Riemann zeta function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">dirichlet_eta</span><span class=\"p\">,</span> <span class=\"n\">zeta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">dirichlet_eta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">zeta</span><span class=\"p\">)</span>\n<span class=\"go\">(1 - 2**(1 - s))*zeta(s)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>zeta</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.zeta_functions.dirichlet_eta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.dirichlet_eta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "dirichlet_eta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.dirichlet_eta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "dirichlet_eta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.zeta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "zeta", "type": "class", "doc": "<p>Hurwitz zeta function (or Riemann zeta function).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For $\\operatorname{Re}(a) &gt; 0$ and $\\operatorname{Re}(s) &gt; 1$, this\nfunction is defined as</p>\n\n<p>$$\\zeta(s, a) = \\sum_{n=0}^\\infty \\frac{1}{(n + a)^s},$$</p>\n\n<p>where the standard choice of argument for $n + a$ is used. For fixed\n$a$ with $\\operatorname{Re}(a) &gt; 0$ the Hurwitz zeta function admits a\nmeromorphic continuation to all of $\\mathbb{C}$, it is an unbranched\nfunction with a simple pole at $s = 1$.</p>\n\n<p>Analytic continuation to other $a$ is possible under some circumstances,\nbut this is not typically done.</p>\n\n<p>The Hurwitz zeta function is a special case of the Lerch transcendent:</p>\n\n<p>$$\\zeta(s, a) = \\Phi(1, s, a).$$</p>\n\n<p>This formula defines an analytic continuation for all possible values of\n$s$ and $a$ (also $\\operatorname{Re}(a) &lt; 0$), see the documentation of\n<code>lerchphi</code> for a description of the branching behavior.</p>\n\n<p>If no value is passed for $a$, by this function assumes a default value\nof $a = 1$, yielding the Riemann zeta function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>For $a = 1$ the Hurwitz zeta function reduces to the famous Riemann\nzeta function:</p>\n\n<p>$$\\zeta(s, 1) = \\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}.$$</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">zeta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">zeta(s)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n<span class=\"go\">zeta(s)</span>\n</code></pre></div>\n\n<p>The Riemann zeta function can also be expressed using the Dirichlet eta\nfunction:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">dirichlet_eta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">dirichlet_eta</span><span class=\"p\">)</span>\n<span class=\"go\">dirichlet_eta(s)/(1 - 2**(1 - s))</span>\n</code></pre></div>\n\n<p>The Riemann zeta function at positive even integer and negative odd integer\nvalues is related to the Bernoulli numbers:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">pi**2/6</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">pi**4/90</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-1/12</span>\n</code></pre></div>\n\n<p>The specific formulae are:</p>\n\n<p>$$\\zeta(2n) = (-1)^{n+1} \\frac{B_{2n} (2\\pi)^{2n}}{2(2n)!}$$</p>\n\n<p>$$\\zeta(-n) = -\\frac{B_{n+1}}{n+1}$$</p>\n\n<p>At negative even integers the Riemann zeta function is zero:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>No closed-form expressions are known at positive odd integers, but\nnumerical evaluation is possible:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">()</span>\n<span class=\"go\">1.20205690315959</span>\n</code></pre></div>\n\n<p>The derivative of $\\zeta(s, a)$ with respect to $a$ can be computed:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">-s*zeta(s + 1, a)</span>\n</code></pre></div>\n\n<p>However the derivative with respect to $s$ has no useful closed form\nexpression:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n<span class=\"go\">Derivative(zeta(s, a), s)</span>\n</code></pre></div>\n\n<p>The Hurwitz zeta function can be expressed in terms of the Lerch\ntranscendent, <code>~.lerchphi</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lerchphi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">zeta</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">)</span>\n<span class=\"go\">lerchphi(1, s, a)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>dirichlet_eta, lerchphi, polylog</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.zeta_functions.zeta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.zeta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "zeta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.zeta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "zeta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.lerchphi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lerchphi", "type": "class", "doc": "<p>Lerch transcendent (Lerch phi function).</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For $\\operatorname{Re}(a) &gt; 0$, $|z| &lt; 1$ and $s \\in \\mathbb{C}$, the\nLerch transcendent is defined as</p>\n\n<p>.. math :: \\Phi(z, s, a) = \\sum_{n=0}^\\infty \\frac{z^n}{(n + a)^s},</p>\n\n<p>where the standard branch of the argument is used for $n + a$,\nand by analytic continuation for other values of the parameters.</p>\n\n<p>A commonly used related function is the Lerch zeta function, defined by</p>\n\n<p>$$L(q, s, a) = \\Phi(e^{2\\pi i q}, s, a).$$</p>\n\n<p><strong>Analytic Continuation and Branching Behavior</strong></p>\n\n<p>It can be shown that</p>\n\n<p>$$\\Phi(z, s, a) = z\\Phi(z, s, a+1) + a^{-s}.$$</p>\n\n<p>This provides the analytic continuation to $\\operatorname{Re}(a) \\le 0$.</p>\n\n<p>Assume now $\\operatorname{Re}(a) &gt; 0$. The integral representation</p>\n\n<p>$$\\Phi_0(z, s, a) = \\int_0^\\infty \\frac{t^{s-1} e^{-at}}{1 - ze^{-t}}\\frac{\\mathrm{d}t}{\\Gamma(s)}$$</p>\n\n<p>provides an analytic continuation to $\\mathbb{C} - [1, \\infty)$.\nFinally, for $x \\in (1, \\infty)$ we find</p>\n\n<p>$$\\lim_{\\epsilon \\to 0^+} \\Phi_0(x + i\\epsilon, s, a)-\\lim_{\\epsilon \\to 0^+} \\Phi_0(x - i\\epsilon, s, a)\n= \\frac{2\\pi i \\log^{s-1}{x}}{x^a \\Gamma(s)},$$</p>\n\n<p>using the standard branch for both $\\log{x}$ and\n$\\log{\\log{x}}$ (a branch of $\\log{\\log{x}}$ is needed to\nevaluate $\\log{x}^{s-1}$).\nThis concludes the analytic continuation. The Lerch transcendent is thus\nbranched at $z \\in {0, 1, \\infty}$ and\n$a \\in \\mathbb{Z}_{\\le 0}$. For fixed $z, a$ outside these\nbranch points, it is an entire function of $s$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>The Lerch transcendent is a fairly general function, for this reason it does\nnot automatically evaluate to simpler functions. Use <code>expand_func()</code> to\nachieve this.</p>\n\n<p>If $z=1$, the Lerch transcendent reduces to the Hurwitz zeta function:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lerchphi</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">))</span>\n<span class=\"go\">zeta(s, a)</span>\n</code></pre></div>\n\n<p>More generally, if $z$ is a root of unity, the Lerch transcendent\nreduces to a sum of Hurwitz zeta functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">))</span>\n<span class=\"go\">zeta(s, a/2)/2**s - zeta(s, a/2 + 1/2)/2**s</span>\n</code></pre></div>\n\n<p>If $a=1$, the Lerch transcendent reduces to the polylogarithm:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">polylog(s, z)/z</span>\n</code></pre></div>\n\n<p>More generally, if $a$ is rational, the Lerch transcendent reduces\nto a sum of polylogarithms:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -</span>\n<span class=\"go\">            polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">-2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -</span>\n<span class=\"go\">                      polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z</span>\n</code></pre></div>\n\n<p>The derivatives with respect to $z$ and $a$ can be computed in\nclosed form:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">(-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">lerchphi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">-s*lerchphi(z, s + 1, a)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>polylog, zeta</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.zeta_functions.lerchphi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.lerchphi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lerchphi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.lerchphi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "lerchphi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.polylog", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polylog", "type": "class", "doc": "<p>Polylogarithm function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For $|z| &lt; 1$ and $s \\in \\mathbb{C}$, the polylogarithm is\ndefined by</p>\n\n<p>$$\\operatorname{Li}_s(z) = \\sum_{n=1}^\\infty \\frac{z^n}{n^s},$$</p>\n\n<p>where the standard branch of the argument is used for $n$. It admits\nan analytic continuation which is branched at $z=1$ (notably not on the\nsheet of initial definition), $z=0$ and $z=\\infty$.</p>\n\n<p>The name polylogarithm comes from the fact that for $s=1$, the\npolylogarithm is related to the ordinary logarithm (see examples), and that</p>\n\n<p>$$\\operatorname{Li}_{s+1}(z) =\\int_0^z \\frac{\\operatorname{Li}_s(t)}{t} \\mathrm{d}t.$$</p>\n\n<p>The polylogarithm is a special case of the Lerch transcendent:</p>\n\n<p>$$\\operatorname{Li}_{s}(z) = z \\Phi(z, s, 1).$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>For $z \\in {0, 1, -1}$, the polylogarithm is automatically expressed\nusing other functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">polylog</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">zeta(s)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">-dirichlet_eta(s)</span>\n</code></pre></div>\n\n<p>If $s$ is a negative integer, $0$ or $1$, the polylogarithm can be\nexpressed using elementary functions. This can be done using\n<code>expand_func()</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">-log(1 - z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">z/(1 - z)</span>\n</code></pre></div>\n\n<p>The derivative with respect to $z$ can be computed in closed form:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">polylog(s - 1, z)/z</span>\n</code></pre></div>\n\n<p>The polylogarithm can be expressed in terms of the lerch transcendent:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">lerchphi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">polylog</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">lerchphi</span><span class=\"p\">)</span>\n<span class=\"go\">z*lerchphi(z, s, 1)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>zeta, lerchphi</p>\n", "bases": "sympy.functions.special.zeta_functions.polylog, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.polylog.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polylog.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.polylog.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "polylog.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.stieltjes", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "stieltjes", "type": "class", "doc": "<p>Represents Stieltjes constants, $\\gamma_{k}$ that occur in\nLaurent Series expansion of the Riemann zeta function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">stieltjes</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">stieltjes(n)</span>\n</code></pre></div>\n\n<p>The zero'th stieltjes constant:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">EulerGamma</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">EulerGamma</span>\n</code></pre></div>\n\n<p>For generalized stieltjes constants:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">)</span>\n<span class=\"go\">stieltjes(n, m)</span>\n</code></pre></div>\n\n<p>Constants are only defined for integers &gt;= 0:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">stieltjes</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">zoo</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.zeta_functions.stieltjes, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.stieltjes.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "stieltjes.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.stieltjes.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "stieltjes.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita", "type": "class", "doc": "<p>Represent the Levi-Civita symbol.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>For even permutations of indices it returns 1, for odd permutations -1, and\nfor everything else (a repeated index) it returns 0.</p>\n\n<p>Thus it represents an alternating pseudotensor.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">LeviCivita</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">,</span> <span class=\"n\">k</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">,</span> <span class=\"n\">k</span><span class=\"p\">)</span>\n<span class=\"go\">LeviCivita(i, j, k)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">LeviCivita</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">,</span> <span class=\"n\">i</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Eijk</p>\n", "bases": "sympy.functions.special.tensor_functions.LeviCivita, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_integer", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_integer", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'integer': True, 'transcendental': False, 'imaginary': False, 'commutative': True, 'infinite': False, 'irrational': False, 'real': True, 'hermitian': True, 'finite': True, 'extended_real': True, 'algebraic': True, 'complex': True, 'noninteger': False, 'rational': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_transcendental", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_transcendental", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_irrational", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_irrational", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_algebraic", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_algebraic", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_noninteger", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_noninteger", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.LeviCivita.is_rational", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "LeviCivita.is_rational", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta", "type": "class", "doc": "<p>The discrete, or Kronecker, delta function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>A function that takes in two integers $i$ and $j$. It returns $0$ if $i$\nand $j$ are not equal, or it returns $1$ if $i$ and $j$ are equal.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>An example with integer indices:</p>\n\n<pre><code>&gt;&gt;&gt; from sympy import KroneckerDelta\n&gt;&gt;&gt; KroneckerDelta(1, 2)\n0\n&gt;&gt;&gt; KroneckerDelta(3, 3)\n1\n</code></pre>\n\n<p>Symbolic indices:</p>\n\n<pre><code>&gt;&gt;&gt; from sympy.abc import i, j, k\n&gt;&gt;&gt; KroneckerDelta(i, j)\nKroneckerDelta(i, j)\n&gt;&gt;&gt; KroneckerDelta(i, i)\n1\n&gt;&gt;&gt; KroneckerDelta(i, i + 1)\n0\n&gt;&gt;&gt; KroneckerDelta(i, i + 1 + k)\nKroneckerDelta(i, i + k + 1)\n</code></pre>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>i : Number, Symbol\n    The first index of the delta function.\nj : Number, Symbol\n    The second index of the delta function.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>eval\nDiracDelta</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.tensor_functions.KroneckerDelta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_integer", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_integer", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'integer': True, 'transcendental': False, 'imaginary': False, 'commutative': True, 'infinite': False, 'irrational': False, 'real': True, 'hermitian': True, 'finite': True, 'extended_real': True, 'algebraic': True, 'complex': True, 'noninteger': False, 'rational': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_transcendental", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_transcendental", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_irrational", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_irrational", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_algebraic", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_algebraic", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_noninteger", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_noninteger", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.KroneckerDelta.is_rational", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "KroneckerDelta.is_rational", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction", "type": "class", "doc": "<p>Singularity functions are a class of discontinuous functions.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Singularity functions take a variable, an offset, and an exponent as\narguments. These functions are represented using Macaulay brackets as:</p>\n\n<p>SingularityFunction(x, a, n) := <x - a>^n</p>\n\n<p>The singularity function will automatically evaluate to\n<code>Derivative(DiracDelta(x - a), x, -n - 1)</code> if <code>n &lt; 0</code>\nand <code>(x - a)**n*Heaviside(x - a)</code> if <code>n &gt;= 0</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">,</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">Piecewise</span><span class=\"p\">,</span> <span class=\"n\">DiracDelta</span><span class=\"p\">,</span> <span class=\"n\">Heaviside</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">SingularityFunction(x, a, n)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;y&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">nonnegative</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">(y + 10)**n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;y&#39;</span><span class=\"p\">,</span> <span class=\"n\">negative</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">10</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">243</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">SingularityFunction(x, 4, -2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Piecewise</span><span class=\"p\">)</span>\n<span class=\"go\">Piecewise(((x - 4)**5, x &gt; 4), (0, True))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span> <span class=\"o\">=</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;y&#39;</span><span class=\"p\">,</span> <span class=\"n\">positive</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s1\">&#39;n&#39;</span><span class=\"p\">,</span> <span class=\"n\">nonnegative</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">({</span><span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">:</span> <span class=\"o\">-</span><span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">:</span> <span class=\"n\">n</span><span class=\"p\">})</span>\n<span class=\"go\">(y + 10)**n</span>\n</code></pre></div>\n\n<p>The methods <code>rewrite(DiracDelta)</code>, <code>rewrite(Heaviside)</code>, and\n<code>rewrite('HeavisideDiracDelta')</code> returns the same output. One can use any\nof these methods according to their choice.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span> <span class=\"o\">=</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">SingularityFunction</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">Heaviside</span><span class=\"p\">)</span>\n<span class=\"go\">(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">DiracDelta</span><span class=\"p\">)</span>\n<span class=\"go\">(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expr</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"s1\">&#39;HeavisideDiracDelta&#39;</span><span class=\"p\">)</span>\n<span class=\"go\">(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>DiracDelta, Heaviside</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.singularity_functions.SingularityFunction, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'real': True, 'imaginary': False, 'commutative': True, 'infinite': False, 'finite': True, 'hermitian': True, 'extended_real': True, 'complex': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.SingularityFunction.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "SingularityFunction.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta", "type": "class", "doc": "<p>The DiracDelta function and its derivatives.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>DiracDelta is not an ordinary function. It can be rigorously defined either\nas a distribution or as a measure.</p>\n\n<p>DiracDelta only makes sense in definite integrals, and in particular,\nintegrals of the form <code>Integral(f(x)*DiracDelta(x - x0), (x, a, b))</code>,\nwhere it equals <code>f(x0)</code> if <code>a &lt;= x0 &lt;= b</code> and <code>0</code> otherwise. Formally,\nDiracDelta acts in some ways like a function that is <code>0</code> everywhere except\nat <code>0</code>, but in many ways it also does not. It can often be useful to treat\nDiracDelta in formal ways, building up and manipulating expressions with\ndelta functions (which may eventually be integrated), but care must be taken\nto not treat it as a real function. SymPy's <code>oo</code> is similar. It only\ntruly makes sense formally in certain contexts (such as integration limits),\nbut SymPy allows its use everywhere, and it tries to be consistent with\noperations on it (like <code>1/oo</code>), but it is easy to get into trouble and get\nwrong results if <code>oo</code> is treated too much like a number. Similarly, if\nDiracDelta is treated too much like a function, it is easy to get wrong or\nnonsensical results.</p>\n\n<p>DiracDelta function has the following properties:</p>\n\n<p>1) $\\frac{d}{d x} \\theta(x) = \\delta(x)$\n2) $\\int_{-\\infty}^\\infty \\delta(x - a)f(x)\\, dx = f(a)$ and $\\int_{a-\n   \\epsilon}^{a+\\epsilon} \\delta(x - a)f(x)\\, dx = f(a)$\n3) $\\delta(x) = 0$ for all $x \\neq 0$\n4) $\\delta(g(x)) = \\sum_i \\frac{\\delta(x - x_i)}{\\|g'(x_i)\\|}$ where $x_i$\n   are the roots of $g$\n5) $\\delta(-x) = \\delta(x)$</p>\n\n<p>Derivatives of <code>k</code>-th order of DiracDelta have the following properties:</p>\n\n<p>6) $\\delta(x, k) = 0$ for all $x \\neq 0$\n7) $\\delta(-x, k) = -\\delta(x, k)$ for odd $k$\n8) $\\delta(-x, k) = \\delta(x, k)$ for even $k$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">DiracDelta</span><span class=\"p\">,</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">DiracDelta(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">4</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">subs</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">DiracDelta(0)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">DiracDelta(x, 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">),</span><span class=\"n\">x</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">DiracDelta(x - 1, 2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">),</span><span class=\"n\">x</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">2*(2*x**2*DiracDelta(x**2 - 1, 2) + DiracDelta(x**2 - 1, 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">x</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">is_simple</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">is_simple</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">False</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">DiracDelta</span><span class=\"p\">((</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">y</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">diracdelta</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">,</span> <span class=\"n\">wrt</span><span class=\"o\">=</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">DiracDelta(x - 1)/(2*Abs(y)) + DiracDelta(x + 1)/(2*Abs(y))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Heaviside\nsympy.simplify.simplify.simplify, is_simple\nsympy.functions.special.tensor_functions.KroneckerDelta</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.delta_functions.DiracDelta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'real': True, 'imaginary': False, 'commutative': True, 'infinite': False, 'finite': True, 'hermitian': True, 'extended_real': True, 'complex': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.DiracDelta.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "DiracDelta.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside", "type": "class", "doc": "<p>Heaviside step function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Heaviside step function has the following properties:</p>\n\n<p>1) $\\frac{d}{d x} \\theta(x) = \\delta(x)$\n2) $\\theta(x) = \\begin{cases} 0 &amp; \\text{for}\\: x &lt; 0 \\ \\frac{1}{2} &amp;\n   \\text{for}\\: x = 0 \\1 &amp; \\text{for}\\: x &gt; 0 \\end{cases}$\n3) $\\frac{d}{d x} \\max(x, 0) = \\theta(x)$</p>\n\n<p>Heaviside(x) is printed as $\\theta(x)$ with the SymPy LaTeX printer.</p>\n\n<p>The value at 0 is set differently in different fields. SymPy uses 1/2,\nwhich is a convention from electronics and signal processing, and is\nconsistent with solving improper integrals by Fourier transform and\nconvolution.</p>\n\n<p>To specify a different value of Heaviside at <code>x=0</code>, a second argument\ncan be given. Using <code>Heaviside(x, nan)</code> gives an expression that will\nevaluate to nan for x=0.</p>\n\n<p><em>Changed in version 1.9 <code>Heaviside(0)</code> now returns 1/2 (before: undefined).</em></p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Heaviside</span><span class=\"p\">,</span> <span class=\"n\">nan</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"mi\">9</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">9</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">nan</span><span class=\"p\">)</span>\n<span class=\"go\">nan</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"p\">(</span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">replace</span><span class=\"p\">(</span><span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">Heaviside</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">))</span>\n<span class=\"go\">Heaviside(x, 1) + 1</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>DiracDelta</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.delta_functions.Heaviside, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'real': True, 'imaginary': False, 'commutative': True, 'infinite': False, 'finite': True, 'hermitian': True, 'extended_real': True, 'complex': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_imaginary", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_imaginary", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_infinite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_infinite", "type": "variable", "doc": "<p></p>\n", "default_value": " = False"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_finite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_finite", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_hermitian", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_hermitian", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_extended_real", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_extended_real", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.Heaviside.is_complex", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Heaviside.is_complex", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselj", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselj", "type": "class", "doc": "<p>Bessel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Bessel $J$ function of order $\\nu$ is defined to be the function\nsatisfying Bessel's differential equation</p>\n\n<p>.. math ::\n    z^2 \\frac{\\mathrm{d}^2 w}{\\mathrm{d}z^2}\n    + z \\frac{\\mathrm{d}w}{\\mathrm{d}z} + (z^2 - \\nu^2) w = 0,</p>\n\n<p>with Laurent expansion</p>\n\n<p>.. math ::\n    J_\\nu(z) = z^\\nu \\left(\\frac{1}{\\Gamma(\\nu + 1) 2^\\nu} + O(z^2) \\right),</p>\n\n<p>if $\\nu$ is not a negative integer. If $\\nu=-n \\in \\mathbb{Z}_{&lt;0}$\n<em>is</em> a negative integer, then the definition is</p>\n\n<p>.. math ::\n    J_{-n}(z) = (-1)^n J_n(z).</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create a Bessel function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">besselj</span><span class=\"p\">,</span> <span class=\"n\">jn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span> <span class=\"o\">=</span> <span class=\"n\">besselj</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<p>Differentiate it:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">besselj(n - 1, z)/2 - besselj(n + 1, z)/2</span>\n</code></pre></div>\n\n<p>Rewrite in terms of spherical Bessel functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)</span>\n</code></pre></div>\n\n<p>Access the parameter and argument:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">order</span>\n<span class=\"go\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">argument</span>\n<span class=\"go\">z</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>bessely, besseli, besselk</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.besselj, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselj.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselj.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselj.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselj.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.bessely", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bessely", "type": "class", "doc": "<p>Bessel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Bessel $Y$ function of order $\\nu$ is defined as</p>\n\n<p>.. math ::\n    Y_\\nu(z) = \\lim_{\\mu \\to \\nu} \\frac{J_\\mu(z) \\cos(\\pi \\mu)\n                                        - J_{-\\mu}(z)}{\\sin(\\pi \\mu)},</p>\n\n<p>where $J_\\mu(z)$ is the Bessel function of the first kind.</p>\n\n<p>It is a solution to Bessel's equation, and linearly independent from\n$J_\\nu$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">bessely</span><span class=\"p\">,</span> <span class=\"n\">yn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span> <span class=\"o\">=</span> <span class=\"n\">bessely</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">bessely(n - 1, z)/2 - bessely(n + 1, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">b</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, besseli, besselk</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.bessely, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.bessely.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bessely.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.bessely.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "bessely.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.besseli", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besseli", "type": "class", "doc": "<p>Modified Bessel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Bessel $I$ function is a solution to the modified Bessel equation</p>\n\n<p>.. math ::\n    z^2 \\frac{\\mathrm{d}^2 w}{\\mathrm{d}z^2}\n    + z \\frac{\\mathrm{d}w}{\\mathrm{d}z} + (z^2 + \\nu^2)^2 w = 0.</p>\n\n<p>It can be defined as</p>\n\n<p>.. math ::\n    I_\\nu(z) = i^{-\\nu} J_\\nu(iz),</p>\n\n<p>where $J_\\nu(z)$ is the Bessel function of the first kind.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">besseli</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">besseli</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">besseli(n - 1, z)/2 + besseli(n + 1, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, bessely, besselk</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.besseli, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.besseli.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besseli.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.besseli.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besseli.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselk", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselk", "type": "class", "doc": "<p>Modified Bessel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Bessel $K$ function of order $\\nu$ is defined as</p>\n\n<p>.. math ::\n    K_\\nu(z) = \\lim_{\\mu \\to \\nu} \\frac{\\pi}{2}\n               \\frac{I_{-\\mu}(z) -I_\\mu(z)}{\\sin(\\pi \\mu)},</p>\n\n<p>where $I_\\mu(z)$ is the modified Bessel function of the first kind.</p>\n\n<p>It is a solution of the modified Bessel equation, and linearly independent\nfrom $Y_\\nu$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">besselk</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">besselk</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-besselk(n - 1, z)/2 - besselk(n + 1, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, besseli, bessely</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.besselk, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselk.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselk.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.besselk.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "besselk.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel1", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel1", "type": "class", "doc": "<p>Hankel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>.. math ::\n    H_\\nu^{(1)} = J_\\nu(z) + iY_\\nu(z),</p>\n\n<p>where $J_\\nu(z)$ is the Bessel function of the first kind, and\n$Y_\\nu(z)$ is the Bessel function of the second kind.</p>\n\n<p>It is a solution to Bessel's equation.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hankel1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hankel1</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hankel2, besselj, bessely</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.hankel1, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel1.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel1.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel1.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel1.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel2", "type": "class", "doc": "<p>Hankel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>.. math ::\n    H_\\nu^{(2)} = J_\\nu(z) - iY_\\nu(z),</p>\n\n<p>where $J_\\nu(z)$ is the Bessel function of the first kind, and\n$Y_\\nu(z)$ is the Bessel function of the second kind.</p>\n\n<p>It is a solution to Bessel's equation, and linearly independent from\n$H_\\nu^{(1)}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hankel2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hankel2</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hankel1, besselj, bessely</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.hankel2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hankel2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hankel2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.jn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jn", "type": "class", "doc": "<p>Spherical Bessel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is a solution to the spherical Bessel equation</p>\n\n<p>.. math ::\n    z^2 \\frac{\\mathrm{d}^2 w}{\\mathrm{d}z^2}\n      + 2z \\frac{\\mathrm{d}w}{\\mathrm{d}z} + (z^2 - \\nu(\\nu + 1)) w = 0.</p>\n\n<p>It can be defined as</p>\n\n<p>.. math ::\n    j_\\nu(z) = \\sqrt{\\frac{\\pi}{2z}} J_{\\nu + \\frac{1}{2}}(z),</p>\n\n<p>where $J_\\nu(z)$ is the Bessel function of the first kind.</p>\n\n<p>The spherical Bessel functions of integral order are\ncalculated using the formula:</p>\n\n<p>$$j_n(z) = f_n(z) \\sin{z} + (-1)^{n+1} f_{-n-1}(z) \\cos{z},$$</p>\n\n<p>where the coefficients $f_n(z)$ are available as\n<code>sympy.polys.orthopolys.spherical_bessel_fn</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">jn</span><span class=\"p\">,</span> <span class=\"n\">sin</span><span class=\"p\">,</span> <span class=\"n\">cos</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">besselj</span><span class=\"p\">,</span> <span class=\"n\">bessely</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;z&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">nu</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;nu&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">sin(z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span> <span class=\"o\">==</span> <span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">z</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">(-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">besselj</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">bessely</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jn</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mf\">5.2</span><span class=\"o\">+</span><span class=\"mf\">0.3</span><span class=\"n\">j</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">0.099419756723640344491 - 0.054525080242173562897*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, bessely, besselk, yn</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.jn, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.jn.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jn.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.jn.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jn.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.yn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "yn", "type": "class", "doc": "<p>Spherical Bessel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is another solution to the spherical Bessel equation, and\nlinearly independent from $j_n$. It can be defined as</p>\n\n<p>.. math ::\n    y_\\nu(z) = \\sqrt{\\frac{\\pi}{2z}} Y_{\\nu + \\frac{1}{2}}(z),</p>\n\n<p>where $Y_\\nu(z)$ is the Bessel function of the second kind.</p>\n\n<p>For integral orders $n$, $y_n$ is calculated using the formula:</p>\n\n<p>$$y_n(z) = (-1)^{n+1} j_{-n-1}(z)$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">yn</span><span class=\"p\">,</span> <span class=\"n\">sin</span><span class=\"p\">,</span> <span class=\"n\">cos</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">besselj</span><span class=\"p\">,</span> <span class=\"n\">bessely</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;z&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">nu</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;nu&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">-cos(z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">))</span> <span class=\"o\">==</span> <span class=\"o\">-</span><span class=\"n\">cos</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">-</span><span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"n\">z</span>\n<span class=\"go\">True</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">besselj</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">bessely</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">yn</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mf\">5.2</span><span class=\"o\">+</span><span class=\"mf\">0.3</span><span class=\"n\">j</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">0.18525034196069722536 + 0.014895573969924817587*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>besselj, bessely, besselk, jn</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.yn, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.yn.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "yn.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.yn.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "yn.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn1", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn1", "type": "class", "doc": "<p>Spherical Hankel function of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>$$h_\\nu^(1)(z) = j_\\nu(z) + i y_\\nu(z),$$</p>\n\n<p>where $j_\\nu(z)$ and $y_\\nu(z)$ are the spherical\nBessel function of the first and second kinds.</p>\n\n<p>For integral orders $n$, $h_n^(1)$ is calculated using the formula:</p>\n\n<p>$$h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z)$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">hn1</span><span class=\"p\">,</span> <span class=\"n\">hankel1</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">yn</span><span class=\"p\">,</span> <span class=\"n\">jn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;z&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">nu</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;nu&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">jn(nu, z) + I*yn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">sin(z)/z - I*cos(z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">-I*sin(z)/z - cos(z)/z + sin(z)/z**2 - I*cos(z)/z**2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**nu*yn(-nu - 1, z) + I*yn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn1</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hankel1</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel1(nu, z)/2</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hn2, jn, yn, hankel1, hankel2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.hn1, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn1.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn1.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn1.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn1.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn2", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn2", "type": "class", "doc": "<p>Spherical Hankel function of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is defined as</p>\n\n<p>$$h_\\nu^(2)(z) = j_\\nu(z) - i y_\\nu(z),$$</p>\n\n<p>where $j_\\nu(z)$ and $y_\\nu(z)$ are the spherical\nBessel function of the first and second kinds.</p>\n\n<p>For integral orders $n$, $h_n^(2)$ is calculated using the formula:</p>\n\n<p>$$h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z)$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">hn2</span><span class=\"p\">,</span> <span class=\"n\">hankel2</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">jn</span><span class=\"p\">,</span> <span class=\"n\">yn</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;z&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">nu</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;nu&quot;</span><span class=\"p\">,</span> <span class=\"n\">integer</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">jn(nu, z) - I*yn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">sin(z)/z + I*cos(z)/z</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)))</span>\n<span class=\"go\">I*sin(z)/z - cos(z)/z + sin(z)/z**2 + I*cos(z)/z**2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hankel2</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel2(nu, z)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">jn</span><span class=\"p\">)</span>\n<span class=\"go\">-(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hn2</span><span class=\"p\">(</span><span class=\"n\">nu</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">yn</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**nu*yn(-nu - 1, z) - I*yn(nu, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hn1, jn, yn, hankel1, hankel2</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.hn2, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn2.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn2.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hn2.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hn2.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyai", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyai", "type": "class", "doc": "<p>The Airy function $\\operatorname{Ai}$ of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Airy function $\\operatorname{Ai}(z)$ is defined to be the function\nsatisfying Airy's differential equation</p>\n\n<p>$$\\frac{\\mathrm{d}^2 w(z)}{\\mathrm{d}z^2} - z w(z) = 0.$$</p>\n\n<p>Equivalently, for real $z$</p>\n\n<p>$$\\operatorname{Ai}(z) := \\frac{1}{\\pi}\n\\int_0^\\infty \\cos\\left(\\frac{t^3}{3} + z t\\right) \\mathrm{d}t.$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create an Airy function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">airyai</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airyai(z)</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/3)/(3*gamma(2/3))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>The Airy function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">airyai(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airyaiprime(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">z*airyai(z)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Airy function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">50</span><span class=\"p\">)</span>\n<span class=\"go\">0.22740742820168557599192443603787379946077222541710</span>\n</code></pre></div>\n\n<p>Rewrite $\\operatorname{Ai}(z)$ in terms of hypergeometric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyai</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">-3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>airybi: Airy function of the second kind.\nairyaiprime: Derivative of the Airy function of the first kind.\nairybiprime: Derivative of the Airy function of the second kind.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.airyai, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyai.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyai.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyai.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyai.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybi", "type": "class", "doc": "<p>The Airy function $\\operatorname{Bi}$ of the second kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Airy function $\\operatorname{Bi}(z)$ is defined to be the function\nsatisfying Airy's differential equation</p>\n\n<p>$$\\frac{\\mathrm{d}^2 w(z)}{\\mathrm{d}z^2} - z w(z) = 0.$$</p>\n\n<p>Equivalently, for real $z$</p>\n\n<p>$$\\operatorname{Bi}(z) := \\frac{1}{\\pi}\n         \\int_0^\\infty\n           \\exp\\left(-\\frac{t^3}{3} + z t\\right)\n           + \\sin\\left(\\frac{t^3}{3} + z t\\right) \\mathrm{d}t.$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create an Airy function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">airybi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airybi(z)</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">3**(5/6)/(3*gamma(2/3))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>The Airy function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">airybi(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airybiprime(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">z*airybi(z)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Airy function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">50</span><span class=\"p\">)</span>\n<span class=\"go\">-0.41230258795639848808323405461146104203453483447240</span>\n</code></pre></div>\n\n<p>Rewrite $\\operatorname{Bi}(z)$ in terms of hypergeometric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybi</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>airyai: Airy function of the first kind.\nairyaiprime: Derivative of the Airy function of the first kind.\nairybiprime: Derivative of the Airy function of the second kind.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.airybi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyaiprime", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyaiprime", "type": "class", "doc": "<p>The derivative $\\operatorname{Ai}^\\prime$ of the Airy function of the first\nkind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Airy function $\\operatorname{Ai}^\\prime(z)$ is defined to be the\nfunction</p>\n\n<p>$$\\operatorname{Ai}^\\prime(z) := \\frac{\\mathrm{d} \\operatorname{Ai}(z)}{\\mathrm{d} z}.$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create an Airy function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">airyaiprime</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airyaiprime(z)</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">-3**(2/3)/(3*gamma(1/3))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>The Airy function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">airyaiprime(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z*airyai(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">z*airyaiprime(z) + airyai(z)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">-3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Airy function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">50</span><span class=\"p\">)</span>\n<span class=\"go\">0.61825902074169104140626429133247528291577794512415</span>\n</code></pre></div>\n\n<p>Rewrite $\\operatorname{Ai}^\\prime(z)$ in terms of hypergeometric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airyaiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>airyai: Airy function of the first kind.\nairybi: Airy function of the second kind.\nairybiprime: Derivative of the Airy function of the second kind.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.airyaiprime, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyaiprime.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyaiprime.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.airyaiprime.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airyaiprime.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybiprime", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybiprime", "type": "class", "doc": "<p>The derivative $\\operatorname{Bi}^\\prime$ of the Airy function of the first\nkind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Airy function $\\operatorname{Bi}^\\prime(z)$ is defined to be the\nfunction</p>\n\n<p>$$\\operatorname{Bi}^\\prime(z) := \\frac{\\mathrm{d} \\operatorname{Bi}(z)}{\\mathrm{d} z}.$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>Create an Airy function object:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">airybiprime</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">airybiprime(z)</span>\n</code></pre></div>\n\n<p>Several special values are known:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/6)/gamma(1/3)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">oo</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">oo</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<p>The Airy function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">))</span>\n<span class=\"go\">airybiprime(conjugate(z))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $z$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z*airybi(z)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">z*airybiprime(z) + airybi(z)</span>\n</code></pre></div>\n\n<p>Series expansion is also supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">series</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Airy function to arbitrary precision\non the whole complex plane:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">50</span><span class=\"p\">)</span>\n<span class=\"go\">0.27879516692116952268509756941098324140300059345163</span>\n</code></pre></div>\n\n<p>Rewrite $\\operatorname{Bi}^\\prime(z)$ in terms of hypergeometric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">airybiprime</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>airyai: Airy function of the first kind.\nairybi: Airy function of the second kind.\nairyaiprime: Derivative of the Airy function of the first kind.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.airybiprime, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybiprime.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybiprime.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.airybiprime.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "airybiprime.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.marcumq", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "marcumq", "type": "class", "doc": "<p>The Marcum Q-function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Marcum Q-function is defined by the meromorphic continuation of</p>\n\n<p>$$Q_m(a, b) = a^{- m + 1} \\int_{b}^{\\infty} x^{m} e^{- \\frac{a^{2}}{2} - \\frac{x^{2}}{2}} I_{m - 1}\\left(a x\\right)\\, dx$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">marcumq</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">marcumq(m, a, b)</span>\n</code></pre></div>\n\n<p>Special values:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">uppergamma(m, b**2/2)/gamma(m)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">0</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">1 - exp(-a**2/2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">1/2 + exp(-a**2)*besseli(0, a**2)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2)</span>\n</code></pre></div>\n\n<p>Differentiation with respect to $a$ and $b$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">),</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">a*(-marcumq(m, a, b) + marcumq(m + 1, a, b))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">marcumq</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">),</span> <span class=\"n\">b</span><span class=\"p\">)</span>\n<span class=\"go\">-a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b)</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.bessel.marcumq, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.marcumq.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "marcumq.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.marcumq.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "marcumq.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hyper", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hyper", "type": "class", "doc": "<p>The generalized hypergeometric function is defined by a series where\nthe ratios of successive terms are a rational function of the summation\nindex. When convergent, it is continued analytically to the largest\npossible domain.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The hypergeometric function depends on two vectors of parameters, called\nthe numerator parameters $a_p$, and the denominator parameters\n$b_q$. It also has an argument $z$. The series definition is</p>\n\n<p>.. math ::\n    {}_pF_q\\left(\\begin{matrix} a_1, \\cdots, a_p \\ b_1, \\cdots, b_q \\end{matrix}\n                 \\middle| z \\right)\n    = \\sum_{n=0}^\\infty \\frac{(a_1)_n \\cdots (a_p)_n}{(b_1)_n \\cdots (b_q)_n}\n                        \\frac{z^n}{n!},</p>\n\n<p>where $(a)_n = (a)(a+1)\\cdots(a+n-1)$ denotes the rising factorial.</p>\n\n<p>If one of the $b_q$ is a non-positive integer then the series is\nundefined unless one of the $a_p$ is a larger (i.e., smaller in\nmagnitude) non-positive integer. If none of the $b_q$ is a\nnon-positive integer and one of the $a_p$ is a non-positive\ninteger, then the series reduces to a polynomial. To simplify the\nfollowing discussion, we assume that none of the $a_p$ or\n$b_q$ is a non-positive integer. For more details, see the\nreferences.</p>\n\n<p>The series converges for all $z$ if $p \\le q$, and thus\ndefines an entire single-valued function in this case. If $p =\nq+1$ the series converges for $|z| &lt; 1$, and can be continued\nanalytically into a half-plane. If $p &gt; q+1$ the series is\ndivergent for all $z$.</p>\n\n<p>Please note the hypergeometric function constructor currently does <em>not</em>\ncheck if the parameters actually yield a well-defined function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>The parameters $a_p$ and $b_q$ can be passed as arbitrary\niterables, for example:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyper</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">),</span> <span class=\"p\">[</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">hyper((1, 2, 3), (3, 4), x)</span>\n</code></pre></div>\n\n<p>There is also pretty printing (it looks better using Unicode):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">pprint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">pprint</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">),</span> <span class=\"p\">[</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">use_unicode</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\">  _</span>\n<span class=\"go\"> |_  /1, 2, 3 |  \\</span>\n<span class=\"go\"> |   |        | x|</span>\n<span class=\"go\">3  2 \\  3, 4  |  /</span>\n</code></pre></div>\n\n<p>The parameters must always be iterables, even if they are vectors of\nlength one or zero:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyper</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"p\">),</span> <span class=\"p\">[],</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">hyper((1,), (), x)</span>\n</code></pre></div>\n\n<p>But of course they may be variables (but if they depend on $x$ then you\nshould not expect much implemented functionality):</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyper</span><span class=\"p\">((</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">,),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">hyper((n, a), (n**2,), x)</span>\n</code></pre></div>\n\n<p>The hypergeometric function generalizes many named special functions.\nThe function <code>hyperexpand()</code> tries to express a hypergeometric function\nusing named special functions. For example:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyperexpand</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">([],</span> <span class=\"p\">[],</span> <span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">exp(x)</span>\n</code></pre></div>\n\n<p>You can also use <code>expand_func()</code>:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">hyper</span><span class=\"p\">([</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">log(x + 1)</span>\n</code></pre></div>\n\n<p>More examples:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">([],</span> <span class=\"p\">[</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"o\">/</span><span class=\"mi\">4</span><span class=\"p\">))</span>\n<span class=\"go\">cos(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">*</span><span class=\"n\">hyper</span><span class=\"p\">([</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">asin(x)</span>\n</code></pre></div>\n\n<p>We can also sometimes <code>hyperexpand()</code> parametric functions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">([</span><span class=\"o\">-</span><span class=\"n\">a</span><span class=\"p\">],</span> <span class=\"p\">[],</span> <span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">(1 - x)**a</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>sympy.simplify.hyperexpand\ngamma\nmeijerg</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.hyper.hyper, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hyper.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hyper.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hyper.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hyper.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.hyper.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hyper.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'commutative': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.meijerg", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "meijerg", "type": "class", "doc": "<p>The Meijer G-function is defined by a Mellin-Barnes type integral that\nresembles an inverse Mellin transform. It generalizes the hypergeometric\nfunctions.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Meijer G-function depends on four sets of parameters. There are\n\"<em>numerator parameters</em>\"\n$a_1, \\ldots, a_n$ and $a_{n+1}, \\ldots, a_p$, and there are\n\"<em>denominator parameters</em>\"\n$b_1, \\ldots, b_m$ and $b_{m+1}, \\ldots, b_q$.\nConfusingly, it is traditionally denoted as follows (note the position\nof $m$, $n$, $p$, $q$, and how they relate to the lengths of the four\nparameter vectors):</p>\n\n<p>.. math ::\n    G_{p,q}^{m,n} \\left(\\begin{matrix}a_1, \\cdots, a_n &amp; a_{n+1}, \\cdots, a_p \\\n                                    b_1, \\cdots, b_m &amp; b_{m+1}, \\cdots, b_q\n                      \\end{matrix} \\middle| z \\right).</p>\n\n<p>However, in SymPy the four parameter vectors are always available\nseparately (see examples), so that there is no need to keep track of the\ndecorating sub- and super-scripts on the G symbol.</p>\n\n<p>The G function is defined as the following integral:</p>\n\n<p>.. math ::\n     \\frac{1}{2 \\pi i} \\int_L \\frac{\\prod_{j=1}^m \\Gamma(b_j - s)\n     \\prod_{j=1}^n \\Gamma(1 - a_j + s)}{\\prod_{j=m+1}^q \\Gamma(1- b_j +s)\n     \\prod_{j=n+1}^p \\Gamma(a_j - s)} z^s \\mathrm{d}s,</p>\n\n<p>where $\\Gamma(z)$ is the gamma function. There are three possible\ncontours which we will not describe in detail here (see the references).\nIf the integral converges along more than one of them, the definitions\nagree. The contours all separate the poles of $\\Gamma(1-a_j+s)$\nfrom the poles of $\\Gamma(b_k-s)$, so in particular the G function\nis undefined if $a_j - b_k \\in \\mathbb{Z}_{&gt;0}$ for some\n$j \\le n$ and $k \\le m$.</p>\n\n<p>The conditions under which one of the contours yields a convergent integral\nare complicated and we do not state them here, see the references.</p>\n\n<p>Please note currently the Meijer G-function constructor does <em>not</em> check any\nconvergence conditions.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>You can pass the parameters either as four separate vectors:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">meijerg</span><span class=\"p\">,</span> <span class=\"n\">Tuple</span><span class=\"p\">,</span> <span class=\"n\">pprint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">pprint</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">((</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">,),</span> <span class=\"p\">[],</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">use_unicode</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\"> __1, 2 /1, 2  a, 4 |  \\</span>\n<span class=\"go\">/__     |           | x|</span>\n<span class=\"go\">\\_|4, 1 \\ 5         |  /</span>\n</code></pre></div>\n\n<p>Or as two nested vectors:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">pprint</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">([(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">),</span> <span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">)],</span> <span class=\"p\">([</span><span class=\"mi\">5</span><span class=\"p\">],</span> <span class=\"n\">Tuple</span><span class=\"p\">()),</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">use_unicode</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\"> __1, 2 /1, 2  3, 4 |  \\</span>\n<span class=\"go\">/__     |           | x|</span>\n<span class=\"go\">\\_|4, 1 \\ 5         |  /</span>\n</code></pre></div>\n\n<p>As with the hypergeometric function, the parameters may be passed as\narbitrary iterables. Vectors of length zero and one also have to be\npassed as iterables. The parameters need not be constants, but if they\ndepend on the argument then not much implemented functionality should be\nexpected.</p>\n\n<p>All the subvectors of parameters are available:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">pprint</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span> <span class=\"o\">=</span> <span class=\"n\">meijerg</span><span class=\"p\">([</span><span class=\"mi\">1</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">3</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">4</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">pprint</span><span class=\"p\">(</span><span class=\"n\">g</span><span class=\"p\">,</span> <span class=\"n\">use_unicode</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">)</span>\n<span class=\"go\"> __1, 1 /1  2 |  \\</span>\n<span class=\"go\">/__     |     | x|</span>\n<span class=\"go\">\\_|2, 2 \\3  4 |  /</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">an</span>\n<span class=\"go\">(1,)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">ap</span>\n<span class=\"go\">(1, 2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">aother</span>\n<span class=\"go\">(2,)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">bm</span>\n<span class=\"go\">(3,)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">bq</span>\n<span class=\"go\">(3, 4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">bother</span>\n<span class=\"go\">(4,)</span>\n</code></pre></div>\n\n<p>The Meijer G-function generalizes the hypergeometric functions.\nIn some cases it can be expressed in terms of hypergeometric functions,\nusing Slater's theorem. For example:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyperexpand</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">c</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">([</span><span class=\"n\">a</span><span class=\"p\">],</span> <span class=\"p\">[],</span> <span class=\"p\">[</span><span class=\"n\">c</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"n\">b</span><span class=\"p\">],</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">allow_hyper</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"go\">x**c*gamma(-a + c + 1)*hyper((-a + c + 1,),</span>\n<span class=\"go\">                             (-b + c + 1,), -x)/gamma(-b + c + 1)</span>\n</code></pre></div>\n\n<p>Thus the Meijer G-function also subsumes many named functions as special\ncases. You can use <code>expand_func()</code> or <code>hyperexpand()</code> to (try to)\nrewrite a Meijer G-function in terms of named special functions. For\nexample:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">expand_func</span><span class=\"p\">,</span> <span class=\"n\">S</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">([[],[]],</span> <span class=\"p\">[[</span><span class=\"mi\">0</span><span class=\"p\">],[]],</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">exp(x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hyperexpand</span><span class=\"p\">(</span><span class=\"n\">meijerg</span><span class=\"p\">([[],[]],</span> <span class=\"p\">[[</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">],[</span><span class=\"mi\">0</span><span class=\"p\">]],</span> <span class=\"p\">(</span><span class=\"n\">x</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">**</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">sin(x)/sqrt(pi)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>hyper\nsympy.simplify.hyperexpand</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.hyper.meijerg, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.meijerg.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "meijerg.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.meijerg.is_commutative", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "meijerg.is_commutative", "type": "variable", "doc": "<p></p>\n", "default_value": " = True"}, {"fullname": "algebra_with_sympy.algebraic_equation.meijerg.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "meijerg.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {'commutative': True}"}, {"fullname": "algebra_with_sympy.algebraic_equation.appellf1", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "appellf1", "type": "class", "doc": "<p>This is the Appell hypergeometric function of two variables as:</p>\n\n<p>.. math ::\n    F_1(a,b_1,b_2,c,x,y) = \\sum_{m=0}^{\\infty} \\sum_{n=0}^{\\infty}\n    \\frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}}\n    \\frac{x^m y^n}{m! n!}.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">appellf1</span><span class=\"p\">,</span> <span class=\"n\">symbols</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b1</span><span class=\"p\">,</span> <span class=\"n\">b2</span><span class=\"p\">,</span> <span class=\"n\">c</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;x y a b1 b2 c&#39;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mf\">2.</span><span class=\"p\">,</span> <span class=\"mf\">1.</span><span class=\"p\">,</span> <span class=\"mf\">6.</span><span class=\"p\">,</span> <span class=\"mf\">4.</span><span class=\"p\">,</span> <span class=\"mf\">5.</span><span class=\"p\">,</span> <span class=\"mf\">6.</span><span class=\"p\">)</span>\n<span class=\"go\">0.0063339426292673</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mf\">12.</span><span class=\"p\">,</span> <span class=\"mf\">12.</span><span class=\"p\">,</span> <span class=\"mf\">6.</span><span class=\"p\">,</span> <span class=\"mf\">4.</span><span class=\"p\">,</span> <span class=\"mf\">0.5</span><span class=\"p\">,</span> <span class=\"mf\">0.12</span><span class=\"p\">)</span>\n<span class=\"go\">172870711.659936</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">6</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">15</span><span class=\"p\">,</span> <span class=\"mi\">60</span><span class=\"p\">)</span>\n<span class=\"go\">appellf1(40, 2, 6, 4, 15, 60)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mf\">20.</span><span class=\"p\">,</span> <span class=\"mf\">12.</span><span class=\"p\">,</span> <span class=\"mf\">10.</span><span class=\"p\">,</span> <span class=\"mf\">3.</span><span class=\"p\">,</span> <span class=\"mf\">0.5</span><span class=\"p\">,</span> <span class=\"mf\">0.12</span><span class=\"p\">)</span>\n<span class=\"go\">15605338197184.4</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">6</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">appellf1(40, 2, 6, 4, x, y)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">appellf1</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b1</span><span class=\"p\">,</span> <span class=\"n\">b2</span><span class=\"p\">,</span> <span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">appellf1(a, b1, b2, c, x, y)</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.hyper.appellf1, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.appellf1.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "appellf1.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.appellf1.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "appellf1.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.legendre", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "legendre", "type": "class", "doc": "<p><code>legendre(n, x)</code> gives the nth Legendre polynomial of x, $P_n(x)$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Legendre polynomials are orthogonal on [-1, 1] with respect to\nthe constant weight 1. They satisfy $P_n(1) = 1$ for all n; further,\n$P_n$ is odd for odd n and even for even n.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">legendre</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">3*x**2/2 - 1/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">legendre(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">legendre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nassoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.legendre, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.legendre.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "legendre.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.legendre.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "legendre.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_legendre", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_legendre", "type": "class", "doc": "<p><code>assoc_legendre(n, m, x)</code> gives $P_n^m(x)$, where n and m are\nthe degree and order or an expression which is related to the nth\norder Legendre polynomial, $P_n(x)$ in the following manner:</p>\n\n<p>$$P_n^m(x) = (-1)^m (1 - x^2)^{\\frac{m}{2}}\n           \\frac{\\mathrm{d}^m P_n(x)}{\\mathrm{d} x^m}$$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Associated Legendre polynomials are orthogonal on [-1, 1] with:</p>\n\n<ul>\n<li>weight = 1            for the same m, and different n.</li>\n<li>weight = 1/(1-x**2)   for the same n, and different m.</li>\n</ul>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">assoc_legendre</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_legendre</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_legendre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_legendre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(1 - x**2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_legendre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">assoc_legendre(n, m, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.assoc_legendre, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_legendre.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_legendre.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_legendre.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_legendre.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.hermite", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hermite", "type": "class", "doc": "<p><code>hermite(n, x)</code> gives the nth Hermite polynomial in x, $H_n(x)$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Hermite polynomials are orthogonal on $(-\\infty, \\infty)$\nwith respect to the weight $\\exp\\left(-x^2\\right)$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hermite</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">4*x**2 - 2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">hermite(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*n*hermite(n - 1, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">hermite</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*hermite(n, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.hermite, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.hermite.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hermite.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.hermite.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "hermite.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt", "type": "class", "doc": "<p>Chebyshev polynomial of the first kind, $T_n(x)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>chebyshevt(n, x)</code> gives the nth Chebyshev polynomial (of the first\nkind) in x, $T_n(x)$.</p>\n\n<p>The Chebyshev polynomials of the first kind are orthogonal on\n$[-1, 1]$ with respect to the weight $\\frac{1}{\\sqrt{1-x^2}}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">chebyshevt</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x**2 - 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">chebyshevt(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*chebyshevt(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">chebyshevt(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">cos(pi*n/2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">n*chebyshevu(n - 1, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.chebyshevt, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu", "type": "class", "doc": "<p>Chebyshev polynomial of the second kind, $U_n(x)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>chebyshevu(n, x)</code> gives the nth Chebyshev polynomial of the second\nkind in x, $U_n(x)$.</p>\n\n<p>The Chebyshev polynomials of the second kind are orthogonal on\n$[-1, 1]$ with respect to the weight $\\sqrt{1-x^2}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">chebyshevu</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">4*x**2 - 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">chebyshevu(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*chebyshevu(n, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-chebyshevu(n - 2, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">cos(pi*n/2)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">n + 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.chebyshevu, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu_root", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu_root", "type": "class", "doc": "<p><code>chebyshevu_root(n, k)</code> returns the kth root (indexed from zero) of the\nnth Chebyshev polynomial of the second kind; that is, if 0 &lt;= k &lt; n,\n<code>chebyshevu(n, chebyshevu_root(n, k)) == 0</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">chebyshevu</span><span class=\"p\">,</span> <span class=\"n\">chebyshevu_root</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu_root</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(2)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevu</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">chebyshevu_root</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>chebyshevt, chebyshevt_root, chebyshevu,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n", "bases": "sympy.functions.special.polynomials.chebyshevu_root, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu_root.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu_root.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevu_root.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevu_root.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt_root", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt_root", "type": "class", "doc": "<p><code>chebyshev_root(n, k)</code> returns the kth root (indexed from zero) of\nthe nth Chebyshev polynomial of the first kind; that is, if\n0 &lt;= k &lt; n, <code>chebyshevt(n, chebyshevt_root(n, k)) == 0</code>.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">chebyshevt</span><span class=\"p\">,</span> <span class=\"n\">chebyshevt_root</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt_root</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(3)/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">chebyshevt</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">chebyshevt_root</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">))</span>\n<span class=\"go\">0</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n", "bases": "sympy.functions.special.polynomials.chebyshevt_root, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt_root.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt_root.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.chebyshevt_root.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "chebyshevt_root.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.laguerre", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "laguerre", "type": "class", "doc": "<p>Returns the nth Laguerre polynomial in x, $L_n(x)$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">laguerre</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1 - x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**2/2 - 2*x + 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-x**3/6 + 3*x**2/2 - 3*x + 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">laguerre(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-assoc_laguerre(n - 1, 1, x)</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>n : int\n    Degree of Laguerre polynomial. Must be <code>n &gt;= 0</code>.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nassoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.laguerre, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.laguerre.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "laguerre.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.laguerre.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "laguerre.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_laguerre", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_laguerre", "type": "class", "doc": "<p>Returns the nth generalized Laguerre polynomial in x, $L_n(x)$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">assoc_laguerre</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a - x + 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +</span>\n<span class=\"go\">    x*(-a**2/2 - 5*a/2 - 3) + 1</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">binomial(a + n, a)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">assoc_laguerre(n, a, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">laguerre(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-assoc_laguerre(n - 1, a + 1, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">assoc_laguerre</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n<span class=\"go\">Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))</span>\n</code></pre></div>\n\n<h1 id=\"parameters\">Parameters</h1>\n\n<p>n : int\n    Degree of Laguerre polynomial. Must be <code>n &gt;= 0</code>.</p>\n\n<p>alpha : Expr\n    Arbitrary expression. For <code>alpha=0</code> regular Laguerre\n    polynomials will be generated.</p>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi, gegenbauer,\nchebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.assoc_laguerre, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_laguerre.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_laguerre.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.assoc_laguerre.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "assoc_laguerre.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.gegenbauer", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gegenbauer", "type": "class", "doc": "<p>Gegenbauer polynomial $C_n^{\\left(\\alpha\\right)}(x)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>gegenbauer(n, alpha, x)</code> gives the nth Gegenbauer polynomial\nin x, $C_n^{\\left(\\alpha\\right)}(x)$.</p>\n\n<p>The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with\nrespect to the weight $\\left(1-x^2\\right)^{\\alpha-\\frac{1}{2}}$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">gegenbauer</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*a*x</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">-a + x**2*(2*a**2 + 2*a)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">gegenbauer(n, a, x)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*gegenbauer(n, a, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">gegenbauer(n, conjugate(a), conjugate(x))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">gegenbauer</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*a*gegenbauer(n - 1, a + 1, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>jacobi,\nchebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.gegenbauer, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.gegenbauer.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gegenbauer.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.gegenbauer.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "gegenbauer.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.jacobi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jacobi", "type": "class", "doc": "<p>Jacobi polynomial $P_n^{\\left(\\alpha, \\beta\\right)}(x)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>jacobi(n, alpha, beta, x)</code> gives the nth Jacobi polynomial\nin x, $P_n^{\\left(\\alpha, \\beta\\right)}(x)$.</p>\n\n<p>The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect\nto the weight $\\left(1-x\\right)^\\alpha \\left(1+x\\right)^\\beta$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">jacobi</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">x</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">1</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a/2 - b/2 + x*(a/2 + b/2 + 1)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">jacobi(n, a, b, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">RisingFactorial(a + 1, n)*gegenbauer(n,</span>\n<span class=\"go\">    a + 1/2, x)/RisingFactorial(2*a + 1, n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">legendre(n, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">S</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**n*jacobi(n, b, a, x)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(2**n*factorial(n)*gamma(a + 1))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"go\">RisingFactorial(a + 1, n)/factorial(n)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">))</span>\n<span class=\"go\">jacobi(n, conjugate(a), conjugate(b), conjugate(x))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">jacobi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">a</span><span class=\"p\">,</span><span class=\"n\">b</span><span class=\"p\">,</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gegenbauer,\nchebyshevt_root, chebyshevu, chebyshevu_root,\nlegendre, assoc_legendre,\nhermite,\nlaguerre, assoc_laguerre,\nsympy.polys.orthopolys.jacobi_poly,\nsympy.polys.orthopolys.gegenbauer_poly\nsympy.polys.orthopolys.chebyshevt_poly\nsympy.polys.orthopolys.chebyshevu_poly\nsympy.polys.orthopolys.hermite_poly\nsympy.polys.orthopolys.legendre_poly\nsympy.polys.orthopolys.laguerre_poly</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.polynomials.jacobi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.jacobi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jacobi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.jacobi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "jacobi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ynm", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ynm", "type": "class", "doc": "<p>Spherical harmonics defined as</p>\n\n<p>$$Y_n^m(\\theta, \\varphi) := \\sqrt{\\frac{(2n+1)(n-m)!}{4\\pi(n+m)!}}\n                          \\exp(i m \\varphi)\n                          \\mathrm{P}_n^m\\left(\\cos(\\theta)\\right)$$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p><code>Ynm()</code> gives the spherical harmonic function of order $n$ and $m$\nin $\\theta$ and $\\varphi$, $Y_n^m(\\theta, \\varphi)$. The four\nparameters are as follows: $n \\geq 0$ an integer and $m$ an integer\nsuch that $-n \\leq m \\leq n$ holds. The two angles are real-valued\nwith $\\theta \\in [0, \\pi]$ and $\\varphi \\in [0, 2\\pi]$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ynm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">simplify</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>Several symmetries are known, for the order:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">(-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>As well as for the angles:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">exp(-2*I*m*phi)*Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>For specific integers $n$ and $m$ we can evaluate the harmonics\nto more useful expressions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">1/(2*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(3)*cos(theta)/(2*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">-sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">-sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))</span>\n</code></pre></div>\n\n<p>We can differentiate the functions with respect\nto both angles:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ynm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">),</span> <span class=\"n\">theta</span><span class=\"p\">)</span>\n<span class=\"go\">m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">),</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">I*m*Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>Further we can compute the complex conjugation:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ynm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">))</span>\n<span class=\"go\">(-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>To get back the well known expressions in spherical\ncoordinates, we use full expansion:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Ynm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">expand_func</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span><span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expand_func</span><span class=\"p\">(</span><span class=\"n\">Ynm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">))</span>\n<span class=\"go\">sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Ynm_c, Znm</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.spherical_harmonics.Ynm, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ynm.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ynm.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Ynm.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Ynm.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.Znm", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Znm", "type": "class", "doc": "<p>Real spherical harmonics defined as</p>\n\n<p>$$Z_n^m(\\theta, \\varphi) :=\n\\begin{cases}\n  \\frac{Y_n^m(\\theta, \\varphi) + \\overline{Y_n^m(\\theta, \\varphi)}}{\\sqrt{2}} &amp;\\quad m &gt; 0 \\\n  Y_n^m(\\theta, \\varphi) &amp;\\quad m = 0 \\\n  \\frac{Y_n^m(\\theta, \\varphi) - \\overline{Y_n^m(\\theta, \\varphi)}}{i \\sqrt{2}} &amp;\\quad m &lt; 0 \\\n\\end{cases}$$</p>\n\n<p>which gives in simplified form</p>\n\n<p>$$Z_n^m(\\theta, \\varphi) =\n\\begin{cases}\n  \\frac{Y_n^m(\\theta, \\varphi) + (-1)^m Y_n^{-m}(\\theta, \\varphi)}{\\sqrt{2}} &amp;\\quad m &gt; 0 \\\n  Y_n^m(\\theta, \\varphi) &amp;\\quad m = 0 \\\n  \\frac{Y_n^m(\\theta, \\varphi) - (-1)^m Y_n^{-m}(\\theta, \\varphi)}{i \\sqrt{2}} &amp;\\quad m &lt; 0 \\\n\\end{cases}$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">Znm</span><span class=\"p\">,</span> <span class=\"n\">Symbol</span><span class=\"p\">,</span> <span class=\"n\">simplify</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">theta</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;theta&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">phi</span> <span class=\"o\">=</span> <span class=\"n\">Symbol</span><span class=\"p\">(</span><span class=\"s2\">&quot;phi&quot;</span><span class=\"p\">)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">Znm</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span>\n<span class=\"go\">Znm(n, m, theta, phi)</span>\n</code></pre></div>\n\n<p>For specific integers n and m we can evaluate the harmonics\nto more useful expressions:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Znm</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">1/(2*sqrt(pi))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Znm</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">-sqrt(3)*sin(theta)*cos(phi)/(2*sqrt(pi))</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">simplify</span><span class=\"p\">(</span><span class=\"n\">Znm</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">theta</span><span class=\"p\">,</span> <span class=\"n\">phi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">func</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">))</span>\n<span class=\"go\">-sqrt(15)*sin(2*theta)*cos(phi)/(4*sqrt(pi))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>Ynm, Ynm_c</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.spherical_harmonics.Znm, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.Znm.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Znm.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.Znm.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Znm.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_k", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_k", "type": "class", "doc": "<p>The complete elliptic integral of the first kind, defined by</p>\n\n<p>$$K(m) = F\\left(\\tfrac{\\pi}{2}\\middle| m\\right)$$</p>\n\n<p>where $F\\left(z\\middle| m\\right)$ is the Legendre incomplete\nelliptic integral of the first kind.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The function $K(m)$ is a single-valued function on the complex\nplane with branch cut along the interval $(1, \\infty)$.</p>\n\n<p>Note that our notation defines the incomplete elliptic integral\nin terms of the parameter $m$ instead of the elliptic modulus\n(eccentricity) $k$.\nIn this case, the parameter $m$ is defined as $m=k^2$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">elliptic_k</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_k</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_k</span><span class=\"p\">(</span><span class=\"mf\">1.0</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">1.50923695405127 + 0.625146415202697*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_k</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>elliptic_f</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.elliptic_integrals.elliptic_k, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_k.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_k.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_k.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_k.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_f", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_f", "type": "class", "doc": "<p>The Legendre incomplete elliptic integral of the first\nkind, defined by</p>\n\n<p>$$F\\left(z\\middle| m\\right) =\\int_0^z \\frac{dt}{\\sqrt{1 - m \\sin^2 t}}$$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function reduces to a complete elliptic integral of\nthe first kind, $K(m)$, when $z = \\pi/2$.</p>\n\n<p>Note that our notation defines the incomplete elliptic integral\nin terms of the parameter $m$ instead of the elliptic modulus\n(eccentricity) $k$.\nIn this case, the parameter $m$ is defined as $m=k^2$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">elliptic_f</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_f</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_f</span><span class=\"p\">(</span><span class=\"mf\">3.0</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mf\">1.0</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">2.909449841483 + 1.74720545502474*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>elliptic_k</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.elliptic_integrals.elliptic_f, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_f.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_f.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_f.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_f.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_e", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_e", "type": "class", "doc": "<p>Called with two arguments $z$ and $m$, evaluates the\nincomplete elliptic integral of the second kind, defined by</p>\n\n<p>$$E\\left(z\\middle| m\\right) = \\int_0^z \\sqrt{1 - m \\sin^2 t} dt$$</p>\n\n<p>Called with a single argument $m$, evaluates the Legendre complete\nelliptic integral of the second kind</p>\n\n<p>$$E(m) = E\\left(\\tfrac{\\pi}{2}\\middle| m\\right)$$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The function $E(m)$ is a single-valued function on the complex\nplane with branch cut along the interval $(1, \\infty)$.</p>\n\n<p>Note that our notation defines the incomplete elliptic integral\nin terms of the parameter $m$ instead of the elliptic modulus\n(eccentricity) $k$.\nIn this case, the parameter $m$ is defined as $m=k^2$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">elliptic_e</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"o\">=</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"mi\">2</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">n</span><span class=\"p\">()</span>\n<span class=\"go\">1.55203744279187 + 0.290764986058437*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_e</span><span class=\"p\">(</span><span class=\"mf\">2.0</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">0.991052601328069 + 0.81879421395609*I</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.elliptic_integrals.elliptic_e, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_e.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_e.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_e.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_e.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_pi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_pi", "type": "class", "doc": "<p>Called with three arguments $n$, $z$ and $m$, evaluates the\nLegendre incomplete elliptic integral of the third kind, defined by</p>\n\n<p>$$\\Pi\\left(n; z\\middle| m\\right) = \\int_0^z \\frac{dt}{\\left(1 - n \\sin^2 t\\right) \\sqrt{1 - m \\sin^2 t}}$$</p>\n\n<p>Called with two arguments $n$ and $m$, evaluates the complete\nelliptic integral of the third kind:</p>\n\n<p>$$\\Pi\\left(n\\middle| m\\right) =\\Pi\\left(n; \\tfrac{\\pi}{2}\\middle| m\\right)$$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>Note that our notation defines the incomplete elliptic integral\nin terms of the parameter $m$ instead of the elliptic modulus\n(eccentricity) $k$.\nIn this case, the parameter $m$ is defined as $m=k^2$.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">elliptic_pi</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">m</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_pi</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">m</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">series</span><span class=\"p\">(</span><span class=\"n\">z</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"o\">=</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"go\">z + z**3*(m/6 + n/3) + O(z**4)</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_pi</span><span class=\"p\">(</span><span class=\"mf\">0.5</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"mf\">1.0</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"mf\">1.2</span><span class=\"p\">)</span>\n<span class=\"go\">2.50232379629182 - 0.760939574180767*I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_pi</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">)</span>\n<span class=\"go\">pi/2</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">elliptic_pi</span><span class=\"p\">(</span><span class=\"mf\">1.0</span> <span class=\"o\">-</span> <span class=\"n\">I</span><span class=\"o\">/</span><span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mf\">2.0</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span>\n<span class=\"go\">3.29136443417283 + 0.32555634906645*I</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.elliptic_integrals.elliptic_pi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_pi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_pi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.elliptic_pi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "elliptic_pi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.beta", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "beta", "type": "class", "doc": "<p>The beta integral is called the Eulerian integral of the first kind by\nLegendre:</p>\n\n<p>$$\\mathrm{B}(x,y)  \\int^{1}_{0} t^{x-1} (1-t)^{y-1} \\mathrm{d}t.$$</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>The Beta function or Euler's first integral is closely associated\nwith the gamma function. The Beta function is often used in probability\ntheory and mathematical statistics. It satisfies properties like:</p>\n\n<p>$$\\mathrm{B}(a,1) = \\frac{1}{a} \\\n\\mathrm{B}(a,b) = \\mathrm{B}(b,a)  \\\n\\mathrm{B}(a,b) = \\frac{\\Gamma(a) \\Gamma(b)}{\\Gamma(a+b)}$$</p>\n\n<p>Therefore for integral values of $a$ and $b$:</p>\n\n<p>$$\\mathrm{B} = \\frac{(a-1)! (b-1)!}{(a+b-1)!}$$</p>\n\n<p>A special case of the Beta function when <code>x = y</code> is the\nCentral Beta function. It satisfies properties like:</p>\n\n<p>$$\\mathrm{B}(x) = 2^{1 - 2x}\\mathrm{B}(x, \\frac{1}{2})\n\\mathrm{B}(x) = 2^{1 - 2x} cos(\\pi x) \\mathrm{B}(\\frac{1}{2} - x, x)\n\\mathrm{B}(x) = \\int_{0}^{1} \\frac{t^x}{(1 + t)^{2x}} dt\n\\mathrm{B}(x) = \\frac{2}{x} \\prod_{n = 1}^{\\infty} \\frac{n(n + 2x)}{(n + x)^2}$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"n\">pi</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span>\n</code></pre></div>\n\n<p>The Beta function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">beta</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">))</span>\n<span class=\"go\">beta(conjugate(x), conjugate(y))</span>\n</code></pre></div>\n\n<p>Differentiation with respect to both $x$ and $y$ is supported:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">beta</span><span class=\"p\">,</span> <span class=\"n\">diff</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">(polygamma(0, x) - polygamma(0, x + y))*beta(x, y)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"n\">y</span><span class=\"p\">)</span>\n<span class=\"go\">(polygamma(0, y) - polygamma(0, x + y))*beta(x, y)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">),</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">2*(polygamma(0, x) - polygamma(0, 2*x))*beta(x, x)</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Beta function to\narbitrary precision for any complex numbers x and y:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">beta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">40</span><span class=\"p\">)</span>\n<span class=\"go\">0.02671848900111377452242355235388489324562</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">beta</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">+</span> <span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n<span class=\"go\">-0.2112723729365330143 - 0.7655283165378005676*I</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>gamma: Gamma function.\nuppergamma: Upper incomplete gamma function.\nlowergamma: Lower incomplete gamma function.\npolygamma: Polygamma function.\nloggamma: Log Gamma function.\ndigamma: Digamma function.\ntrigamma: Trigamma function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.beta_functions.beta, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.beta.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "beta.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.beta.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "beta.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieus", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieus", "type": "class", "doc": "<p>The Mathieu Sine function $S(a,q,z)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is one solution of the Mathieu differential equation:</p>\n\n<p>.. math ::\n    y(x)^{\\prime\\prime} + (a - 2 q \\cos(2 x)) y(x) = 0</p>\n\n<p>The other solution is the Mathieu Cosine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">mathieus</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieus</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieus(a, q, z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieus</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sin(sqrt(a)*z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">mathieus</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieusprime(a, q, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>mathieuc: Mathieu cosine function.\nmathieusprime: Derivative of Mathieu sine function.\nmathieucprime: Derivative of Mathieu cosine function.</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.mathieu_functions.mathieus, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieus.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieus.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieus.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieus.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieuc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieuc", "type": "class", "doc": "<p>The Mathieu Cosine function $C(a,q,z)$.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is one solution of the Mathieu differential equation:</p>\n\n<p>.. math ::\n    y(x)^{\\prime\\prime} + (a - 2 q \\cos(2 x)) y(x) = 0</p>\n\n<p>The other solution is the Mathieu Sine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">mathieuc</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieuc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieuc(a, q, z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieuc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">cos(sqrt(a)*z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">mathieuc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieucprime(a, q, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>mathieus: Mathieu sine function\nmathieusprime: Derivative of Mathieu sine function\nmathieucprime: Derivative of Mathieu cosine function</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.mathieu_functions.mathieuc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieuc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieuc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieuc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieuc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieusprime", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieusprime", "type": "class", "doc": "<p>The derivative $S^{\\prime}(a,q,z)$ of the Mathieu Sine function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is one solution of the Mathieu differential equation:</p>\n\n<p>.. math ::\n    y(x)^{\\prime\\prime} + (a - 2 q \\cos(2 x)) y(x) = 0</p>\n\n<p>The other solution is the Mathieu Cosine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">mathieusprime</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieusprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieusprime(a, q, z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieusprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">sqrt(a)*cos(sqrt(a)*z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">mathieusprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">(-a + 2*q*cos(2*z))*mathieus(a, q, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>mathieus: Mathieu sine function\nmathieuc: Mathieu cosine function\nmathieucprime: Derivative of Mathieu cosine function</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.mathieu_functions.mathieusprime, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieusprime.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieusprime.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieusprime.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieusprime.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieucprime", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieucprime", "type": "class", "doc": "<p>The derivative $C^{\\prime}(a,q,z)$ of the Mathieu Cosine function.</p>\n\n<h1 id=\"explanation\">Explanation</h1>\n\n<p>This function is one solution of the Mathieu differential equation:</p>\n\n<p>.. math ::\n    y(x)^{\\prime\\prime} + (a - 2 q \\cos(2 x)) y(x) = 0</p>\n\n<p>The other solution is the Mathieu Sine function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">diff</span><span class=\"p\">,</span> <span class=\"n\">mathieucprime</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieucprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">mathieucprime(a, q, z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">mathieucprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">-sqrt(a)*sin(sqrt(a)*z)</span>\n</code></pre></div>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">diff</span><span class=\"p\">(</span><span class=\"n\">mathieucprime</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">,</span> <span class=\"n\">z</span><span class=\"p\">),</span> <span class=\"n\">z</span><span class=\"p\">)</span>\n<span class=\"go\">(-a + 2*q*cos(2*z))*mathieuc(a, q, z)</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>mathieus: Mathieu sine function\nmathieuc: Mathieu cosine function\nmathieusprime: Derivative of Mathieu sine function</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.mathieu_functions.mathieucprime, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieucprime.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieucprime.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.mathieucprime.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "mathieucprime.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.riemann_xi", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "riemann_xi", "type": "class", "doc": "<p>Riemann Xi function.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<p>The Riemann Xi function is closely related to the Riemann zeta function.\nThe zeros of Riemann Xi function are precisely the non-trivial zeros\nof the zeta function.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">riemann_xi</span><span class=\"p\">,</span> <span class=\"n\">zeta</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy.abc</span> <span class=\"kn\">import</span> <span class=\"n\">s</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">riemann_xi</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">zeta</span><span class=\"p\">)</span>\n<span class=\"go\">s*(s - 1)*gamma(s/2)*zeta(s)/(2*pi**(s/2))</span>\n</code></pre></div>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.zeta_functions.riemann_xi, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.riemann_xi.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "riemann_xi.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.riemann_xi.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "riemann_xi.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc", "type": "class", "doc": "<p>The Generalized Incomplete Beta function is defined as</p>\n\n<p>$$\\mathrm{B}_{(x_1, x_2)}(a, b) = \\int_{x_1}^{x_2} t^{a - 1} (1 - t)^{b - 1} dt$$</p>\n\n<p>The Incomplete Beta function is a special case\nof the Generalized Incomplete Beta function :</p>\n\n<p>$$\\mathrm{B}_z (a, b) = \\mathrm{B}_{(0, z)}(a, b)$$</p>\n\n<p>The Incomplete Beta function satisfies :</p>\n\n<p>$$\\mathrm{B}_z (a, b) = (-1)^a \\mathrm{B}_{\\frac{z}{z - 1}} (a, 1 - a - b)$$</p>\n\n<p>The Beta function is a special case of the Incomplete Beta function :</p>\n\n<p>$$\\mathrm{B}(a, b) = \\mathrm{B}_{1}(a, b)$$</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">betainc</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;a b x x1 x2&#39;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<p>The Generalized Incomplete Beta function is given by:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">)</span>\n<span class=\"go\">betainc(a, b, x1, x2)</span>\n</code></pre></div>\n\n<p>The Incomplete Beta function can be obtained as follows:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">betainc(a, b, 0, x)</span>\n</code></pre></div>\n\n<p>The Incomplete Beta function obeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">))</span>\n<span class=\"go\">betainc(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Incomplete Beta function to\narbitrary precision for any complex numbers a, b, x1 and x2:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">betainc</span><span class=\"p\">,</span> <span class=\"n\">I</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">3</span><span class=\"p\">,</span> <span class=\"mi\">4</span><span class=\"p\">,</span> <span class=\"mi\">5</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)</span>\n<span class=\"go\">56.08333333</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"mf\">0.75</span><span class=\"p\">,</span> <span class=\"mi\">1</span> <span class=\"o\">-</span> <span class=\"mi\">4</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">2</span> <span class=\"o\">+</span> <span class=\"mi\">3</span><span class=\"o\">*</span><span class=\"n\">I</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">25</span><span class=\"p\">)</span>\n<span class=\"go\">0.2241657956955709603655887 + 0.3619619242700451992411724*I</span>\n</code></pre></div>\n\n<p>The Generalized Incomplete Beta function can be expressed\nin terms of the Generalized Hypergeometric function.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">(-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/a</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>beta: Beta function\nhyper: Generalized Hypergeometric function</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.beta_functions.betainc, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc_regularized", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc_regularized", "type": "class", "doc": "<p>The Generalized Regularized Incomplete Beta function is given by</p>\n\n<p>$$\\mathrm{I}_{(x_1, x_2)}(a, b) = \\frac{\\mathrm{B}_{(x_1, x_2)}(a, b)}{\\mathrm{B}(a, b)}$$</p>\n\n<p>The Regularized Incomplete Beta function is a special case\nof the Generalized Regularized Incomplete Beta function :</p>\n\n<p>$$\\mathrm{I}_z (a, b) = \\mathrm{I}_{(0, z)}(a, b)$$</p>\n\n<p>The Regularized Incomplete Beta function is the cumulative distribution\nfunction of the beta distribution.</p>\n\n<h1 id=\"examples\">Examples</h1>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">betainc_regularized</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">conjugate</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">(</span><span class=\"s1\">&#39;a b x x1 x2&#39;</span><span class=\"p\">)</span>\n</code></pre></div>\n\n<p>The Generalized Regularized Incomplete Beta\nfunction is given by:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">)</span>\n<span class=\"go\">betainc_regularized(a, b, x1, x2)</span>\n</code></pre></div>\n\n<p>The Regularized Incomplete Beta function\ncan be obtained as follows:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">)</span>\n<span class=\"go\">betainc_regularized(a, b, 0, x)</span>\n</code></pre></div>\n\n<p>The Regularized Incomplete Beta function\nobeys the mirror symmetry:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">conjugate</span><span class=\"p\">(</span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">))</span>\n<span class=\"go\">betainc_regularized(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))</span>\n</code></pre></div>\n\n<p>We can numerically evaluate the Regularized Incomplete Beta function\nto arbitrary precision for any complex numbers a, b, x1 and x2:</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">betainc_regularized</span><span class=\"p\">,</span> <span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">E</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mf\">0.25</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)</span>\n<span class=\"go\">0.4375000000</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">E</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">evalf</span><span class=\"p\">(</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"go\">1.00000</span>\n</code></pre></div>\n\n<p>The Generalized Regularized Incomplete Beta function can be\nexpressed in terms of the Generalized Hypergeometric function.</p>\n\n<div class=\"pdoc-code codehilite\"><pre><span></span><code><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"kn\">from</span> <span class=\"nn\">sympy</span> <span class=\"kn\">import</span> <span class=\"n\">hyper</span>\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">betainc_regularized</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">,</span> <span class=\"n\">x1</span><span class=\"p\">,</span> <span class=\"n\">x2</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">rewrite</span><span class=\"p\">(</span><span class=\"n\">hyper</span><span class=\"p\">)</span>\n<span class=\"go\">(-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/(a*beta(a, b))</span>\n</code></pre></div>\n\n<h1 id=\"see-also\">See Also</h1>\n\n<p>beta: Beta function\nhyper: Generalized Hypergeometric function</p>\n\n<h1 id=\"references\">References</h1>\n\n<div class=\"footnotes\">\n<hr />\n<ol>\n</ol>\n</div>\n", "bases": "sympy.functions.special.beta_functions.betainc_regularized, EqnFunction"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc_regularized.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc_regularized.__init__", "type": "function", "doc": "<p></p>\n", "signature": "()", "funcdef": "def"}, {"fullname": "algebra_with_sympy.algebraic_equation.betainc_regularized.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "betainc_regularized.default_assumptions", "type": "variable", "doc": "<p></p>\n", "default_value": " = {}"}, {"fullname": "algebra_with_sympy.preparser", "modulename": "algebra_with_sympy.preparser", "type": "module", "doc": "<p></p>\n"}, {"fullname": "algebra_with_sympy.preparser.algebra_with_sympy_preparser", "modulename": "algebra_with_sympy.preparser", "qualname": "algebra_with_sympy_preparser", "type": "function", "doc": "<p>In IPython compatible environments (Jupyter, IPython, etc...) this supports\na special compact input method for equations.</p>\n\n<p>The syntax supported is <code>equation_name =@ equation.lhs = equation.rhs</code>,\nwhere <code>equation_name</code> is a valid Python name that can be used to refer to\nthe equation later. <code>equation.lhs</code> is the left-hand side of the equation\nand <code>equation.rhs</code> is the right-hand side of the equation. Each side of the\nequation must parse into a valid Sympy expression.</p>\n\n<p><strong>Note</strong>: If the <code>equation_name</code> is omitted the equation will be formed,\nbut it will\nnot be assigned to a name that can be used to refer to it later. You may be\nable to access it through one of the special IPython underscore names. This\nis not recommended.</p>\n\n<p><strong>THIS FUNCTION IS USED BY THE IPYTHON ENVIRONMENT TO PREPARSE THE INPUT\nBEFORE IT IS PASSED TO THE PYTHON INTERPRETER. IT IS NOT MEANT TO BE USED\nDIRECTLY BY A USER</strong></p>\n", "signature": "(lines)", "funcdef": "def"}];
 
     // mirrored in build-search-index.js (part 1)
     // Also split on html tags. this is a cheap heuristic, but good enough.
diff --git a/setup.py b/setup.py
index 6608fa4..1d195ff 100644
--- a/setup.py
+++ b/setup.py
@@ -6,7 +6,7 @@
 setuptools.setup(
     name="Algebra_with_SymPy",
     url = "https://gutow.github.io/Algebra_with_Sympy/",
-    version="0.9.1.1",
+    version="0.9.2",
     description="Equations that can be algebraicly manipulated.",
     long_description=long_description,
     long_description_content_type="text/markdown",
