ch03-code-listing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 3: Symbolic computing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Robert Johansson\n",
    "\n",
    "Source code listings for [Numerical Python - Scientific Computing and Data Science Applications with Numpy, SciPy and Matplotlib](https://link.springer.com/book/10.1007/979-8-8688-0413-7) (ISBN 979-8-8688-0412-0)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import I, pi, oo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = sympy.Symbol(\"y\", real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.is_real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.is_real is None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Symbol(\"z\", imaginary=True).is_real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = sympy.Symbol(\"y\", positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{x^{2}}$"
      ],
      "text/plain": [
       "   ____\n",
       "  ╱  2 \n",
       "╲╱  x  "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sqrt(x ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAANCAYAAAB/9ZQ7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8klEQVQoFXWS4Q2CQAyFi3EAZ9ANnEE3UEdwBcMv+GdwA3ECoxvoBkY20A00bIDva+4AE23Sa/t4LY8eSdM0luf5yMzW8on8oXqn6BaeHRSXg4ClgXBRnQYshpWSBcVApLHijUI2lz896w6wKyWT32o4U8iYcvSsO2ZKeaNPrknUAIj2khoTNlUAayeDY0t5JYI3O2LGgFpYRR0/kBztf/VC6JO/iJrG1FYv5CFHsI0i+ywUX3J2jrlez7iUX55lWSG/95+5DE3by+/erUM5G+BGtxEjRs1o6+/3oLpUU9w/XEvCv8F1sg0MrScRO60Om30Aa7RlfSPkRy8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle y$"
      ],
      "text/plain": [
       "y"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sqrt(y ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "n1, n2, n3 = sympy.Symbol(\"n\"), sympy.Symbol(\"n\", integer=True), sympy.Symbol(\"n\", odd=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(\\pi n \\right)}$"
      ],
      "text/plain": [
       "cos(π⋅n)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(n1 * pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(-1\\right)^{n}$"
      ],
      "text/plain": [
       "    n\n",
       "(-1) "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(n2 * pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAOCAYAAAA1+Nx+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiElEQVQ4EWP8//8/A7VAQ0ODEdCs1UBsDGR/AJnLAiIoAUCDBID6ZwPxOyA2AWIlIIYDalgAcmkoyESgZWVACuQLOGCCs2jEGLWAYMDSPIhYoMlsL9ApoORGLAgF6jtHjGKQBaBkZkyMYnLU0DyIhr4FjNQo7IDxCCrgQIkEVBaB6HtADEoEpwEsISASVdEGngAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle -1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(n3 * pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = sympy.symbols(\"a, b, c\", negative=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "d, e, f = sympy.symbols(\"d, e, f\", positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "i = sympy.Integer(19)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"i = 19 [type <class 'sympy.core.numbers.Integer'>]\""
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"i = {} [type {}]\".format(i, type(i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, True, True)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i.is_Integer, i.is_real, i.is_odd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sympy.Float(2.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"f = 2.30000000000000 [type <class 'sympy.core.numbers.Float'>]\""
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"f = {} [type {}]\".format(f, type(f))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(False, True, False)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.is_Integer, f.is_real, f.is_odd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "i, f = sympy.sympify(19), sympy.sympify(2.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.numbers.Integer"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.numbers.Float"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = sympy.Symbol(\"n\", integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, False, None, True)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n.is_integer, n.is_Integer, n.is_positive, n.is_Symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "i = sympy.Integer(19)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, True, True, False)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i.is_integer, i.is_Integer, i.is_positive, i.is_Symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle 8663234049605954426644038200675212212900743262211018069459689001$"
      ],
      "text/plain": [
       "8663234049605954426644038200675212212900743262211018069459689001"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i ** 50"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000$"
      ],
      "text/plain": [
       "93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factorial(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.2999999999999999888977698'"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"%.25f\" % 0.3  # create a string represention with 25 decimals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.2999999999999999888977698$"
      ],
      "text/plain": [
       "0.2999999999999999888977698"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Float(0.3, 25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.3$"
      ],
      "text/plain": [
       "0.3000000000000000000000000"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Float('0.3', 25)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rationals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{11}{13}$"
      ],
      "text/plain": [
       "11\n",
       "──\n",
       "13"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Rational(11, 13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 = sympy.Rational(2, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "r2 = sympy.Rational(4, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{8}{15}$"
      ],
      "text/plain": [
       "8/15"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1 * r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{5}{6}$"
      ],
      "text/plain": [
       "5/6"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1 / r2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, z = sympy.symbols(\"x, y, z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sympy.Function(\"f\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.function.UndefinedFunction"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = sympy.Function(\"g\")(x, y, z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAVCAYAAAAD1GMqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEHklEQVRYCd2Y31HcMBCHHYYCLkkFOTogoYJAB/ypAOgAhid4y5AOgAoS6ABSAYEOoIMAHZDvE5LiE7bxnV8gO6OTtNr9eXe1WtlXPT4+VkPb/v7+eCjGa9Vv8m2uGkgHBwc7QCwOhHnN6uPoY7ZxUNAAWwXpI/1ZRvzPBvh2oUvR1+DdO4/FLATICL1f9J9n0X9rOvh5hc1f6R+GZNohIEdvzfkB9uqrPldDMu2eqL8fYMSbU8Xfe4z+NFOmobyM8u2b83q4wfq8Pl/iEBCL+wrtJq5Z5JfhH8e53RotFMgaLw+Rtd5tRcYS/SZtTNuIvEtkel8eNbwF9G+Yf484VVw7odemqQgdb/3TqLTG/JqmnaW/CVefV+bSzB4Fz+we/TZNw3TMAlga9AVeCirDZ3SofsS4ZPWEpiG79AY01Ab6vqRN2nNO2yuU1pm70bOQdlmr9FObfX3apa8nCKxM+jzOmYagD1Yp1yl4tzSd1Ng6yburM9IYeTOsDIrYBkz6QCvXw0LTD3juvIGXPAFlWZDXmvUqNVHEDbdhWodn4LbTvKHX51EOGhOjfoHSQxJmnF5ay6Ok41kuycf+N3p1x3TKtA88+jJrC/Vn0zt00vPNqm+FhPW15BUiz6fJHlcYmwRuZNpY2U2kD0+ZhpLBUbF8hdCgh/oDmHcSsteFgBgvGVOo/JuCFzaHXhxtzEcHXrJ76kxLTwDDTLYcdWVYEg/JkmqaNUoqHW5LfSOuA50UHVUmZUoFb2TrVGxeNEPN2BDEKJI2tbS7GaHggqX+Kn3e1Bds0+67FLQAh0I+VlFZ0LKeKevZdocmSB3aOU09Kewe84wLz6Jed7xi3ieIPq+OI37jpvbBQ0Yb/a7MN7GAkGWqjcy02xS0n0oBUA9EUg5rBYo7u1TwnGqIzToUdoVxDhA811JRZ5jriS/K3tJdNBGwiCXexKbG53biIeOxNrsWGLvJOzQvAW/HrvroJ+N5CBrCOuauhWtXAMZmk7dndpp5Ig31wSVZW6w5OrOFrpm2SX9E81Y1E/NRZV4xF9+AvHRswxFCPr0apAtlop71xNtAboUmpnXc1xhttrZ1HXVlzlr/S+N/pBvaUdv/XHF9sW19Wj54q7RRXz1kD2lXbfLT4rXhJD54Y9qN8zkiu0g7pY2IYiB5DDyq5W36JPD0azb2uXHqOl3jJZ7blNUVfDM1H1/G2mrmdh2lVrwuIzrWzMrwfunxTKkZ5KNB1jPfjFtTlTWPoYW0XgcDxrQ/8Zl/OvQ8Fj9q69p3jN7EUU/rPfCSaK8++qiv4XVnHi2zyfq1DtNvO3fRgE3UCnhNZF3x2816OIR8dnmL1fHcZY32i0Ubzbwu+17Cq2P3GRujVENn/2soPQnjzbRl+vzSmdb+hx6/LAN+KeXb+y/Ofz+cuFGSDQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle g{\\left(x,y,z \\right)}$"
      ],
      "text/plain": [
       "g(x, y, z)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{x, y, z\\right\\}$"
      ],
      "text/plain": [
       "{x, y, z}"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.free_symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sin"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAOCAYAAAA1+Nx+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiElEQVQ4EWP8//8/A7VAQ0ODEdCs1UBsDGR/AJnLAiIoAUCDBID6ZwPxOyA2AWIlIIYDalgAcmkoyESgZWVACuQLOGCCs2jEGLWAYMDSPIhYoMlsL9ApoORGLAgF6jtHjGKQBaBkZkyMYnLU0DyIhr4FjNQo7IDxCCrgQIkEVBaB6HtADEoEpwEsISASVdEGngAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle -1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(pi * 1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = sympy.Symbol(\"n\", integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(pi * n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = sympy.Lambda(x, x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( x \\mapsto x^{2} \\right)$"
      ],
      "text/plain": [
       "     2\n",
       "x ↦ x "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 25$"
      ],
      "text/plain": [
       "25"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "       2\n",
       "(x + 1) "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(1+x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "e = 1 + 2 * x**2 + 3 * x**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3 x^{3} + 2 x^{2} + 1$"
      ],
      "text/plain": [
       "   3      2    \n",
       "3⋅x  + 2⋅x  + 1"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 1, \\  2 x^{2}, \\  3 x^{3}\\right)$"
      ],
      "text/plain": [
       "⎛      2     3⎞\n",
       "⎝1, 2⋅x , 3⋅x ⎠"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x^{2}$"
      ],
      "text/plain": [
       "   2\n",
       "2⋅x "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2}$"
      ],
      "text/plain": [
       " 2\n",
       "x "
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args[1].args[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJCAYAAADkZNYtAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxklEQVQYGV2QsQ3CMBBFD8gAQEnJCCD6FIyAxARASRmlSlqzAhMg2ACK9BFsACVlxAbm/ciWopz09O++ndN3Bt57U5VlOUb27WC2Qncwh23w6mFoJI4PToK+hjOsmTNUi1x7GUMbHXRrw3ALxhR1phhFUSykEeY7POMcNW5+dVfSr+HS8yzpG0TSRVWMYHjKbElorvR64AM9gC58pKFy5kwxtEk0GNrQwA/awtOZ/o6N0jT9onrtBJYc5lVVveEIMzyP10b6A5YUWgllggWIAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle x$"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args[1].args[1].args[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAVCAYAAAByrA+0AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABGUlEQVQ4EY2S4XHCMAyFc0yQKxt4BKAb0A2gI8AG8DP52xHYgQ3SFWCDMELbDcL7FIvz+VKM7mRb0nt69lnVMAxV6k3TBI/Ts+dmVWJt2x4ULpJUiLlH6kFQYaPsXPvZqzp/c461MY2UpGv5xWXznRoY8q7wJfppbDG5UgNT2YPF/s275jEYVGa631q8m7GfL2A+udJWbo97jjfMB4SVvC+AKYMJEGr5j7xkYGoIb/K/Elp13mAKL2ANYo1RgMm1SmZXh8DdQgmtOgo3CFf5u7xkSwE6CJ08ndD/iHzw2Uej17cv8nHwWLUg74lRwBisvZ2ml2PEjMMHUx06OnlX32P3zmNXoC8zNTXi5KiN5kz22G3nOcW7XPUOBnnnsLTbW+8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left( \\right)$"
      ],
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args[1].args[1].args[0].args"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simplification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = 2 * (x**2 - x) - x * (x + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x^{2} - x \\left(x + 1\\right) - 2 x$"
      ],
      "text/plain": [
       "   2                  \n",
       "2⋅x  - x⋅(x + 1) - 2⋅x"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x \\left(x - 3\\right)$"
      ],
      "text/plain": [
       "x⋅(x - 3)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.simplify(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x \\left(x - 3\\right)$"
      ],
      "text/plain": [
       "x⋅(x - 3)"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x^{2} - x \\left(x + 1\\right) - 2 x$"
      ],
      "text/plain": [
       "   2                  \n",
       "2⋅x  - x⋅(x + 1) - 2⋅x"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = 2 * sympy.cos(x) * sympy.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 \\sin{\\left(x \\right)} \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "2⋅sin(x)⋅cos(x)"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 x \\right)}$"
      ],
      "text/plain": [
       "sin(2⋅x)"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.trigsimp(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sympy.exp(x) * sympy.exp(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{x} e^{y}$"
      ],
      "text/plain": [
       " x  y\n",
       "ℯ ⋅ℯ "
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{x + y}$"
      ],
      "text/plain": [
       " x + y\n",
       "ℯ     "
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.powsimp(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = (x + 1) * (x + 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + 3 x + 2$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 3⋅x + 2"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.expand(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)} \\cos{\\left(y \\right)} + \\sin{\\left(y \\right)} \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)⋅cos(y) + sin(y)⋅cos(x)"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x + y).expand(trig=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b = sympy.symbols(\"a, b\", positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\log{\\left(a \\right)} + \\log{\\left(b \\right)}$"
      ],
      "text/plain": [
       "log(a) + log(b)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.log(a * b).expand(log=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle i e^{b} \\sin{\\left(a \\right)} + e^{b} \\cos{\\left(a \\right)}$"
      ],
      "text/plain": [
       "   b           b       \n",
       "ⅈ⋅ℯ ⋅sin(a) + ℯ ⋅cos(a)"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(I*a + b).expand(complex=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{x} b^{x}$"
      ],
      "text/plain": [
       " x  x\n",
       "a ⋅b "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.expand((a * b)**x, power_exp=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{i a x} e^{- i b x}$"
      ],
      "text/plain": [
       " ⅈ⋅a⋅x  -ⅈ⋅b⋅x\n",
       "ℯ     ⋅ℯ      "
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(I*(a-b)*x).expand(power_exp=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{a x} e^{- b x}$"
      ],
      "text/plain": [
       " a⋅x  -b⋅x\n",
       "ℯ   ⋅ℯ    "
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp((a-b)*x).expand(power_exp=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x - 1\\right) \\left(x + 1\\right)$"
      ],
      "text/plain": [
       "(x - 1)⋅(x + 1)"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(x**2 - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x \\left(\\sin{\\left(z \\right)} + \\cos{\\left(y \\right)}\\right)$"
      ],
      "text/plain": [
       "x⋅(sin(z) + cos(y))"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(x * sympy.cos(y) + sympy.sin(z) * x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\log{\\left(\\frac{a}{b} \\right)}$"
      ],
      "text/plain": [
       "   ⎛a⎞\n",
       "log⎜─⎟\n",
       "   ⎝b⎠"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.logcombine(sympy.log(a) - sympy.log(b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x + y + x * y * z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x y z + x + y$"
      ],
      "text/plain": [
       "x⋅y⋅z + x + y"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x \\left(y z + 1\\right) + y$"
      ],
      "text/plain": [
       "x⋅(y⋅z + 1) + y"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.collect(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAAVCAYAAABmOZFVAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEFUlEQVRoBeWa7VEbMRCGL4wLcJIOTAcBVxDTAZgKEncAwy/7H2M6AEogHZhUQKADu4N43IHzPrKkOcSdT765sy/2zog9rb72XWl3dYeT5XKZHEIZDoedQ8DpMGbhPUoOgEaj0ZVgfjsAqGmIHYvbyz5xEvaZBPhc+Lri1/uGU5g4wE8qJ3pehPgk45DPxH/RttebLZBtYfwtfgLYfSCL6VFY5iqnKmz4Z8k/bLbkieSvYt9pbyGoi7RAT3MTTh7qWqNg3rHa7wv67KS5rG3YNCl8gdJ6jklP4McOg7pzNp5F2RX1ZZBdHbQizFuxjcWPHdp1b3YR4NraBY6oMqttgf9rYuzQ92GcnZfgp8XQFf+h0lG5tLIX9TGJ3tZrZSl9jrXQVPU7t6BtexQ34czJA07bcyDz1dT8yKLwaoy7EDHmQvU3FWzUE29qBEFX7HCW9uyxFL6jqOFFhUsAILjFchCI+9ukG6vLRIveBAv3VeeWvY64vEzXdCiDF5uQAzn0jCdnXos3eaOlorFDx3i2lMWjw83EmO515UtGu0T1kPTBWzhw0JlKGI6R5Xotg0QcUG6sH6gMXquTudW6CSVjwweu3mCOHdoujP+R0mmDYkxClJGJrwuXido57eTIkDgktGcZhPnz5p2rzaUMvPg2mJi1QlnQJWHtRSi09Y3xSh9vHz27SOecIWcZg71q2+SutaYB3b1nvwUdMWYhEDdG4LM2M5Gc6MCrl8+3bsw6rv5mk8TRA8P6MCkZeRNZkWerSzZpjtJ4NZaoMxDPxByumNdP8lK2CeePrJuDn87ZZpyUcB7qPCuRrE2JnLjKbng+EcBsvp0Y/RaShRsWrstpLtRZ80TjtX3Pxb0j6LlwjVCxHdTRcd6yyvLJjfyDt5gTq2cftiTjsuQBqr4twovSerBuTL6m31yF8e+oLF6N41BkRSkubXnp6N3aO6zg2bOW/gCCwq3SnAA9L1QMWZDusuTE2+JsNIoasrqga0wIxfO7q5Hv/m6MV+uSOjjsRBneDihfVQjFTd9oqZjwuXjS0h+8mZyIERKBIR9NVLhY8F01fVmiyzYJA/M+zZvCXxXeuaGYfM2GgCGkMngvpQMRJRFng/FmDiK2Kkon6lYtaU0iMY55amd+tXrwLSTrfsTejmv9R4gWrvQSovnY9J541D821I/3bPPxwxqlMaxq2+QB0zqkMpz3+CivU0Vy0oFPCZvMKeXuVYgshvTMSeZ7wO1KEvWXwxET8qMmq7hTadtsqAfRcfUNxf2yoWlcv7SYqlw5vfT8pDJ29ViuMROVg/qVirMNuMHv6uTsphIn0v3aglyNp8fk6hAPFyhynMm5YeOe17mz+AtkrTm7KYbUISFvkev9x5mm6FaXHsJKynsW96+u/wB/XGz7ZQs3lQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle x + y \\left(x z + 1\\right)$"
      ],
      "text/plain": [
       "x + y⋅(x⋅z + 1)"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.collect(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sympy.cos(x + y) + sympy.sin(x - y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}\\right) \\left(- \\sin{\\left(y \\right)} + \\cos{\\left(y \\right)}\\right)$"
      ],
      "text/plain": [
       "(sin(x) + cos(x))⋅(-sin(y) + cos(y))"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.expand(trig=True).collect([sympy.cos(x), sympy.sin(x)]).collect(sympy.cos(y) - sympy.sin(y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Together, apart, cancel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{1}{x + 2} + \\frac{1}{x + 1}$"
      ],
      "text/plain": [
       "    1       1  \n",
       "- ───── + ─────\n",
       "  x + 2   x + 1"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.apart(1/(x**2 + 3*x + 2), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{y + 1}{y \\left(x + 1\\right)}$"
      ],
      "text/plain": [
       "  y + 1  \n",
       "─────────\n",
       "y⋅(x + 1)"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.together(1 / (y * x + y) + 1 / (1+x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{x + 1}$"
      ],
      "text/plain": [
       "  1  \n",
       "─────\n",
       "x + 1"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cancel(y / (y * x + y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Substitutions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 y$"
      ],
      "text/plain": [
       "2⋅y"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + y).subs(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(y e^{y} \\right)}$"
      ],
      "text/plain": [
       "   ⎛   y⎞\n",
       "sin⎝y⋅ℯ ⎠"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x * sympy.exp(x)).subs(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(y e^{y} \\right)}$"
      ],
      "text/plain": [
       "   ⎛   y⎞\n",
       "cos⎝y⋅ℯ ⎠"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x * z).subs({z: sympy.exp(y), x: y, sympy.sin: sympy.cos})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x * y + z**2 *x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [],
   "source": [
    "values = {x: 1.25, y: 0.4, z: 3.2}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 13.3$"
      ],
      "text/plain": [
       "13.3000000000000"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(values)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical evaluation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 4.14159265358979$"
      ],
      "text/plain": [
       "4.14159265358979"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.N(1 + pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3.1415926535897932384626433832795028841971693993751$"
      ],
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.N(pi, 50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x + 0.3183099$"
      ],
      "text/plain": [
       "x + 0.3183099"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + 1/pi).evalf(7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sympy.sin(pi * x * sympy.exp(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 0, \\  0.774, \\  0.642, \\  0.722, \\  0.944, \\  0.205, \\  0.974, \\  0.977, \\  -0.87, \\  -0.695\\right]$"
      ],
      "text/plain": [
       "[0, 0.774, 0.642, 0.722, 0.944, 0.205, 0.974, 0.977, -0.87, -0.695]"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[expr.subs(x, xx).evalf(3) for xx in range(0, 10)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr_func = sympy.lambdify(x, expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.773942685266709$"
      ],
      "text/plain": [
       "0.773942685266709"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_func(1.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr_func = sympy.lambdify(x, expr, 'numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [],
   "source": [
    "xvalues = np.arange(0, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  0.77394269,  0.64198244,  0.72163867,  0.94361635,\n",
       "        0.20523391,  0.97398794,  0.97734066, -0.87034418, -0.69512687])"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_func(xvalues)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sympy.Function('f')(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{d}{d x} f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "d       \n",
       "──(f(x))\n",
       "dx      "
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(f, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "  2      \n",
       " d       \n",
       "───(f(x))\n",
       "  2      \n",
       "dx       "
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(f, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{d^{3}}{d x^{3}} f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "  3      \n",
       " d       \n",
       "───(f(x))\n",
       "  3      \n",
       "dx       "
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(f, x, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = sympy.Function('g')(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\partial^{2}}{\\partial y\\partial x} g{\\left(x,y \\right)}$"
      ],
      "text/plain": [
       "   2          \n",
       "  ∂           \n",
       "─────(g(x, y))\n",
       "∂y ∂x         "
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.diff(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\partial^{5}}{\\partial y^{2}\\partial x^{3}} g{\\left(x,y \\right)}$"
      ],
      "text/plain": [
       "    5           \n",
       "   ∂            \n",
       "───────(g(x, y))\n",
       "  2   3         \n",
       "∂y  ∂x          "
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.diff(x, 3, y, 2)         # equivalent to s.diff(g, x, x, x, y, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x**4 + x**3 + x**2 + x + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 4 x^{3} + 3 x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       "   3      2          \n",
       "4⋅x  + 3⋅x  + 2⋅x + 1"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAAAaCAYAAACgjS7xAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGC0lEQVRoBe2a63UTORSAh5wUECghdBBCBRs6ALYCoIPl5F/+7YEOgAqySwdABTw6WDrY4A6y3ydLs/JYHtuT8Xjs5J6jSHP1ui/deyXn3vX1dbXvcHFxcQSP55HP41i/AD/Zd97HxN/BNolB2U8pZwPQ8IZ9XsfyjP2uKN8H2PdWboGc35QY35qxQdBTCHpC/blEWM+4l+yTG7XCOAZ30vM+d8tNJXCJbD81hbEVY4OQENaoXzUJ2tC3+3zb0Np3yzYkgF5/gPqb+o+8ayvGJiGUdzkhm2zD9HtKnp9pfD+jUDa59a1dW5nD/CtqHUuAQ/+COKZ6HTBVdUptTmOOo4X2CnGvU+onbQvT34z7uuYb08Mahk5D+KO2/TfZBw3KO3l1leG3eeUQKUUvrEU56jQe0c4Pcr6+DuUDxTy5OoyMy2hAiKStor9TbyKn0qi1+iKwp4I33nsqgvCpZcrykNIZ4try1iagzuuvMhEaNC4PcjK2ivZLcJ+on1E+rrLONsZAm7RrPDojnZK6agP1rG0dUSYHflBe5DPo0CC0VhXcNwTBtizqnp+hIT/lMvmzZc7SLtZL3sMDNKGcRNzSuc0BzDujyEcXcJ4XFr1rgr9iIz3PJPxG6q70M0+5eSA8KJfLiHM8Y4xGzx2rsXlL+0WHCs1BZWuRKqkXYK10+ysm6/SrAMekkB72Ba+BtIbdNgKZKw+69D9pa2TuocA8oV1AWTXlteo6Cl8lWAJAT91OuA3X0m4ZAtT1NIzS0KhUwCKG+yRKwzYxX7SXJ9vTs6if7k7gm5p8zLytsU8dyjqt2mESeyrv+/lUcMnLeSD2Df6BoeDZzNnqXK3BZfBC9HsS+4LHLNRmSO75gz2tf6f8SzFP8xqdh9WKb40nhTLXNRXQgzlP+MqYkP9Qzyh32j2Ov9DmATSVMUedyWX5XpnHcXBTpEL7MUIeGUbngA6VreJmwtncwPURCq8YuqJg04reVk2i31L0PhpbOv1pjImn/W9BfKWYuJpLSbP7qMDRAnQaTXyHkj8VUkotdprHKPyk7+OisTHIJN23KRXZJzxgscmSBVXCzAlnvAn0B/AaUUWtR2sak8YYPBm1+zT7QY0H4EEP7mExslxSvP3XB4r2zvMYpZ30/SC8s+UqgEnzBm+DS/MZxqj8XxQFt8q7leOTpdP8H5hvriaidOs011L4pxTD6TfG5uO8PEhDwFEvSg0YtjqwjrIwzDVBY67oL8lIOtban/EfKSpFD34/tm/MI+sMQn9TOI3vWt8zxgZxIQeiLgmxsUYQtgaigkshYG48CAUaFFXqBOdajlkEhvaKPQ07OWgQfYd89ynKAbweyN9W1/b8zDmRcOomD8pQPiwaX7N/bR5Zo3f6oW1dSPq+OkwzIUwBPswJpJ2Um3uRNCXUjLnRQ+vMYlNjC3s28Olzjg72VwlCCqEVOD2odZvhOmRQiHSFGzHt5MGW0sDYneGxwEzQBfjwqFvBjKftMXXTO2iAtRssLLQuSmNJm5fmmmeVjM0QrRc1vHuz8bU9KSCcXr5zQzzne1SGJrORJulKYVN0AlMEYad5nLIw87f2bAcIQOV6IVCJ77Iizut4n0rz1piESnMW2Mt8TGHXyT1tjdN3mvQrh0ZmuYp9HoaaRnD2uc9YwQPtz3E1QLOHWj6TvHedx5o3GtqXjmJySEOjEhHyNeocmnlD3tel7XoataU2kHwh8P5a4JU/PXB6Mn7jO9GiQXpbVSEVeBWkp3O8IUojrEOqY8YE0OYt/yzSm0hT/vnv0KPlEbq1Fw9GchreotWN75qlHNaoJD/VvaH/LRyCvL36+1ogQCJ2DaBdT9TpgjAGXoekn710AEbM9wdbYN43s86/c26B3tKWeuWiZy4NHiFuEPoxMD2g9wF1XhlGhwbD3RdK8zIyNB2d90OIO+uVZXpA+s2168vQ4J4NRo3vXgIMRXew3xLQodROZXBji7L1Znm+33K+3dzhTLxwmqvVT1KDXxCSCqJn8wZWfOVO4+7q3ZMAOvV2raHN5Obb8mwVhPg8kT/O7p5U7yheJAEdyNzvw/8B5miPW5sOlGwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 2 \\cdot \\left(6 x^{2} + 3 x + 1\\right)$"
      ],
      "text/plain": [
       "  ⎛   2          ⎞\n",
       "2⋅⎝6⋅x  + 3⋅x + 1⎠"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = (x + 1)**3 * y ** 2 * (z - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAaCAYAAACn4zKhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEr0lEQVRoBe2Z7VEbMRCGD4YCICVABwYqADogpAKgAxh+wb8MdABUQEgHkAr46MB0EOIOyPsISbPIsk/Gd7aH8c4sklbSrrRf2jMLb29v1Rza18DZ2dmypJx4Sau+3Re9t9S++LkEr4FzKfwwaEP9S/WfhGuLgThvW9fAgRS/baScq78qWmduBKOVlrtEwWNOxsL8TcippX2aIoBI2FU73XSkA4QHqv1btyhh1HtofQcDCNc51tTSkQ5yJPkc5isAuZ371II3GFGwrn6PDX3pSBMssHAj2rMljNsXP7xgU+3xuLwmtV9nxWFuhVF5VrbmMcKL2t+WbvuaI/Kpkr5DVwvPXjSCX3An4qH6934RQjsarzFuAsSLevmPWheKTfBsi4c/67X4vwo3hChtRXTnwep/ANEpObdy86JhAMpS63g81sf2OwGF32uxM4D6AAp7cb3m/hBpHGYioPtQFpIurkYVqD0oO3htSfrkXtwvfg8YmRgIfdJGkIxDZwR1SA9YeSvOqiP6jh031N9DcEO8SthwcbB10L2uhP+Ex0IMGEHjlThIOuFhPhG9l25M1o49FH+8sunIGvtcDTPgfnuj8AzpiCh4lpJofwj/CnkHbkWL6Ul9POrAz3U1vlDfgZ+7VuvCN9CTlrnIL5mrPA/4A5vCfSG5lDMBD1oz8OF7XzL1v9yPDFKc/hb9xcPJNzQmlC6EpAyMQKoKcMKcBjzgRI8FrG/X2rnQ53HrhkGmpXJANjIehDyK2xrzmOEAaeUm0swB98NxiiFEAhuoglLr/RId78a634QoBsDSaVqBNtDL2SRAkVQafSAZRECqZIwaqgnkp/N9fGaAwP24ZzEs6fK8BWxIlQqNlxzl4MGPWhdSAV7/U2iBfJ/S7Dx9FNlLiX4Mf3sGjEqKdDS1w9JcpXkqE86QAjKZzxUD8B/KN2VWMOa8n4oENg5SDnIp8ZyXq+WiWDpGjWgdT6uLBC3Lg3ikH4TICVGQ32So2p9TciU60cT54/tltrXRHeZoWXmLnooRhoWQ80a/Fs/Bg6zRUBgRlSrSb4lNnRy3UHyCR4fIq0RbBiOn2e1wxmzKHXTkJT9BruWxTYGvWpRrPZxQs0ZhT8l7wDoO1xeqXrl8LPIwI8t5tfpWDkVBcWSIx7SASLDnrj2HiwR/cb6W48OnPhYl9+8nXD4I0Dq8FswZMdlaESmUnikEHq9eLsaKkeZlhKIg3TtrYxy3RBfx3CESKl10R4gn8sABWJTfQVCcBbyRigmDhe8J5m20MM4Bhwv87Tx7eWMwRiXe/H51J2QtxQHGiamJNZMCySVCcUiKE+BJNHTCN0vuneEO0ZnZUAv8U2ccPD09PRc+lfLQ2q6wU7p+3HWStSs8GpdPyX7JWRV2S9baNeFhrjUWC2T5SyGe6UB9PIQStq40fd/w/hcvyVYydlGDfdJaTG0N8s2xIkuMFgXaMJIRtJ5QuxEGuFaHH62KU4XWknYoGfse6MC0yVZyeOtiOd0kb8vL34d7jSwrvgmW4ZA+lkbQkVp+WyIySt6ClCVlLrmWquqrAO/Xpz784j91Jq0J7znbn/GcSZ+1Tp7uQEom4j5UjnX7wvx/FXKkqh55L4UAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 6 y \\left(x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "           2\n",
       "6⋅y⋅(x + 1) "
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x, y, z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sympy.sin(x * y) * sympy.cos(x / 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle y \\cos{\\left(\\frac{x}{2} \\right)} \\cos{\\left(x y \\right)} - \\frac{\\sin{\\left(\\frac{x}{2} \\right)} \\sin{\\left(x y \\right)}}{2}$"
      ],
      "text/plain": [
       "                       ⎛x⎞         \n",
       "                    sin⎜─⎟⋅sin(x⋅y)\n",
       "     ⎛x⎞               ⎝2⎠         \n",
       "y⋅cos⎜─⎟⋅cos(x⋅y) - ───────────────\n",
       "     ⎝2⎠                   2       "
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sympy.functions.special.polynomials.hermite(x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2^{x} \\sqrt{\\pi} \\operatorname{polygamma}{\\left(0,\\frac{1}{2} - \\frac{x}{2} \\right)}}{2 \\Gamma\\left(\\frac{1}{2} - \\frac{x}{2}\\right)} + \\frac{2^{x} \\sqrt{\\pi} \\log{\\left(2 \\right)}}{\\Gamma\\left(\\frac{1}{2} - \\frac{x}{2}\\right)}$"
      ],
      "text/plain": [
       " x             ⎛   1   x⎞               \n",
       "2 ⋅√π⋅polygamma⎜0, ─ - ─⎟    x          \n",
       "               ⎝   2   2⎠   2 ⋅√π⋅log(2)\n",
       "───────────────────────── + ────────────\n",
       "           ⎛1   x⎞             ⎛1   x⎞  \n",
       "        2⋅Γ⎜─ - ─⎟            Γ⎜─ - ─⎟  \n",
       "           ⎝2   2⎠             ⎝2   2⎠  "
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [],
   "source": [
    "d = sympy.Derivative(sympy.exp(sympy.cos(x)), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{d}{d x} e^{\\cos{\\left(x \\right)}}$"
      ],
      "text/plain": [
       "d ⎛ cos(x)⎞\n",
       "──⎝ℯ      ⎠\n",
       "dx         "
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - e^{\\cos{\\left(x \\right)}} \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "  cos(x)       \n",
       "-ℯ      ⋅sin(x)"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integrals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b = sympy.symbols(\"a, b\")\n",
    "x, y = sympy.symbols('x, y')\n",
    "f = sympy.Function('f')(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFkAAAAuCAYAAABDLLXWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFoUlEQVR4Ae2aTXLUMBCFJ2EOkMAJgB1LIPtUATfg5wTADaCyy45KbgCcIMCOZbLInpAbkC2rhLlBeJ9H7fJoLFuSBZNK3FUaybb6qfXUakn2rF1eXk5yZXd396t0z5TOVd7Pxbnueuu5HXQE/5D+odKert/kYl13vSySRehDEfNc6ZPShdJMCY8epYWBacu9mFs7qnQqsiH3VGkzRumm1snyZJGFFx/cVNJS+51Msrz3qWsEDx4lgoFkkoX5wuGeROCPVcRADsl48szF45HECAaSSBaxG8K8pzR6cQS5ViWJZClZPGZvPEokA6kkP3O446IXSTDVUkk2Tx7Dxb8guRGPV7LoqX3WgiwZopvVoKeU4skr82KR9E52c5TPlXsOI1d/kN40QXvL1S0Wj9Xxj8LkaI580LWV53f0q3ucLu8oz37LJ90jpYdgKX2rwRMK6Kv6ZyVm1Bddv41Vz/HkIjsLGbknI28rcTznDZ7NFBXnojpsGXeUv7d7ubkwGCSwwEwW6fGu5pEU0U/iIIVkm66lFj2IhWCIBvNIyRcGAm8vJWCBmSUi2RyhzdYg5lrMS3uBQ/BPJRa9wW/cnLF4w6bKSyHCrNWzPyXaMzxyMJXdVR5st1m/WZYOA0TIud+831eO9eTHDqiUF7PfPuvqqJ7hNf/iHTWYL11/UjNsSvJiGphGtlLkECLiCBFgYeyFrvl8Rd62iPAiKtgh6RAb7WsMi/JrJRalV0rID9VpW+TAxAY+OARFumCxFuDx50roMaM/KNUSY0esJ1s85nNTtsigT0qQB0EfKSu1EUwbzJ5fFALCJ699kp5jFyv/U11DDPih2AsmBAZFGDgB4ZE23rs2DM8f+F47Ykk2owZv32SwYfnG+p2GqAv/JtfCwIOt01aFrZ55Loup/9zqgQl2qwibZ8wwyG2GK8r2NajSjbWjN1wIyLx44jVaNZTxU+EJq2/AIGoWwD/xbGH6Q0BFinJmS0ioYwPdVofBgegv3kO82wbRHkXZEePJtuj1kWIN9+XEz6aH9NVfet4yQBBwsFSx/UbX4KHBosjhpR5glSGdgVnYH8fa0evJAmYDjpTaWeDJMQPGQNC5TlFHIRipvcyRMlFeEzWvUv2CSchYEqfHc9++ajei58EQp2dBO2JINk9mISgh4C2s0AFQiFia1o4IYiYLDp2uFk6Vm7Oj65SIJzfr6nJJ/AWe8FMRr3aI/ZSxL8qOmHCB5yGDPVkGQhqeEvQIGnJCRwgtvuAxJLZ+5pUzq6R7PPNJssfkzMyFaW8PpQsO7daDq3ssslxb/7d0j0GKtmOqykERWLMxGh8q1YAJNwYLItqO1AwQe1w6ORHWW6VDJeoy2yC/Dh3U8QS90M6Dqk+UPguDN393lIj1LKzMHO5Z7I+2o/NYLVCmBlOCldtisy7zRBgQcVt51+pfg6see1r20jGDUuuFCsLBaRiQpGNxCC/2/npPRZuuMdO7FUod4qzPQCHEY/OE6kbPDx4XOqz0qLY+5qAC5n+VPpItHrfGsEhL6dSGiCZ+TpR3TeUFSNUlLPDCvQ5bCxUSLhwGWJ3H6QTI6KqdMVkoeB5iQX9+lfZbkSwV8qgw4cGjw0wgLg4RQlVO+0ParHSDMVkjjufxWpC3Zf81hvm9cl7Ie4ksL5QeOwQOGH1bN7/pItddJLMKEya+ybiVeECRHl4BkK6YXCIeX4Eurt6ELpK3nHnZO4vVd+9qWNBFMuGCz00riWNXg54yVrSSLGJZ9EijFxfguZVk4drWbcj+uIB51wMiRLIteqMnFxjnEMkLXxoKtHOjIUIks+gd3GhmCnZ+iWQtehCMZJ2u5qrjb5OBdZHKe9LmVw/eenHKmzUrjuV8BqZSZZG7AELEUsaT73I9ShkGbm1vb38X1OPj4+MHyvn3DS/Jf5eBH1Fg4C/HFeoLBJjCRwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\int f{\\left(x \\right)}\\, dx$"
      ],
      "text/plain": [
       "⌠        \n",
       "⎮ f(x) dx\n",
       "⌡        "
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\int\\limits_{a}^{b} f{\\left(x \\right)}\\, dx$"
      ],
      "text/plain": [
       "b        \n",
       "⌠        \n",
       "⎮ f(x) dx\n",
       "⌡        \n",
       "a        "
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(f, (x, a, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(a \\right)} - \\cos{\\left(b \\right)}$"
      ],
      "text/plain": [
       "cos(a) - cos(b)"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x), (x, a, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{\\pi}}{2}$"
      ],
      "text/plain": [
       "√π\n",
       "──\n",
       "2 "
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.exp(-x**2), (x, 0, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = sympy.symbols(\"a, b, c\", positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{\\pi} a c$"
      ],
      "text/plain": [
       "√π⋅a⋅c"
      ]
     },
     "execution_count": 141,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(a * sympy.exp(-((x-b)/c)**2), (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\int \\sin{\\left(x \\cos{\\left(x \\right)} \\right)}\\, dx$"
      ],
      "text/plain": [
       "⌠                 \n",
       "⎮ sin(x⋅cos(x)) dx\n",
       "⌡                 "
      ]
     },
     "execution_count": 142,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x * sympy.cos(x)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sympy.sin(x*sympy.exp(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - e^{- y} \\cos{\\left(x e^{y} \\right)}$"
      ],
      "text/plain": [
       "  -y    ⎛   y⎞\n",
       "-ℯ  ⋅cos⎝x⋅ℯ ⎠"
      ]
     },
     "execution_count": 144,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = (x + y)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{3}}{3} + x^{2} y + x y^{2}$"
      ],
      "text/plain": [
       " 3              \n",
       "x     2        2\n",
       "── + x ⋅y + x⋅y \n",
       "3               "
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKgAAAAuCAYAAAC8olJxAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGzElEQVR4Ae2c73UUNxTF1zkuwCQdmA7ArgDoAJIKbDoIx5/sbz7QAVCBEzogqQBDB5AKYtyBc3/DaI52st6VNE872tG8c2RptJLm3afrp38zs3d3d7eIlYuLi0PVea5wq3CgcKxwqfwviictwgjesxYkdkBOlI8tosW6vWgFRqog3EEc2k/U77XqfdNN3lBf8e+K/lZ4wPXE5bXwvnQYlX6r9GeFhy4vMrZuL/L2oxUP4tBPiepdqt6VV/cXpb9511NOnoqUTz2AGPpQeY+8vJikdXsx9x6zbBCHkjyoOqM/lDPc01E1CN7z2hCodXuGquVrKpRDeylzUKe2bsLQ/pvCldLNcO9+qyUWbv4xnytOHeKXTGXd3lLjBV4I71oOpQ7xDVQ1DimfKDxrb1SgCfKpJMwM64wejy3uYt2ehU6529jEoUEe1CnfGpaFwmOl+8O/KzapWDhZhbJAeqF00greN4h1e37bu5AWfv7Z/8eh6DmoGmKb5R+FJ0o7MroOOlK+y1NymtKSidX3MxAqxri3ipMWiqoH2c3aQ6eSRXiDORQ9xKtxyHij4HcGq1ry/1CYtLRkwnOy7/uIoDQLHWwSLdbtRSswQgVhDuZQ50FVCVaftvqy8X6iwH82iyDkk8p8+JFcvFB8put/22vKM7xz4068NllAfNV1t5Bqf3uvmLZGFU9P9NiEnWEIWxF3ojYgaSNeeyG4N7bn2h0r9vCgwlr7eGU3YaffN3KoI6gKdxvGugkrq/cKkPKVAh6D1WpDUF0zjIcM5ShAfRYStNcRVOlfFcgvQWKwhxxGBOOWbULaG9tGwfaRokHYQznUDPEqjOfs72NCHucxf17xu7LuF7WJ9/3UlmCu5k8JyCbvLxJjijX2XcEdavMY++TA7jzotRr3CQR5vrg8xSnD8I3qOYLjLS97RmHe2s/rFdnKpTX2XcEdatwY+5hjdx60P1xDnqtQBKvKiZzNfFQxbTFne+fKKY+FBXkleFBT7LuC2/XFplh4gu2TA3tDUF9J3QRCIc77LZR3QPiRHf0X74s3bgjb1uYebMv0wUc3bllB+lhi3xncoTaMsI8Z9n3dFOL9qcBEGI/WrEaV9of8ZuIbCqRXjrmo3xY/lzL/zIm9WNy9/rn3cgA3zLDjQfEaBOYPdNiNQuftlMdvbrGjZLQskbNtjzY/RrdkXyEn9pJxh1oy1T5m2PelKV6T+SHKLESglwofFdxzjv7ElyKx8koV2O9kl4B9U/bHkNHnn60OubCXjLvpgIA/qdwww25yFh8AtCvSEvWpYpMHLLqGC0/UiptuGYKdIT6bSLG3Ct2Ji9JMIdhzLWF7acadwQLWfZ6VoMLPtMHfruI06Z1AdDsEQ2ykdvDE7nh2SFPWdbPiDlF2RNuYYt8PATugDHMRXofg6JS5Jx7Vcu6JRyaUJrlxh+Adyzam2LMSVGQ08ZQhvVFSmVpx0wfW2HMP8SXxZtZlBy0wE3QHO60mlWeC1tTbO4h17/z8PP7TIsZANW/ZW9ekfufQgNVhX3gMEOH0qy+c/699Cku/j469r/Sq63X2yWUb9CjBPlvfqF/VAal5MiDPrLJL4D8IndrcpOpNxTbzED8pWk4PzEzQ6fXppBDNBJ1Ud04PzEzQ6fXppBAlnSRpAn4oKzQPNivmSI1r98CzkvVIawuO95AjBXYUeJO1qLcFUG4bYs2NaIJKAQhJBziCLpTmgQ2eIeUzMNs83rzVfQmjiLC6f8xuO0t5PPf6WTHfq7J87iAW49ZtI7zm3EgZ4iHjqZTx32l3XxQ5i7XikPIQQKF7GW9IW4l1IeOJX1f64E0hB6/RjCYj2cacGykEZeha+u+UMbiuUTg8+C78eA5f8Jy8aIiHrUnMuZEyxGP8pa9hqCOcN+XEpybBFnyf6b5/0D5xJ20b2cGcG4NPkqQUXgRiskgac7gtpvNlh69ShhOutUe4xSicSRELbkR7UIdFN+fjC5DzWAHXfq1QvbR2YWh3K/vqbGLJjcEeFOtLIYZ4FgXbXsVz+6JEtsB7snjrdjmKUnDLygzlhglBwSxFvitizvVA6fvmZBSdrAh3MwdXPJPT6+Uh3Ihexetm7qOtngpN0g3xDPvViezSvLxXMzlzcCOKoFIAD8lrxGxEV7VCXfcfJ1swxXmouPOcSrNIqmabSVizcCOWoAzdhA9SiNiXo/aCrYZqRHZgsXisuL8ogrQ3tRii5YM5N1JW8f2OWEg5OuNAgc/m9Ik72T4SVjwki0MWRf4eME/6MxWq7UFqc24kLZJkeOaZ3fmz0nRUdQ+LyA5Md/Cgq4RXTqr6vA9GsObGf3WFn6tu5dsbAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{3} y}{3} + \\frac{x^{2} y^{2}}{2} + \\frac{x y^{3}}{3}$"
      ],
      "text/plain": [
       " 3      2  2      3\n",
       "x ⋅y   x ⋅y    x⋅y \n",
       "──── + ───── + ────\n",
       " 3       2      3  "
      ]
     },
     "execution_count": 147,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr, x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{7}{6}$"
      ],
      "text/plain": [
       "7/6"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr, (x, 0, 1), (y, 0, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Series"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sympy.Function(\"f\")(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(0 \\right)} + x \\left. \\frac{d}{d \\xi} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }} + \\frac{x^{2} \\left. \\frac{d^{2}}{d \\xi^{2}} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }}}{2} + \\frac{x^{3} \\left. \\frac{d^{3}}{d \\xi^{3}} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }}}{6} + \\frac{x^{4} \\left. \\frac{d^{4}}{d \\xi^{4}} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }}}{24} + \\frac{x^{5} \\left. \\frac{d^{5}}{d \\xi^{5}} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "                             ⎛  2      ⎞│         ⎛  3      ⎞│         ⎛  4      ⎞│         ⎛  5      ⎞│           \n",
       "                           2 ⎜ d       ⎟│       3 ⎜ d       ⎟│       4 ⎜ d       ⎟│       5 ⎜ d       ⎟│           \n",
       "                          x ⋅⎜───(f(ξ))⎟│      x ⋅⎜───(f(ξ))⎟│      x ⋅⎜───(f(ξ))⎟│      x ⋅⎜───(f(ξ))⎟│           \n",
       "                             ⎜  2      ⎟│         ⎜  3      ⎟│         ⎜  4      ⎟│         ⎜  5      ⎟│           \n",
       "         ⎛d       ⎞│         ⎝dξ       ⎠│ξ=0      ⎝dξ       ⎠│ξ=0      ⎝dξ       ⎠│ξ=0      ⎝dξ       ⎠│ξ=0    ⎛ 6⎞\n",
       "f(0) + x⋅⎜──(f(ξ))⎟│    + ────────────────── + ────────────────── + ────────────────── + ────────────────── + O⎝x ⎠\n",
       "         ⎝dξ      ⎠│ξ=0           2                    6                    24                  120                "
      ]
     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.series(f, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "x0 = sympy.Symbol(\"{x_0}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle f{\\left({x_0} \\right)} + \\left(x - {x_0}\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}={x_0} }} + O\\left(\\left(x - {x_0}\\right)^{2}; x\\rightarrow {x_0}\\right)$"
      ],
      "text/plain": [
       "                       ⎛ d        ⎞│            ⎛           2           ⎞\n",
       "f({x_0}) + (x - {x_0})⋅⎜───(f(ξ₁))⎟│         + O⎝(x - {x_0}) ; x → {x_0}⎠\n",
       "                       ⎝dξ₁       ⎠│ξ₁={x_0}                             "
      ]
     },
     "execution_count": 153,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.series(x, x0, n=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x - {x_0}\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}={x_0} }} + f{\\left({x_0} \\right)}$"
      ],
      "text/plain": [
       "            ⎛ d        ⎞│                   \n",
       "(x - {x_0})⋅⎜───(f(ξ₁))⎟│         + f({x_0})\n",
       "            ⎝dξ₁       ⎠│ξ₁={x_0}           "
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.series(x, x0, n=2).removeO()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "     2    4        \n",
       "    x    x     ⎛ 6⎞\n",
       "1 - ── + ── + O⎝x ⎠\n",
       "    2    24        "
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "     3     5        \n",
       "    x     x     ⎛ 6⎞\n",
       "x - ── + ─── + O⎝x ⎠\n",
       "    6    120        "
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "         2    3    4     5        \n",
       "        x    x    x     x     ⎛ 6⎞\n",
       "1 + x + ── + ── + ── + ─── + O⎝x ⎠\n",
       "        2    6    24   120        "
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 - x + x^{2} - x^{3} + x^{4} - x^{5} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "         2    3    4    5    ⎛ 6⎞\n",
       "1 - x + x  - x  + x  - x  + O⎝x ⎠"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1/(1+x)).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sympy.cos(x) / (1 + sympy.sin(x * y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 - x y + x^{2} \\left(y^{2} - \\frac{1}{2}\\right) + x^{3} \\left(- \\frac{5 y^{3}}{6} + \\frac{y}{2}\\right) + O\\left(x^{4}\\right)$"
      ],
      "text/plain": [
       "                           ⎛     3    ⎞        \n",
       "           2 ⎛ 2   1⎞    3 ⎜  5⋅y    y⎟    ⎛ 4⎞\n",
       "1 - x⋅y + x ⋅⎜y  - ─⎟ + x ⋅⎜- ──── + ─⎟ + O⎝x ⎠\n",
       "             ⎝     2⎠      ⎝   6     2⎠        "
      ]
     },
     "execution_count": 160,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.series(x, n=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x \\right)} - x y \\cos{\\left(x \\right)} + x^{2} y^{2} \\cos{\\left(x \\right)} - \\frac{5 x^{3} y^{3} \\cos{\\left(x \\right)}}{6} + O\\left(y^{4}\\right)$"
      ],
      "text/plain": [
       "                                        3  3               \n",
       "                       2  2          5⋅x ⋅y ⋅cos(x)    ⎛ 4⎞\n",
       "cos(x) - x⋅y⋅cos(x) + x ⋅y ⋅cos(x) - ────────────── + O⎝y ⎠\n",
       "                                           6               "
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.series(y, n=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{61 x^{5} y^{5}}{120} + \\frac{5 x^{5} y^{3}}{12} - \\frac{x^{5} y}{24} + \\frac{2 x^{4} y^{4}}{3} - \\frac{x^{4} y^{2}}{2} + \\frac{x^{4}}{24} - \\frac{5 x^{3} y^{3}}{6} + \\frac{x^{3} y}{2} + x^{2} y^{2} - \\frac{x^{2}}{2} - x y + 1$"
      ],
      "text/plain": [
       "      5  5      5  3    5        4  4    4  2    4      3  3    3              2          \n",
       "  61⋅x ⋅y    5⋅x ⋅y    x ⋅y   2⋅x ⋅y    x ⋅y    x    5⋅x ⋅y    x ⋅y    2  2   x           \n",
       "- ──────── + ─────── - ──── + ─────── - ───── + ── - ─────── + ──── + x ⋅y  - ── - x⋅y + 1\n",
       "    120         12      24       3        2     24      6       2             2           "
      ]
     },
     "execution_count": 162,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.series(y).removeO().series(x).removeO().expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUAChmA4MIIiCxjEoAgCxQkoLkD0PSA+B8SnAeDgGGsGGU5UAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.limit(sympy.sin(x) / x, x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sympy.Function('f')\n",
    "x, h = sympy.symbols(\"x, h\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [],
   "source": [
    "diff_limit = (f(x + h) - f(x))/h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.limit(diff_limit.subs(f, sympy.cos), h, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 167,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.limit(diff_limit.subs(f, sympy.sin), h, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = (x**2 - 3*x) / (2*x - 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = sympy.limit(expr/x, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {},
   "outputs": [],
   "source": [
    "q = sympy.limit(expr - p*x, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{1}{2}, \\  -1\\right)$"
      ],
      "text/plain": [
       "(1/2, -1)"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p, q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sums and products"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = sympy.symbols(\"n\", integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Sum(1/(n**2), (n, 1, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{n=1}^{\\infty} \\frac{1}{n^{2}}$"
      ],
      "text/plain": [
       "  ∞     \n",
       " ____   \n",
       " ╲      \n",
       "  ╲   1 \n",
       "   ╲  ──\n",
       "   ╱   2\n",
       "  ╱   n \n",
       " ╱      \n",
       " ‾‾‾‾   \n",
       "n = 1   "
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAuCAYAAAAsnen7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACY0lEQVRIDe2XPU4DMRCFCXAA4AjLDfjpIwEdJYITADcAUSUdCiVdoKSCcAMocgCgpIOWCkFHGb638hgn2LubDQ1SRprMeDzzdjyeXTuNwWAwU4fa7fYCcScuNnNyH/un9HlnqCM6gBxaIHoX/RFelm1WPzXpALDNILaDnmFbkW0SYGX7IJAoqcZ/wa1WqwO/GNYkGftE3fJ3MKyacWJgQDPAVN9V9LwjcnBLvY5k6Rncs1j0Fdk0buinDrlM1WLHQbw29FiZ+z5mcIRRDa/GT9EFfta76ln5Snqy+TxjBqqR6A6W8zp8Da+58T1yBr8nySo0h5Ma+oug82az+drv97cZXzF+Rt9Avxcgc2/olWneZRFmsoXtzCEo84vKaIHjULsBqOx/WobVYAvHQWixOgSMqzZGtTVSj9YiD0xm2rQDON8oQ8MefmjMXCo9MJ4CfRpZusqwVYoScQiBBXA64jOU/chc4bD2m1eIymSYcZnvWPNTYF+uaSl8KRocJfWOEA8RV6YviK/L/2s3f/z7NSQUvtN2kpvHNbbwrDR7LkuBCc7w1LXgEN2uAT3G4vwujPxFVWosAF0Bwo++jrHXX2iBobCPAdMNUsCL6GOd1mUZ68r1OS6oEi+rse4ZOmAl9+B3WHXtYQtLg2mYkqUgUHX8gLXzXcb+RoQuu/4h3SKjVFYKBek25EEdyg3yErseHqUkMEG2WbHdtyusbqNRSgI7b4HaA2IAWcwoWxXg5HKJj60mf1YZsF7jWFb6d6Q2THZGIbAL1FvnvxPoWsEuvA8nKdluYYQDtpIsMXeKLfkBUuw3KI03rBEJ6zsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\pi^{2}}{6}$"
      ],
      "text/plain": [
       " 2\n",
       "π \n",
       "──\n",
       "6 "
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Product(n, (n, 1, 7))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\prod_{n=1}^{7} n$"
      ],
      "text/plain": [
       "  7    \n",
       "─┬─┬─  \n",
       " │ │  n\n",
       " │ │   \n",
       "n = 1  "
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 5040$"
      ],
      "text/plain": [
       "5040"
      ]
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{x} - 1$"
      ],
      "text/plain": [
       " x    \n",
       "ℯ  - 1"
      ]
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Sum((x)**n/(sympy.factorial(n)), (n, 1, oo)).doit().simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.symbols(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ -3, \\  1\\right]$"
      ],
      "text/plain": [
       "[-3, 1]"
      ]
     },
     "execution_count": 182,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(x**2 + 2*x - 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = sympy.symbols(\"a, b, c\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{- b - \\sqrt{- 4 a c + b^{2}}}{2 a}, \\  \\frac{- b + \\sqrt{- 4 a c + b^{2}}}{2 a}\\right]$"
      ],
      "text/plain": [
       "⎡        _____________          _____________⎤\n",
       "⎢       ╱           2          ╱           2 ⎥\n",
       "⎢-b - ╲╱  -4⋅a⋅c + b    -b + ╲╱  -4⋅a⋅c + b  ⎥\n",
       "⎢─────────────────────, ─────────────────────⎥\n",
       "⎣         2⋅a                    2⋅a         ⎦"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(a * x**2 + b * x + c, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{\\pi}{4}\\right]$"
      ],
      "text/plain": [
       "⎡π⎤\n",
       "⎢─⎥\n",
       "⎣4⎦"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.sin(x) - sympy.cos(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAAzCAYAAABhaa0bAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGk0lEQVR4Ae2c7VHVQBSGwaEA1A6gA9QKlA5kqEDsQIdf8M/RDtQKVDoAK1DpQDsQ6UDfJ+zZiUuS/chN7nWvZ2bvJpv9OPu+e85+JLB5cnKyvdEhp6en1x3J/5MWhIDw7cR9S/V/U9gJ2nmn++dBWjW3AuOTOvNW8cWYTjlQqetA17kDmHJPgvYvNmUh35X4fKxyQcUre6t+AsS5YgbdaFE9e6rkvcJjXeeS4ttX2SPdHNzxKWtwoU5Dxg/FUTKUZ0/hu0KnazG49PxS168U8DSjBZe1FiLgGIGAfNDXYQc+o/1K4aFC6Mo7i6rcmcK+wieF3vo7CweJa2EhAglg3yoMgqV81wCqwPz5IcBq8NaVgXCIL5a1IETonCu8EVi4lynlpSp/rXYG3dyQAtUT4kbsPcWANamojTM18EPhdWlD1RPiwIlO4qUAdpSD+CORg5vMlqoJESgvhAjug1XQLKI22dvgGosssmpCBAqTMyug4v2BypcIAwAryZ5LqiVEYLBhw23MZh0t5uwEIHvFVS0hAudYgWXs1CurFg83l84iIQULzZKaCeGcyEZqFigLysypwI7IwUqTpUpCnLvCf7P/WJZ8dA2HB4iD+lRJiHp86Hq9NAtxbovFxP4gA8HDWs+ymNCZP9ikZYnK4GqwLs6ykG9KYx76ovhNk5L+81VZ0SVZJiVEHWDH+lQBP8poAaCXSm9GrmI6j0kDAM/pgH+3oOesUljPU77ZASstZZMHmNSVLap/8Lwrs0KIXB2Xpc4Bpq003un+gYJ3I67zBjBEcGIKMY3ommcAxF5i193fPOz5VR7IJWRbR0+VY5J517QhnZJJmWMOMWAAqUQO1aGcUYs1IQ0YN5dL+zUrTXZbcxBy5eC41wOLjR4D0mdzIyt3pWT12EDw9S3hwnTYTW17ckIEqrkgA8rrpmecNfGeAumyIFyYd3E32aK/RryBES0wVYZW302naFNb0RyLyQApfwEuZSEI0Ay4++2m9JwJ3chqP4pdWzu0uSpiOkX1mdxCnAa4rXCU8GHFmdINOK+00rneVmxkuWqSIiPWXGVSoQkz0b+w773NzUUISrUBZylsr0gNuLZLOxYZuWt+66RvxxKWHNO/ZJ0GXZYbqZ9zKlRelq+XAQgN6NSnADmc8WAdG9wrcNmMIl0zyedO5JQ3sXpoJ1nU7u/UzMq7mZrX5Uu2kBghdOpBZuNd2c318CqVucH2HpaXdrbdDRN50csdq6wkLgC5pJlomTldFsow+vkuKhy9WJCRNfb9hbdGGlwRaXRK0WUuQn46ZXBnjasKlPMW0kFWkDV6a2Qnu4lojeMyoIfpFK1pLkJMITtGCRVjBCV9URgW7Lg38rc7ni0ryVx2tP25CEEhvovqU4z0PrKinQgyGPlB8tJuGRjJLmtwUl9UF0QEu+3eHbeeL4oMVLbOF7ks6bKjOmxRwakx9XFCfak4S1TGrDR5kMxCSFYvxmc2KwTYLHFk8OWhP8zUNa8QeCdScoxjOpgbjeozl8uKKrKoDALORnLJch3wn7V1UX1YCyOcdze5YoTYIImWr44Q12MAwN3kCsvyXyLBXI2Vx92yqTWALT0WW34bJLH8G7USAgAGRhSEVgaA79onWZaQKEvvix/xQESuvYVw9MKIzgJQ+dkndb272HPAJo908kuwuN7FTJMj+KnVQuxNHYCMEhEEGVibrbyS6lM5yjAgss7lqiREYDCSrxUOFcYKkznfA+SePttg6DqZ6NWpSkJcb/lQzUDpBWDogUjgBdmF4pJ9Et9jMR8lzx/oUjMhzfdVAqSIFJVrPpQuJANseeeT/cazWkIEZLNiEih+kwdKKaKygMlnR94ydJ38na7yNmSqjvA1Q7T5aglxPWejdySAkldbyssk/khxOIlD0pWrNxZRtujvUrZiNf/LzwUqkzGkHCuEAN/qmvLuKBFXx7zRdjeci/EXttGJXXlwkdST9U2v8jdSu4XQSYh4IaBSrAQyABOX0w5YB6u2FDHryJrMreKqLYROigishLkgaiXKV3L+ZVjSFtZBuOsTMy/WwUKAhAND5hJG/5SChbHbT7WmW7qsBSECiI3iKwUAm0TUhs09WRvBUJm1IIROCzAmZDZqTPILFdXJHMOyOHuJHSpi/54JJdnZelHlxWbnK1nBC/WLsyX+SUz2HqGrO6qHZfJ7hax/z6Ry4SKDRcS+WQhLvF+tsPBRpLpXQgQEy1H8PJPvKHGgZpPhGsXF3cL8D5f/BmYTC9WuAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ - W\\left(\\frac{1}{2}\\right)\\right]$"
      ],
      "text/plain": [
       "[-W(1/2)]"
      ]
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.exp(x) + 2 * x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 0\\right)}, \\  \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 1\\right)}, \\  \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 2\\right)}, \\  \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 3\\right)}, \\  \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 4\\right)}\\right]$"
      ],
      "text/plain": [
       "⎡       ⎛ 5    2       ⎞         ⎛ 5    2       ⎞         ⎛ 5    2       ⎞         ⎛ 5    2       ⎞         ⎛ 5    2       ⎞⎤\n",
       "⎣CRootOf⎝x  - x  + 1, 0⎠, CRootOf⎝x  - x  + 1, 1⎠, CRootOf⎝x  - x  + 1, 2⎠, CRootOf⎝x  - x  + 1, 3⎠, CRootOf⎝x  - x  + 1, 4⎠⎦"
      ]
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(x**5 - x**2 + 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUAChmA4MIIiCxjEoAgCxQkoLkD0PSA+B8SnAeDgGGsGGU5UAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 188,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 #s.solve(s.tan(x) - x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {},
   "outputs": [],
   "source": [
    "eq1 = x + 2 * y - 1\n",
    "eq2 = x - y + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKMAAAAzCAYAAAD7CrjDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAIJ0lEQVR4Ae2d7XXVOBCGb/akgLAdbOgAkgpIOtiwFSR0ACe/kn8c6ACogIUOCBUQ6ACoYEM6YN/H8TiKr219+N7rj2jO0ZUsjT5m9EojyXKydXZ2trNooPPz8+uG6ByVNdBbA8JWI+a2VfJXud1aDW/1/KwW1/iogsn7hkSFDxuZcuRsNaA+fyThPsu9VPh1oKAfxHdQ473Y0sz4XZHPVNBFLdH7qDzPxfRKjrxHes6zqVdr82NQvwMsAHYld6jnHzFSiv9E/EfMjEmkAgAhYATIzKSzJcnH6EfZjxWe7YCTbFi5F2VH7skHXC8U/62Ma/SUfiH3lxKZIb8qjJ6iAEnBSWBURYwEgPhW4VkCUXKxrnknR4fQMfWljKLmQ5IX+V7JPzKpFGbCAVzMdp2WU+nXck/E/1OOZVv0ku0PZUqhYvSo8qB1ZUoFQ+cplcvSAxnfD92eDdQP8I7deiQ7/YwlwCp4CZ2JicnpQGGsSRSlgpGZsXOkRLUiM49BA/TpL4EIi+AS/byjeGbOELKBS3lRFA1Gp7GMgkzz0QCg+6H+bevXOkjbJLf8D9sY2uK32xIC4llLZZqJBgTCaq1YE6kwt0rv3MQ4eZJxET0zOpXm4Mw1IAACRMyz7bDXKnEG41rVO/nC2bhwYhJ6mN1L4AzGXuqbb2YBkOMZzg83dmKSwThfPCVLJgDyRmSxSSBSXwYjWshUaUAA/FsPD10gKryLq5jWFMhgXJNip1isAMeGZV9+fcMCQJN3yaG66HO0E1pH5puABsqZjw0L68TiFlbZ7D/lP1Lc2jcxGYwdQFEH0Dkc9vJuGuI9LedtXzbROUWNm/tBVkxxsV6sVRt6xljLFveYwdihLwGu7SC4I9c0kyTr46FbnteMQ/dArr/SQAZjpYocWLEGWGtGUQoYWVdA9kL85in/Zg1IAzL3hgvW2lGUAsaDsoZPUTVl5vukATY8hpNguVM2MKcqnatGQfcZxccIsR3avsJc4GR2/UcOYmf68SbY/Kt0vtPZ6Kup5pasN9bRFdevvuu5Ok4p097J926qxDO0vl6q/R/UjhO54C8BomZGFWznTzFXyrnK/hqnBn6R4yo/N4E5WAWo3DBuJfHBA3jteKWVdwYJp6WesDoMepee6oHD504ag77UBiYXQPhGYfouiILBqEIBIgp5onDQxzbiY0asgw2F2kzIIreerqhbUhnXenogf/Cjh9tWrT4k+eg0BivEYK/rmDivNRqLvtQOLlgASM5mg0y210yrIGYmvq3GTD6QH0OXyuMqFYV+szj5XpNDZeIDkEEkXtrLV2r4ocS3LtXBrsK/QzOKdyuU18N3pbJskDLoMXUu0aH1ODe9CqucYH2RSfxrkVfl8uUoszwm2/tdtReMlYQ3jeZbiGBBxVt1cFkOCq2/93Sr6B0u29drFlUZqwJYsDymV/noiIFUrbUUxztj4rwzo3iiac3ygperkEZ5zTRKkrPvGX6Wigkp+w6P8qFkyEb/QnGAGyVnutUA1gLrQScaoTv6oT64LX2UvtrL0o7XjFieajPW1lgvGC2jCmMN8K/cZ4VZ33SSeADaJzkDIfkXenbNNgt2V+mdZU4hUfL0HVzo1tURYgetF8ekH+mBvQB7BvYYQYMoGIwIqkIBFFMuaPcRIMSxFqKDyFcBT3Gk2YJdwWYir9xvOdatoybaqgbyuWeftt4BosoyPQad69IGuUH1VbaZP/LAmjEIiHRsFBjJIGLq5UoRSuoi1jese+A7ET9APpbPdp8Rg9Iqk63nRhIPAKaDLhsZRhTptBXZAGYKFWtq5edIjA61TV7QetFpw5D6MhmqdW+IIrZDmGo8ppRO01EqpTDNll9xgM8LQOM3X/lszWpRG/FVLybTZABcPAMS08FSO2irnPc8cCljGaG8DD4D4ELPmLvqBKItnxuvPEn6Ur5oed16nTATUPCMaPlSwGhmJHXkW92j9tUxyMcfPTIwLhRmRmcdzIK8a1BxW7orvVF25cHq7MkvTgPk0wbqPG7MsMLIsq5UeauWlOXwbDip0nyBFDPtK3Mu6YCA5YU7y7GBg05vvOXfsjP+W04JimFGee9w8raKT0Wjge2UERpMkrejcPYIUZTB2K4uzAwmE1eQQFGFLa7Bfyo+7zFGQz6ibK31XGUwS7K+Xuu5rNOOVHmdIvoFU8x0vxonklsgYF14542T4myWBCiNJJ6oRbtbiPJuYgZ0q6zCqfJWBawgkGfGQCWqszChbCZm/8dRUckQ8uaZEc13kDqFV3EAcV8OU3YpN1saUt4MRg+s1DkAELdQGDPNLRTfbhr2SdKQ8mYzHQEZdRRrOjYx3ELZicg6SdZNy5vB2AITdQRvmTDRdTIzjemeDY1B3gzGBjipY5j1eL+MSb4PM+Ao5M1gbAbjtaJxHwVGfJf2yofWV4Iu8xTCpYyDy9tnA8MnA3OmpcNmdRobmB05jnfqIJ26LgaXtw8Yp678zvYLbLyGO5BzD7h3lcn7P1E6Cx5p4grlTZ6kosGoRl/LoVJmiFmT5MQUz8Yc+zprRfIaLqItR+qakQ5i1rCKfXLm9PujgYNSVPfCR5D0qWC09UXr7ZWg2jPTrDRQTk5ggvuXxYuCGAGTwFhWxM0UbpfY5YGYejPvzDRQApFPhKHqcvDNY9iv/YtfLgDYXb0ipwr32nzxcC2evCz2q0uoRQH5595oQH2PabaNnvdf/Iq/vrzjLuWhzYwU9MtxAMxLKpTZkSvu/AHyoA+GvIVmhklpQP3OWyo+0OPuJZ9chNzwhn8Jb/8DbiDnWjQoSvIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ x : - \\frac{1}{3}, \\  y : \\frac{2}{3}\\right\\}\\right]$"
      ],
      "text/plain": [
       "[{x: -1/3, y: 2/3}]"
      ]
     },
     "execution_count": 190,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve([eq1, eq2], [x, y], dict=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {},
   "outputs": [],
   "source": [
    "eq1 = x**2 - y\n",
    "eq2 = y**2 - x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {},
   "outputs": [],
   "source": [
    "sols = sympy.solve([eq1, eq2], [x, y], dict=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ x : 0, \\  y : 0\\right\\}, \\  \\left\\{ x : 1, \\  y : 1\\right\\}, \\  \\left\\{ x : \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right)^{2}, \\  y : - \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right\\}, \\  \\left\\{ x : \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right)^{2}, \\  y : - \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right\\}\\right]$"
      ],
      "text/plain": [
       "⎡                            ⎧               2               ⎫  ⎧               2               ⎫⎤\n",
       "⎢                            ⎪   ⎛  1   √3⋅ⅈ⎞        1   √3⋅ⅈ⎪  ⎪   ⎛  1   √3⋅ⅈ⎞        1   √3⋅ⅈ⎪⎥\n",
       "⎢{x: 0, y: 0}, {x: 1, y: 1}, ⎨x: ⎜- ─ - ────⎟ , y: - ─ - ────⎬, ⎨x: ⎜- ─ + ────⎟ , y: - ─ + ────⎬⎥\n",
       "⎢                            ⎪   ⎝  2    2  ⎠        2    2  ⎪  ⎪   ⎝  2    2  ⎠        2    2  ⎪⎥\n",
       "⎣                            ⎩                               ⎭  ⎩                               ⎭⎦"
      ]
     },
     "execution_count": 193,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[True, True, True, True]"
      ]
     },
     "execution_count": 194,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[eq1.subs(sol).simplify() == 0 and eq2.subs(sol).simplify() == 0 for sol in sols]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1⎤\n",
       "⎢ ⎥\n",
       "⎣2⎦"
      ]
     },
     "execution_count": 195,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([1,2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[1  2]"
      ]
     },
     "execution_count": 196,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([[1,2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2⎤\n",
       "⎢    ⎥\n",
       "⎣3  4⎦"
      ]
     },
     "execution_count": 197,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([[1, 2], [3, 4]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1 & 2 & 3\\\\10 & 11 & 12 & 13\\\\20 & 21 & 22 & 23\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0   1   2   3 ⎤\n",
       "⎢              ⎥\n",
       "⎢10  11  12  13⎥\n",
       "⎢              ⎥\n",
       "⎣20  21  22  23⎦"
      ]
     },
     "execution_count": 198,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(3, 4, lambda m,n: 10 * m + n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c, d = sympy.symbols(\"a, b, c, d\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = sympy.Matrix([[a, b], [c, d]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAAzCAYAAADciPtuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADfklEQVRoBe2a7VEbMRCGTYYCkkkHpoOQVADpAOggoYPw0/7HkA6ACoLpADpIQgehgxA6SN7nLN3IOkknO3eMznM7o9OXtbuvdrWn2/HObDZ7PQnQfD5/DgwXNyQ9g/q/kqY/Vf545aI4BHGFFp7uYFnsyGK/1DgV8vv42uHMCMdnaXuMxbaSdktHZc4Q7vZe5Un9vRydi7eYgDyrfBSYR5Xs41I8MMc679S+c/rJ5iCAyWKHBsXWWaxyRdwyaSZnMit4iOFUa85UYPxb5VblSONfVb8EYbEHyftihBFA7tRHjyC1uqIWw5SX+IXaZyqAuTT94Ftfc52R5CGD88XmXhn5bPJCbcaClARmmBJqAURUskT7UWPZrmEXblDb83Vg5dlavI5i/JLAtIirFTt24zFAWNQNvN/+b5fzhRvWm6g2OkFvl1Xz2QbsREvuA0xxgezQ2xS71ggv5h/eCsag78uq+YwCM7vCzjx4ywA70Xx26PXWr9vlfHHGXTo2nagOUWAOF39XYFqBFTgiI9brm+rzLXlsNpvLxb12T1+BKDCzCAC14hrj5kzfusYHjdVCfeYd9TnLtQ5qE8xuJPcqxX83Nam5A5VrMeH9wUH9psJhJvQzRr9v+iQBVh7vr0vJbg1cSWBigKmtP7sAQmPufGdto8Ppugyjrrguo9J+PwIrzSJt+owWa9uh0uZHi5VmkTZ9Rou17VBp81trseSVqjQrWH10zeJT5lplqsKFuHHlGiowvjr2Beiv6uAH72BdUaAOBQoKfmwOFpgAJXONQwaGxYLWwoxrnTGZn6wVCVOo+vDUmJ8TWc52+JQMgoSbsAUQAeQ8JiYLmGHMISW/WH29qq6SphrDJXojycEypAMIFlUaQrUNGFGL5boijG7F0P0kJ6FiBajZPUkeiRtAhRK2K7lGX3qrxcQcy+AKK2bX+BufWQ/9jRO2ORbDMitJ0x4AxFg2ZBsrstFJb0kCM0xwh94DhI8sIRuwE81HzxfzbcBsQpJ/FjRIzDnYfdNGCdskMKMxicl9X3uBwv+f/PGu+uLPpuIpuF1FGstO2OYED1LJJCzddxiCzo3wpdR+nhsnbFuBoa8A8HJ8cTIbF0rOhsZW9MtxxZUFQ+mMwIZiKavnaDG7E0OpbVScKgJxw6jJRKS6X2rD11t6VjisK3LRdf+MyTtrKMTtv6H7P8/oJGMHh2IhAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡a  b⎤\n",
       "⎢    ⎥\n",
       "⎣c  d⎦"
      ]
     },
     "execution_count": 201,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a^{2} + b c & a b + b d\\\\a c + c d & b c + d^{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 2                  ⎤\n",
       "⎢a  + b⋅c   a⋅b + b⋅d⎥\n",
       "⎢                    ⎥\n",
       "⎢                  2 ⎥\n",
       "⎣a⋅c + c⋅d  b⋅c + d  ⎦"
      ]
     },
     "execution_count": 202,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M * M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Matrix(sympy.symbols(\"x_1, x_2\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a x_{1} + b x_{2}\\\\c x_{1} + d x_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡a⋅x₁ + b⋅x₂⎤\n",
       "⎢           ⎥\n",
       "⎣c⋅x₁ + d⋅x₂⎦"
      ]
     },
     "execution_count": 204,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M * x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {},
   "outputs": [],
   "source": [
    "p, q = sympy.symbols(\"p, q\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = sympy.Matrix([[1, p], [q, 1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & p\\\\q & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  p⎤\n",
       "⎢    ⎥\n",
       "⎣q  1⎦"
      ]
     },
     "execution_count": 207,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = sympy.Matrix(sympy.symbols(\"b_1, b_2\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡b₁⎤\n",
       "⎢  ⎥\n",
       "⎣b₂⎦"
      ]
     },
     "execution_count": 209,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{- b_{1} + b_{2} p}{p q - 1}\\\\\\frac{b_{1} q - b_{2}}{p q - 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-b₁ + b₂⋅p⎤\n",
       "⎢──────────⎥\n",
       "⎢ p⋅q - 1  ⎥\n",
       "⎢          ⎥\n",
       "⎢b₁⋅q - b₂ ⎥\n",
       "⎢───────── ⎥\n",
       "⎣ p⋅q - 1  ⎦"
      ]
     },
     "execution_count": 210,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = M.solve(b)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = M.LUsolve(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}b_{1} - \\frac{p \\left(- b_{1} q + b_{2}\\right)}{- p q + 1}\\\\\\frac{- b_{1} q + b_{2}}{- p q + 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡     p⋅(-b₁⋅q + b₂)⎤\n",
       "⎢b₁ - ──────────────⎥\n",
       "⎢        -p⋅q + 1   ⎥\n",
       "⎢                   ⎥\n",
       "⎢    -b₁⋅q + b₂     ⎥\n",
       "⎢    ──────────     ⎥\n",
       "⎣     -p⋅q + 1      ⎦"
      ]
     },
     "execution_count": 212,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = M.inv() * b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{b_{1}}{- p q + 1} - \\frac{b_{2} p}{- p q + 1}\\\\- \\frac{b_{1} q}{- p q + 1} + \\frac{b_{2}}{- p q + 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    b₁        b₂⋅p   ⎤\n",
       "⎢ ──────── - ──────── ⎥\n",
       "⎢ -p⋅q + 1   -p⋅q + 1 ⎥\n",
       "⎢                     ⎥\n",
       "⎢    b₁⋅q        b₂   ⎥\n",
       "⎢- ──────── + ────────⎥\n",
       "⎣  -p⋅q + 1   -p⋅q + 1⎦"
      ]
     },
     "execution_count": 214,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Versions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.10.12 64bit [Clang 14.0.6 ]"
        },
        {
         "module": "IPython",
         "version": "8.12.0"
        },
        {
         "module": "OS",
         "version": "macOS 10.15.7 x86_64 i386 64bit"
        },
        {
         "module": "sympy",
         "version": "1.11.1"
        },
        {
         "module": "numpy",
         "version": "1.22.3"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.10.12 64bit [Clang 14.0.6 ]</td></tr><tr><td>IPython</td><td>8.12.0</td></tr><tr><td>OS</td><td>macOS 10.15.7 x86\\_64 i386 64bit</td></tr><tr><td>sympy</td><td>1.11.1</td></tr><tr><td>numpy</td><td>1.22.3</td></tr><tr><td colspan='2'>Sat Nov 02 18:08:14 2024 JST</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.10.12 64bit [Clang 14.0.6 ] \\\\ \\hline\n",
       "IPython & 8.12.0 \\\\ \\hline\n",
       "OS & macOS 10.15.7 x86\\_64 i386 64bit \\\\ \\hline\n",
       "sympy & 1.11.1 \\\\ \\hline\n",
       "numpy & 1.22.3 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Sat Nov 02 18:08:14 2024 JST} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.10.12 64bit [Clang 14.0.6 ]\n",
       "IPython 8.12.0\n",
       "OS macOS 10.15.7 x86_64 i386 64bit\n",
       "sympy 1.11.1\n",
       "numpy 1.22.3\n",
       "Sat Nov 02 18:08:14 2024 JST"
      ]
     },
     "execution_count": 215,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%reload_ext version_information\n",
    "%version_information sympy, numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}