diff --git a/dev/functions_list/index.html b/dev/functions_list/index.html index 8f28631a..932a81d9 100644 --- a/dev/functions_list/index.html +++ b/dev/functions_list/index.html @@ -20,35 +20,35 @@ true julia> gamma(4 + 1) == prod(1:4) == factorial(4) -true

External links: DLMF 5.2.1, Wikipedia.

See also: loggamma(z) for $\log \Gamma(z)$ and gamma(a,z) for the upper incomplete gamma function $\Gamma(a,z)$.

Implementation by

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gamma(a,x)

Returns the upper incomplete gamma function

\[\Gamma(a,x) = \int_x^\infty t^{a-1} e^{-t} \mathrm{d}t\]

supporting arbitrary real or complex a and x.

(The ordinary gamma function gamma(x) corresponds to $\Gamma(a) = \Gamma(a,0)$. See also the gamma_inc function to compute both the upper and lower ($\gamma(a,x)$) incomplete gamma functions scaled by $\Gamma(a)$.

External links: DLMF 8.2.2, Wikipedia

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SpecialFunctions.loggamma — Function
loggamma(x)

Computes the logarithm of gamma for given x. If x is a Real, then it throws a DomainError if gamma(x) is negative.

If x is complex, then exp(loggamma(x)) matches gamma(x) (up to floating-point error), but loggamma(x) may differ from log(gamma(x)) by an integer multiple of $2πi$ (i.e. it may employ a different branch cut).

See also logabsgamma for real x.

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loggamma(a,x)

Returns the log of the upper incomplete gamma function gamma(a,x):

\[\log \Gamma(a,x) = \log \int_x^\infty t^{a-1} e^{-t} \mathrm{d}t\]

supporting arbitrary real or complex a and x.

If a and/or x is complex, then exp(loggamma(a,x)) matches gamma(a,x) (up to floating-point error), but loggamma(a,x) may differ from log(gamma(a,x)) by an integer multiple of $2\pi i$ (i.e. it may employ a different branch cut).

See also loggamma(x).

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SpecialFunctions.logabsgamma — Function
logabsgamma(x)

Compute the logarithm of absolute value of gamma for Real x and returns a tuple (log(abs(gamma(x))), sign(gamma(x))).

See also loggamma.

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SpecialFunctions.loggamma1p — Function
loggamma1p(x)

Compute $\log(\Gamma(1+x))$ for -1 < x <= 1.

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SpecialFunctions.logfactorial — Function
logfactorial(x)

Compute the logarithmic factorial of a nonnegative integer x. Equivalent to loggamma of x + 1, but loggamma extends this function to non-integer x.

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SpecialFunctions.digamma — Function
digamma(x)

Compute the digamma function of x (the logarithmic derivative of gamma(x)).

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SpecialFunctions.invdigamma — Function
invdigamma(x)

Compute the inverse digamma function of x.

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SpecialFunctions.trigamma — Function
trigamma(x)

Compute the trigamma function of x (the logarithmic second derivative of gamma(x)).

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SpecialFunctions.polygamma — Function
polygamma(m, x)

Compute the polygamma function of order m of argument x (the (m+1)th derivative of the logarithm of gamma(x))

External links: Wikipedia

See also: gamma(z)

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SpecialFunctions.gamma_inc — Function
gamma_inc(a,x,IND=0)

Returns a tuple $(p, q)$ where $p + q = 1$, and $p = P(a,x)$ is the Incomplete gamma function ratio given by:

\[P(a,x) = \frac{1}{\Gamma (a)} \int_{0}^{x} e^{-t}t^{a-1} \mathrm{d}t.\]

and $q = Q(a,x)$ is the Incomplete gamma function ratio given by:

\[Q(a,x) = \frac{1}{\Gamma (a)} \int_{x}^{\infty} e^{-t}t^{a-1} \mathrm{d}t.\]

In terms of these, the lower incomplete gamma function is $\gamma(a,x) = P(a,x) \Gamma(a)$ and the upper incomplete gamma function is $\Gamma(a,x) = Q(a,x) \Gamma(a)$.

IND ∈ [0,1,2] sets accuracy: IND=0 means 14 significant digits accuracy, IND=1 means 6 significant digit, and IND=2 means only 3 digit accuracy.

External links: DLMF 8.2.4, Wikipedia

See also gamma(z), gamma_inc_inv(a,p,q)

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SpecialFunctions.gamma_inc_inv — Function
gamma_inc_inv(a,p,q)

Inverts the gamma_inc(a,x) function, by computing x given a,p,q in $P(a,x) = p$ and $Q(a,x) = q$.

External links: DLMF 8.2.4, Wikipedia

See also: gamma_inc(a,x,ind).

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SpecialFunctions.loggammadiv — Function
loggammadiv(a,b)

Computes $\log(\Gamma(b)/\Gamma(a+b))$ when b >= 8

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SpecialFunctions.gammax — Function
gammax(x)

\[\operatorname{gammax}(x) = \begin{cases} +true

External links: DLMF 5.2.1, Wikipedia.

See also: loggamma(z) for $\log \Gamma(z)$ and gamma(a,z) for the upper incomplete gamma function $\Gamma(a,z)$.

Implementation by

  • Float: C standard math library libm.
  • Complex: by exp(loggamma(z)).
  • BigFloat: C library for multiple-precision floating-point MPFR
source
gamma(a,x)

Returns the upper incomplete gamma function

\[\Gamma(a,x) = \int_x^\infty t^{a-1} e^{-t} \mathrm{d}t\]

supporting arbitrary real or complex a and x.

(The ordinary gamma function gamma(x) corresponds to $\Gamma(a) = \Gamma(a,0)$. See also the gamma_inc function to compute both the upper and lower ($\gamma(a,x)$) incomplete gamma functions scaled by $\Gamma(a)$.

External links: DLMF 8.2.2, Wikipedia

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SpecialFunctions.loggamma — Function
loggamma(x)

Computes the logarithm of gamma for given x. If x is a Real, then it throws a DomainError if gamma(x) is negative.

If x is complex, then exp(loggamma(x)) matches gamma(x) (up to floating-point error), but loggamma(x) may differ from log(gamma(x)) by an integer multiple of $2πi$ (i.e. it may employ a different branch cut).

See also logabsgamma for real x.

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loggamma(a,x)

Returns the log of the upper incomplete gamma function gamma(a,x):

\[\log \Gamma(a,x) = \log \int_x^\infty t^{a-1} e^{-t} \mathrm{d}t\]

supporting arbitrary real or complex a and x.

If a and/or x is complex, then exp(loggamma(a,x)) matches gamma(a,x) (up to floating-point error), but loggamma(a,x) may differ from log(gamma(a,x)) by an integer multiple of $2\pi i$ (i.e. it may employ a different branch cut).

See also loggamma(x).

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SpecialFunctions.logabsgamma — Function
logabsgamma(x)

Compute the logarithm of absolute value of gamma for Real x and returns a tuple (log(abs(gamma(x))), sign(gamma(x))).

See also loggamma.

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SpecialFunctions.loggamma1p — Function
loggamma1p(x)

Compute $\log(\Gamma(1+x))$ for -1 < x <= 1.

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SpecialFunctions.logfactorial — Function
logfactorial(x)

Compute the logarithmic factorial of a nonnegative integer x. Equivalent to loggamma of x + 1, but loggamma extends this function to non-integer x.

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SpecialFunctions.digamma — Function
digamma(x)

Compute the digamma function of x (the logarithmic derivative of gamma(x)).

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SpecialFunctions.invdigamma — Function
invdigamma(x)

Compute the inverse digamma function of x.

source
SpecialFunctions.trigamma — Function
trigamma(x)

Compute the trigamma function of x (the logarithmic second derivative of gamma(x)).

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SpecialFunctions.polygamma — Function
polygamma(m, x)

Compute the polygamma function of order m of argument x (the (m+1)th derivative of the logarithm of gamma(x))

External links: Wikipedia

See also: gamma(z)

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SpecialFunctions.gamma_inc — Function
gamma_inc(a,x,IND=0)

Returns a tuple $(p, q)$ where $p + q = 1$, and $p = P(a,x)$ is the Incomplete gamma function ratio given by:

\[P(a,x) = \frac{1}{\Gamma (a)} \int_{0}^{x} e^{-t}t^{a-1} \mathrm{d}t.\]

and $q = Q(a,x)$ is the Incomplete gamma function ratio given by:

\[Q(a,x) = \frac{1}{\Gamma (a)} \int_{x}^{\infty} e^{-t}t^{a-1} \mathrm{d}t.\]

In terms of these, the lower incomplete gamma function is $\gamma(a,x) = P(a,x) \Gamma(a)$ and the upper incomplete gamma function is $\Gamma(a,x) = Q(a,x) \Gamma(a)$.

IND ∈ [0,1,2] sets accuracy: IND=0 means 14 significant digits accuracy, IND=1 means 6 significant digit, and IND=2 means only 3 digit accuracy.

External links: DLMF 8.2.4, Wikipedia

See also gamma(z), gamma_inc_inv(a,p,q)

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SpecialFunctions.gamma_inc_inv — Function
gamma_inc_inv(a,p,q)

Inverts the gamma_inc(a,x) function, by computing x given a,p,q in $P(a,x) = p$ and $Q(a,x) = q$.

External links: DLMF 8.2.4, Wikipedia

See also: gamma_inc(a,x,ind).

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SpecialFunctions.loggammadiv — Function
loggammadiv(a,b)

Computes $\log(\Gamma(b)/\Gamma(a+b))$ when b >= 8

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SpecialFunctions.gammax — Function
gammax(x)

\[\operatorname{gammax}(x) = \begin{cases} e^{\operatorname{stirling}(x)}\quad\quad\quad \text{if} \quad x>0,\\ \dfrac{\Gamma(x)}{\sqrt{2 \pi}e^{-x + (x-0.5)\log(x)}},\quad \text{if} \quad x\leq 0. -\end{cases}\]

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SpecialFunctions.rgammax — Function
rgammax(a,x)

Evaluation of $1/\Gamma(a) e^{-x} x^{a}$. Based on DRCOMP from the NSWC library.

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SpecialFunctions.rgamma1pm1 — Function
rgamma1pm1(a)

Computation of $1/\Gamma(a+1) - 1$ for -0.5<=a<=1.5: $1/\Gamma (a+1) - 1$.

Uses the relation gamma(a+1) = a*gamma(a).

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SpecialFunctions.gamma_inc_asym — Function
gamma_inc_asym(a, x, ind)

Compute $P(a,x)$ using asymptotic expansion given by:

\[R(a,x)/a * (1 + \sum_{n=1}^{N-1}(a_{n}/x^{n} + \Theta _{n}a_{n}/x^{n}))\]

where R(a,x) = rgammax(a,x). Used when 1 <= a <= BIG and x >= x0.

External links: DLMF 8.11.2

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_cf — Function
gamma_inc_cf(a, x, ind)

Computes $P(a,x)$ by continued fraction expansion given by:

\[R(a,x) \cdot \cfrac{1}{1-\cfrac{z}{a+1+\cfrac{z}{a+2-\cfrac{(a+1)z}{a+3+\cfrac{2z}{a+4-\cfrac{(a+2)z}{a+5+\cfrac{3z}{a+6-\ddots}}}}}}}\]

Used when 1 <= a <= BIG and x < x0.

External links: DLMF 8.9.2

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_fsum — Function
gamma_inc_fsum(a,x)

Compute using Finite Sums for $Q(a,x)$ when a >= 1 && 2a is integer. Used when a <= x <= x0 and a = n/2. Refer Eqn (14) in the paper.

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_minimax — Function
gamma_inc_minimax(a,x,z)

Compute $P(a,x)$ using minimax approximations given by :

\[1/2 * \operatorname{erfc}(\sqrt{y}) - e^{-y}/\sqrt{2\pi a} ⋅ T(a,\lambda)\]

where

\[T(a,\lambda) = \sum_{0}^{N} c_{k}(z)a^{-k}\]

This is a higher accuracy approximation of Temme expansion, which deals with the region near a ≈ x with a large. Refer Appendix F in the paper for the extensive set of coefficients calculated using Brent's multiple precision arithmetic(set at 50 digits) in

Brent, R. P. A Fortran multiple-precision arithmetic package, ACM Trans. Math. Softw. 4(1978), 57-70, doi: 10.1145/355769.355775.

External links: DLMF 8.12.8

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_taylor — Function
gamma_inc_taylor(a, x, ind)

Compute $P(a,x)$ using Taylor Series for P/R given by:

\[R(a,x)/a * (1 + \sum_{n=1}^{\infty} x^{n}/((a+1)(a+2)...(a+n)))\]

Used when 1 <= a <= BIG and x <= max{a, ln 10}.

External links: DLMF 8.11.2

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_taylor_x — Function
gamma_inc_taylor_x(a,x,ind)

Computes $P(a,x)$ based on Taylor expansion of $P(a,x)/x^a$ given by:

\[J = -a * \sum_{1}^{\infty} (-x)^{n}/((a+n)n!)\]

and $P(a,x)/x^a$ is given by:

\[(1 - J) / \Gamma(a+1)\]

resulting from term-by-term integration of gamma_inc(a,x,ind). This is used when a < 1 and x < 1.1 - Refer Eqn (9) in the paper.

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_temme — Function
gamma_inc_temme(a, x, z)

Compute $P(a,x)$ using Temme's expansion given by :

\[1/2 * \operatorname{erfc}(\sqrt{y}) - e^{-y}/\sqrt{2\pi a} ⋅ T(a,\lambda)\]

where

\[T(a,\lambda) = \sum_{0}^{N} c_{k}(z)a^{-k}\]

This mainly solves the problem near the region when a ≈ x with a large, and is a lower accuracy version of the minimax approximation.

External links: DLMF 8.12.8

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_temme_1 — Function
gamma_inc_temme_1(a, x, z, ind)

Computes $P(a,x)$ using simplified Temme expansion near $y=0$ by:

\[E(y) - (1 - y)/\sqrt{2\pi a} ⋅ T(a,\lambda)\]

where

\[E(y) = 1/2 - (1 - y/3) ⋅ \sqrt{y/\pi}\]

Used instead of it's previous function when $\sigma \leq e_{0}/\sqrt{a}$.

External links: DLMF 8.12.8

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SpecialFunctions.gamma_inc_inv_psmall — Function
gamma_inc_inv_psmall(a, logr)

Compute x0 - initial approximation when p is small. Here we invert the series in Eqn (2.20) in the paper and write the inversion problem as:

\[x = r\left[1 + a\sum_{k=1}^{\infty}\frac{(-x)^{n}}{(a+n)n!}\right]^{-1/a},\]

where $r = (p\Gamma(1+a))^{1/a}$ Inverting this relation we obtain $x = r + \sum_{k=2}^{\infty} c_{k} r^{k}$.

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SpecialFunctions.gamma_inc_inv_qsmall — Function
gamma_inc_inv_qsmall(a, q, qgammaxa)

Compute x0 - initial approximation when q is small from $e^{-x_{0}} x_{0}^{a} = q \Gamma(a)$. Asymptotic expansions Eqn (2.29) in the paper is used here and higher approximations are obtained using

\[x \sim x_{0} - L + b \sum_{k=1}^{\infty} d_{k}/x_{0}^{k}\]

where $b = 1-a$, $L = \ln{x_0}$.

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SpecialFunctions.gamma_inc_inv_alarge — Function
gamma_inc_inv_alarge(a, minpq, pcase)

Compute x0 - initial approximation when a is large. The inversion problem is rewritten as:

\[0.5 \operatorname{erfc}(\eta \sqrt{a/2}) + R_{a}(\eta) = q\]

For large values of a we can write: $\eta(q,a) = \eta_{0}(q,a) + \epsilon(\eta_{0},a)$ and it is possible to expand:

\[\epsilon(\eta_{0},a) = \epsilon_{1}(\eta_{0},a)/a + \epsilon_{2}(\eta_{0},a)/a^{2} + \epsilon_{3}(\eta_{0},a)/a^{3} + \cdots\]

which is calculated by coeff1, coeff2 and coeff3 functions below. This returns a tuple (x0,fp), where fp is computed since it's an approximation for the coefficient after inverting the original power series.

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SpecialFunctions.auxgam — Function
auxgam(x)

Compute function g in $1/\Gamma(x+1) = 1 + x (x-1) g(x)$, -1 <= x <= 1.

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Beta Function

SpecialFunctions.beta — Function
beta(x, y)

Euler integral of the first kind $\operatorname{B}(x,y) = \Gamma(x)\Gamma(y)/\Gamma(x+y)$.

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SpecialFunctions.logbeta — Function
logbeta(x, y)

Natural logarithm of the beta function $\log(|\operatorname{B}(x,y)|)$.

See also logabsbeta.

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SpecialFunctions.logabsbeta — Function
logabsbeta(x, y)

Compute the natural logarithm of the absolute value of the beta function, returning a tuple (log(abs(beta(x,y))), sign(beta(x,y)))

See also logbeta.

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SpecialFunctions.logabsbinomial — Function
logabsbinomial(n, k)

Accurate natural logarithm of the absolute value of the binomial coefficient binomial(n, k) for large n and k near n/2.

Returns a tuple (log(abs(binomial(n,k))), sign(binomial(n,k))).

External links Base.binomial

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SpecialFunctions.beta_inc — Function
beta_inc(a, b, x, y=1-x)

Return a tuple $(I_{x}(a,b), 1-I_{x}(a,b))$ where $I_{x}(a,b)$ is the regularized incomplete beta function given by

\[I_{x}(a,b) = \frac{1}{B(a,b)} \int_{0}^{x} t^{a-1}(1-t)^{b-1} \mathrm{d}t,\]

where $B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$.

External links: DLMF 8.17.1, Wikipedia

See also: beta_inc_inv

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SpecialFunctions.beta_inc_inv — Function
beta_inc_inv(a, b, p, q=1-p)

Return a tuple (x, 1-x) where x satisfies $I_{x}(a, b) = p$, i.e., x is the inverse of the regularized incomplete beta function $I_{x}(a, b)$.

See also: beta_inc

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SpecialFunctions.ncbeta — Function
ncbeta(a,b,lambda,x)

Compute the CDF of the noncentral beta distribution given by

\[I_{x}(a,b; \lambda) = \sum_{j=0}^{\infty} q(\lambda/2,j) I_{x}(a+j,b;0)\]

For $\lambda < 54$ : algorithm suggested by Lenth(1987) in ncbeta_tail(a,b,lambda,x). Else for $\lambda \geq 54$: modification in Chattamvelli(1997) in ncbeta_poisson(a,b,lambda,x) by using both forward and backward recurrences.

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SpecialFunctions.ncbeta_poisson — Function
ncbeta_poisson(a,b,lambda,x)

Compute CDF of noncentral beta if lambda >= 54 using: First $\lambda/2$ is calculated and the Poisson term is calculated using $P(j-1) = j/\lambda P(j)$ and $P(j+1) = \lambda/(j+1) P(j)$. Then backward recurrences are used until either the Poisson weights fall below errmax or iterlo is reached.

\[I_{x}(a+j-1,b) = I_{x}(a+j,b) + \Gamma(a+b+j-1)/\Gamma(a+j)\Gamma(b)x^{a+j-1}(1-x)^{b}\]

Then forward recurrences are used until error bound falls below errmax.

\[I_{x}(a+j+1,b) = I_{x}(a+j,b) - \Gamma(a+b+j)/\Gamma(a+j)\Gamma(b)x^{a+j}(1-x)^{b}\]

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SpecialFunctions.ncbeta_tail — Function
ncbeta_tail(x,a,b,lambda)

Compute tail of the noncentral beta distribution. Uses the recursive relation

\[I_{x}(a,b+1;0) = I_{x}(a,b;0) - \Gamma(a+b)/\Gamma(a+1)\Gamma(b) x^a (1-x)^b\]

and $\Gamma(a+1) = a\Gamma(a)$ given in DLMF 8.17.21.

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SpecialFunctions.beta_inc_power_series1 — Function
beta_inc_power_series1(a,b,x,epps)

Another variant of BPSER(A,B,X,EPS).

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

APSER(A,B,X,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_inc_power_series2 — Function
beta_inc_power_series2(a,b,x,epps)

Variant of BPSER(A,B,X,EPS).

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

FPSER(A,B,X,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_inc_asymptotic_asymmetric — Function
beta_inc_asymptotic_asymmetric(a, b, x, y, w, epps)

Evaluation of $I_{x}(a,b)$ using asymptotic expansion. It is assumed a >= 15 and b <= 1, and epps is tolerance used.

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

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SpecialFunctions.beta_inc_asymptotic_symmetric — Function
beta_inc_asymptotic_symmetric(a,b,lambda,epps)

Compute $I_{x}(a,b)$ using asymptotic expansion for a, b >= 15. It is assumed that $\lambda = (a+b)*y - b$.

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

BASYM(A,B,LAMBDA,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_inc_power_series — Function
beta_inc_power_series(a, b, x, epps)

Computes $I_x(a,b)$ using power series:

\[I_{x}(a,b) = G(a,b) x^{a}/a \left[1 + a \sum_{j=1}^{\infty} ((1-b)(2-b)\dots(j-b)/j!(a+j)) x^{j}\right]\]

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

BPSER(A,B,X,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_inc_diff — Function
beta_inc_diff(a, b, x, y, n, epps)

Compute $I_{x}(a,b) - I_{x}(a+n,b)$ where n is positive integer and epps is tolerance. A generalised version of DLMF 8.17.20.

External links: DLMF 8.17.20, Wikipedia

See also: beta_inc

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SpecialFunctions.beta_inc_cont_fraction — Function
beta_inc_cont_fraction(a,b,x,y,lambda,epps)

Compute $I_{x}(a,b)$ using continued fraction expansion when a, b > 1. It is assumed that $\lambda = (a+b)*y - b$

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

BFRAC(A,B,X,Y,LAMBDA,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_integrand — Function
beta_integrand(a, b, x, y, mu=0.0)

Compute $e^{\mu} x^a y^b / B(a,b)$

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Utilities

SpecialFunctions.chepolsum — Function
chepolsum(n,x,a)

Computes a series of Chebyshev Polynomials given by: a[1]/2 + a[2]T1(x) + .... + a[n]T{n-1}(X).

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SpecialFunctions.lambdaeta — Function
lambdaeta(eta)

Compute the value of $\lambda$ satisfying $\eta^{2}/2 = \lambda-1-\log{\lambda}$.

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SpecialFunctions.stirling_corr — Function
stirling_corr(a0,b0)

Compute stirling(a0) + stirling(b0) - stirling(a0 + b0) for a0, b0 >= 8

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SpecialFunctions.stirling_error — Function
stirling_error(x)

Compute $\ln{\Gamma(x)} - (x-0.5)*\ln{x} + x - \ln{(2\pi)}/2$. Adapted from stirling in IncgamFI.

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SpecialFunctions.esum — Function
esum(mu,x)

Compute $e^{\mu+x}$

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SpecialFunctions.coeff1 — Function

Computing the first coefficient for the expansion:

\[\epsilon (\eta_{0},a) = \epsilon_{1}(\eta_{0},a)/a + \epsilon_{2}(\eta_{0},a)/a^{2} + \epsilon_{3}(\eta_{0},a)/a^{3}\]

Refer Eqn (3.12) in the paper

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SpecialFunctions.coeff2 — Function

Computing the second coefficient for the expansion:

\[\epsilon (\eta_{0},a) = \epsilon_{1} (\eta_{0},a)/a + \epsilon_{2} (\eta_{0},a)/a^{2} + \epsilon_{3} (\eta_{0},a)/a^{3}\]

Refer Eqn (3.12) in the paper

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SpecialFunctions.coeff3 — Function

Computing the third and last coefficient for the expansion:

\[\epsilon(\eta_{0},a) = \epsilon_{1}(\eta_{0},a)/a + \epsilon_{2}(\eta_{0},a)/a^{2} + \epsilon_{3}(\eta_{0},a)/a^{3}\]

Refer Eqn (3.12) in the paper

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SpecialFunctions.ncF — Function
ncF(x,v1,v2,lambda)

Compute CDF of noncentral F distribution given by:

\[F(x, v_1, v_2; \lambda) = I_{v_1 x/(v_1 x + v_2)}(v_1/2, v_2/2; \lambda)\]

where $I_{x}(a,b; \lambda)$ is the noncentral beta function computed above.

External links: Wikipedia

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Exponential and Trigonometric Integrals

SpecialFunctions.expint — Function
expint(z)
-expint(ν, z)

Computes the exponential integral

\[\operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t.\]

If $\nu$ is not specified, $\nu=1$ is used. Arbitrary complex $\nu$ and $z$ are supported.

External links: DLMF 8.19, Wikipedia

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SpecialFunctions.expinti — Function
expinti(x::Real)

Computes the exponential integral function

\[\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t,\]

which is equivalent to $-\Re[\operatorname{E}_1(-x)]$ where $\operatorname{E}_1$ is the expint function.

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SpecialFunctions.expintx — Function
expintx(z)
-expintx(ν, z)

Computes the scaled exponential integral

\[\exp(z) \operatorname{E}_\nu(z) = e^z \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t.\]

If $\nu$ is not specified, $\nu = 1$ is used. Arbitrary complex $\nu$ and $z$ are supported.

See also: expint(ν, z)

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SpecialFunctions.sinint — Function
sinint(x)

Compute the sine integral function of $x$, defined by

\[\operatorname{Si}(x) +\end{cases}\]

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SpecialFunctions.rgammax — Function
rgammax(a,x)

Evaluation of $1/\Gamma(a) e^{-x} x^{a}$. Based on DRCOMP from the NSWC library.

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SpecialFunctions.rgamma1pm1 — Function
rgamma1pm1(a)

Computation of $1/\Gamma(a+1) - 1$ for -0.5<=a<=1.5: $1/\Gamma (a+1) - 1$.

Uses the relation gamma(a+1) = a*gamma(a).

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SpecialFunctions.gamma_inc_asym — Function
gamma_inc_asym(a, x, ind)

Compute $P(a,x)$ using asymptotic expansion given by:

\[R(a,x)/a * (1 + \sum_{n=1}^{N-1}(a_{n}/x^{n} + \Theta _{n}a_{n}/x^{n}))\]

where R(a,x) = rgammax(a,x). Used when 1 <= a <= BIG and x >= x0.

External links: DLMF 8.11.2

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_cf — Function
gamma_inc_cf(a, x, ind)

Computes $P(a,x)$ by continued fraction expansion given by:

\[R(a,x) \cdot \cfrac{1}{1-\cfrac{z}{a+1+\cfrac{z}{a+2-\cfrac{(a+1)z}{a+3+\cfrac{2z}{a+4-\cfrac{(a+2)z}{a+5+\cfrac{3z}{a+6-\ddots}}}}}}}\]

Used when 1 <= a <= BIG and x < x0.

External links: DLMF 8.9.2

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_fsum — Function
gamma_inc_fsum(a,x)

Compute using Finite Sums for $Q(a,x)$ when a >= 1 && 2a is integer. Used when a <= x <= x0 and a = n/2. Refer Eqn (14) in the paper.

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_minimax — Function
gamma_inc_minimax(a,x,z)

Compute $P(a,x)$ using minimax approximations given by :

\[1/2 * \operatorname{erfc}(\sqrt{y}) - e^{-y}/\sqrt{2\pi a} ⋅ T(a,\lambda)\]

where

\[T(a,\lambda) = \sum_{0}^{N} c_{k}(z)a^{-k}\]

This is a higher accuracy approximation of Temme expansion, which deals with the region near a ≈ x with a large. Refer Appendix F in the paper for the extensive set of coefficients calculated using Brent's multiple precision arithmetic(set at 50 digits) in

Brent, R. P. A Fortran multiple-precision arithmetic package, ACM Trans. Math. Softw. 4(1978), 57-70, doi: 10.1145/355769.355775.

External links: DLMF 8.12.8

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_taylor — Function
gamma_inc_taylor(a, x, ind)

Compute $P(a,x)$ using Taylor Series for P/R given by:

\[R(a,x)/a * (1 + \sum_{n=1}^{\infty} x^{n}/((a+1)(a+2)...(a+n)))\]

Used when 1 <= a <= BIG and x <= max{a, ln 10}.

External links: DLMF 8.11.2

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_taylor_x — Function
gamma_inc_taylor_x(a,x,ind)

Computes $P(a,x)$ based on Taylor expansion of $P(a,x)/x^a$ given by:

\[J = -a * \sum_{1}^{\infty} (-x)^{n}/((a+n)n!)\]

and $P(a,x)/x^a$ is given by:

\[(1 - J) / \Gamma(a+1)\]

resulting from term-by-term integration of gamma_inc(a,x,ind). This is used when a < 1 and x < 1.1 - Refer Eqn (9) in the paper.

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_temme — Function
gamma_inc_temme(a, x, z)

Compute $P(a,x)$ using Temme's expansion given by :

\[1/2 * \operatorname{erfc}(\sqrt{y}) - e^{-y}/\sqrt{2\pi a} ⋅ T(a,\lambda)\]

where

\[T(a,\lambda) = \sum_{0}^{N} c_{k}(z)a^{-k}\]

This mainly solves the problem near the region when a ≈ x with a large, and is a lower accuracy version of the minimax approximation.

External links: DLMF 8.12.8

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_temme_1 — Function
gamma_inc_temme_1(a, x, z, ind)

Computes $P(a,x)$ using simplified Temme expansion near $y=0$ by:

\[E(y) - (1 - y)/\sqrt{2\pi a} ⋅ T(a,\lambda)\]

where

\[E(y) = 1/2 - (1 - y/3) ⋅ \sqrt{y/\pi}\]

Used instead of it's previous function when $\sigma \leq e_{0}/\sqrt{a}$.

External links: DLMF 8.12.8

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SpecialFunctions.gamma_inc_inv_psmall — Function
gamma_inc_inv_psmall(a, logr)

Compute x0 - initial approximation when p is small. Here we invert the series in Eqn (2.20) in the paper and write the inversion problem as:

\[x = r\left[1 + a\sum_{k=1}^{\infty}\frac{(-x)^{n}}{(a+n)n!}\right]^{-1/a},\]

where $r = (p\Gamma(1+a))^{1/a}$ Inverting this relation we obtain $x = r + \sum_{k=2}^{\infty} c_{k} r^{k}$.

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SpecialFunctions.gamma_inc_inv_qsmall — Function
gamma_inc_inv_qsmall(a, q, qgammaxa)

Compute x0 - initial approximation when q is small from $e^{-x_{0}} x_{0}^{a} = q \Gamma(a)$. Asymptotic expansions Eqn (2.29) in the paper is used here and higher approximations are obtained using

\[x \sim x_{0} - L + b \sum_{k=1}^{\infty} d_{k}/x_{0}^{k}\]

where $b = 1-a$, $L = \ln{x_0}$.

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SpecialFunctions.gamma_inc_inv_alarge — Function
gamma_inc_inv_alarge(a, minpq, pcase)

Compute x0 - initial approximation when a is large. The inversion problem is rewritten as:

\[0.5 \operatorname{erfc}(\eta \sqrt{a/2}) + R_{a}(\eta) = q\]

For large values of a we can write: $\eta(q,a) = \eta_{0}(q,a) + \epsilon(\eta_{0},a)$ and it is possible to expand:

\[\epsilon(\eta_{0},a) = \epsilon_{1}(\eta_{0},a)/a + \epsilon_{2}(\eta_{0},a)/a^{2} + \epsilon_{3}(\eta_{0},a)/a^{3} + \cdots\]

which is calculated by coeff1, coeff2 and coeff3 functions below. This returns a tuple (x0,fp), where fp is computed since it's an approximation for the coefficient after inverting the original power series.

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SpecialFunctions.auxgam — Function
auxgam(x)

Compute function g in $1/\Gamma(x+1) = 1 + x (x-1) g(x)$, -1 <= x <= 1.

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Beta Function

SpecialFunctions.beta — Function
beta(x, y)

Euler integral of the first kind $\operatorname{B}(x,y) = \Gamma(x)\Gamma(y)/\Gamma(x+y)$.

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SpecialFunctions.logbeta — Function
logbeta(x, y)

Natural logarithm of the beta function $\log(|\operatorname{B}(x,y)|)$.

See also logabsbeta.

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SpecialFunctions.logabsbeta — Function
logabsbeta(x, y)

Compute the natural logarithm of the absolute value of the beta function, returning a tuple (log(abs(beta(x,y))), sign(beta(x,y)))

See also logbeta.

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SpecialFunctions.logabsbinomial — Function
logabsbinomial(n, k)

Accurate natural logarithm of the absolute value of the binomial coefficient binomial(n, k) for large n and k near n/2.

Returns a tuple (log(abs(binomial(n,k))), sign(binomial(n,k))).

External links Base.binomial

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SpecialFunctions.beta_inc — Function
beta_inc(a, b, x, y=1-x)

Return a tuple $(I_{x}(a,b), 1-I_{x}(a,b))$ where $I_{x}(a,b)$ is the regularized incomplete beta function given by

\[I_{x}(a,b) = \frac{1}{B(a,b)} \int_{0}^{x} t^{a-1}(1-t)^{b-1} \mathrm{d}t,\]

where $B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$.

External links: DLMF 8.17.1, Wikipedia

See also: beta_inc_inv

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SpecialFunctions.beta_inc_inv — Function
beta_inc_inv(a, b, p, q=1-p)

Return a tuple (x, 1-x) where x satisfies $I_{x}(a, b) = p$, i.e., x is the inverse of the regularized incomplete beta function $I_{x}(a, b)$.

See also: beta_inc

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SpecialFunctions.ncbeta — Function
ncbeta(a,b,lambda,x)

Compute the CDF of the noncentral beta distribution given by

\[I_{x}(a,b; \lambda) = \sum_{j=0}^{\infty} q(\lambda/2,j) I_{x}(a+j,b;0)\]

For $\lambda < 54$ : algorithm suggested by Lenth(1987) in ncbeta_tail(a,b,lambda,x). Else for $\lambda \geq 54$: modification in Chattamvelli(1997) in ncbeta_poisson(a,b,lambda,x) by using both forward and backward recurrences.

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SpecialFunctions.ncbeta_poisson — Function
ncbeta_poisson(a,b,lambda,x)

Compute CDF of noncentral beta if lambda >= 54 using: First $\lambda/2$ is calculated and the Poisson term is calculated using $P(j-1) = j/\lambda P(j)$ and $P(j+1) = \lambda/(j+1) P(j)$. Then backward recurrences are used until either the Poisson weights fall below errmax or iterlo is reached.

\[I_{x}(a+j-1,b) = I_{x}(a+j,b) + \Gamma(a+b+j-1)/\Gamma(a+j)\Gamma(b)x^{a+j-1}(1-x)^{b}\]

Then forward recurrences are used until error bound falls below errmax.

\[I_{x}(a+j+1,b) = I_{x}(a+j,b) - \Gamma(a+b+j)/\Gamma(a+j)\Gamma(b)x^{a+j}(1-x)^{b}\]

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SpecialFunctions.ncbeta_tail — Function
ncbeta_tail(x,a,b,lambda)

Compute tail of the noncentral beta distribution. Uses the recursive relation

\[I_{x}(a,b+1;0) = I_{x}(a,b;0) - \Gamma(a+b)/\Gamma(a+1)\Gamma(b) x^a (1-x)^b\]

and $\Gamma(a+1) = a\Gamma(a)$ given in DLMF 8.17.21.

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SpecialFunctions.beta_inc_power_series1 — Function
beta_inc_power_series1(a,b,x,epps)

Another variant of BPSER(A,B,X,EPS).

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

APSER(A,B,X,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_inc_power_series2 — Function
beta_inc_power_series2(a,b,x,epps)

Variant of BPSER(A,B,X,EPS).

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

FPSER(A,B,X,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_inc_asymptotic_asymmetric — Function
beta_inc_asymptotic_asymmetric(a, b, x, y, w, epps)

Evaluation of $I_{x}(a,b)$ using asymptotic expansion. It is assumed a >= 15 and b <= 1, and epps is tolerance used.

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

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SpecialFunctions.beta_inc_asymptotic_symmetric — Function
beta_inc_asymptotic_symmetric(a,b,lambda,epps)

Compute $I_{x}(a,b)$ using asymptotic expansion for a, b >= 15. It is assumed that $\lambda = (a+b)*y - b$.

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

BASYM(A,B,LAMBDA,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_inc_power_series — Function
beta_inc_power_series(a, b, x, epps)

Computes $I_x(a,b)$ using power series:

\[I_{x}(a,b) = G(a,b) x^{a}/a \left[1 + a \sum_{j=1}^{\infty} ((1-b)(2-b)\dots(j-b)/j!(a+j)) x^{j}\right]\]

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

BPSER(A,B,X,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_inc_diff — Function
beta_inc_diff(a, b, x, y, n, epps)

Compute $I_{x}(a,b) - I_{x}(a+n,b)$ where n is positive integer and epps is tolerance. A generalised version of DLMF 8.17.20.

External links: DLMF 8.17.20, Wikipedia

See also: beta_inc

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SpecialFunctions.beta_inc_cont_fraction — Function
beta_inc_cont_fraction(a,b,x,y,lambda,epps)

Compute $I_{x}(a,b)$ using continued fraction expansion when a, b > 1. It is assumed that $\lambda = (a+b)*y - b$

External links: DLMF 8.17.22, Wikipedia

See also: beta_inc

Implementation

BFRAC(A,B,X,Y,LAMBDA,EPS) from Didonato and Morris (1982)

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SpecialFunctions.beta_integrand — Function
beta_integrand(a, b, x, y, mu=0.0)

Compute $e^{\mu} x^a y^b / B(a,b)$

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Utilities

SpecialFunctions.chepolsum — Function
chepolsum(n,x,a)

Computes a series of Chebyshev Polynomials given by: a[1]/2 + a[2]T1(x) + .... + a[n]T{n-1}(X).

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SpecialFunctions.lambdaeta — Function
lambdaeta(eta)

Compute the value of $\lambda$ satisfying $\eta^{2}/2 = \lambda-1-\log{\lambda}$.

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SpecialFunctions.stirling_corr — Function
stirling_corr(a0,b0)

Compute stirling(a0) + stirling(b0) - stirling(a0 + b0) for a0, b0 >= 8

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SpecialFunctions.stirling_error — Function
stirling_error(x)

Compute $\ln{\Gamma(x)} - (x-0.5)*\ln{x} + x - \ln{(2\pi)}/2$. Adapted from stirling in IncgamFI.

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SpecialFunctions.esum — Function
esum(mu,x)

Compute $e^{\mu+x}$

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SpecialFunctions.coeff1 — Function

Computing the first coefficient for the expansion:

\[\epsilon (\eta_{0},a) = \epsilon_{1}(\eta_{0},a)/a + \epsilon_{2}(\eta_{0},a)/a^{2} + \epsilon_{3}(\eta_{0},a)/a^{3}\]

Refer Eqn (3.12) in the paper

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SpecialFunctions.coeff2 — Function

Computing the second coefficient for the expansion:

\[\epsilon (\eta_{0},a) = \epsilon_{1} (\eta_{0},a)/a + \epsilon_{2} (\eta_{0},a)/a^{2} + \epsilon_{3} (\eta_{0},a)/a^{3}\]

Refer Eqn (3.12) in the paper

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SpecialFunctions.coeff3 — Function

Computing the third and last coefficient for the expansion:

\[\epsilon(\eta_{0},a) = \epsilon_{1}(\eta_{0},a)/a + \epsilon_{2}(\eta_{0},a)/a^{2} + \epsilon_{3}(\eta_{0},a)/a^{3}\]

Refer Eqn (3.12) in the paper

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SpecialFunctions.ncF — Function
ncF(x,v1,v2,lambda)

Compute CDF of noncentral F distribution given by:

\[F(x, v_1, v_2; \lambda) = I_{v_1 x/(v_1 x + v_2)}(v_1/2, v_2/2; \lambda)\]

where $I_{x}(a,b; \lambda)$ is the noncentral beta function computed above.

External links: Wikipedia

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Exponential and Trigonometric Integrals

SpecialFunctions.expint — Function
expint(z)
+expint(ν, z)

Computes the exponential integral

\[\operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t.\]

If $\nu$ is not specified, $\nu=1$ is used. Arbitrary complex $\nu$ and $z$ are supported.

External links: DLMF 8.19, Wikipedia

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SpecialFunctions.expinti — Function
expinti(x::Real)

Computes the exponential integral function

\[\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t,\]

which is equivalent to $-\Re[\operatorname{E}_1(-x)]$ where $\operatorname{E}_1$ is the expint function.

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SpecialFunctions.expintx — Function
expintx(z)
+expintx(ν, z)

Computes the scaled exponential integral

\[\exp(z) \operatorname{E}_\nu(z) = e^z \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t.\]

If $\nu$ is not specified, $\nu = 1$ is used. Arbitrary complex $\nu$ and $z$ are supported.

See also: expint(ν, z)

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SpecialFunctions.sinint — Function
sinint(x)

Compute the sine integral function of $x$, defined by

\[\operatorname{Si}(x) := \int_0^x \frac{\sin t}{t} \, \mathrm{d}t \quad \text{for} \quad -x \in \mathbb{R} \,.\]

External links: DLMF 6.2.9, Wikipedia.

See also: cosint(x).

Implementation

Using the rational approximants tabulated in:

A.J. MacLeod, "Rational approximations, software and test methods for sine and cosine integrals", Numer. Algor. 12, pp. 259–272 (1996). https://doi.org/10.1007/BF02142806, https://link.springer.com/article/10.1007/BF02142806.

Note: the second zero of $\text{Ci}(x)$ has a typo that is fixed: $r_1 = 3.38418 0422\mathbf{8} 51186 42639 78511 46402$ in the article, but is in fact: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

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SpecialFunctions.cosint — Function
cosint(x)

Compute the cosine integral function of $x$, defined by

\[\operatorname{Ci}(x) +x \in \mathbb{R} \,.\]

External links: DLMF 6.2.9, Wikipedia.

See also: cosint(x).

Implementation

Using the rational approximants tabulated in:

A.J. MacLeod, "Rational approximations, software and test methods for sine and cosine integrals", Numer. Algor. 12, pp. 259–272 (1996). https://doi.org/10.1007/BF02142806, https://link.springer.com/article/10.1007/BF02142806.

Note: the second zero of $\text{Ci}(x)$ has a typo that is fixed: $r_1 = 3.38418 0422\mathbf{8} 51186 42639 78511 46402$ in the article, but is in fact: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

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SpecialFunctions.cosint — Function
cosint(x)

Compute the cosine integral function of $x$, defined by

\[\operatorname{Ci}(x) := \gamma + \log x + \int_0^x \frac{\cos (t) - 1}{t} \, \mathrm{d}t \quad \text{for} \quad -x > 0 \,,\]

where $\gamma$ is the Euler-Mascheroni constant.

External links: DLMF 6.2.13, Wikipedia.

See also: sinint(x).

Implementation

Using the rational approximants tabulated in:

A.J. MacLeod, "Rational approximations, software and test methods for sine and cosine integrals", Numer. Algor. 12, pp. 259–272 (1996). https://doi.org/10.1007/BF02142806, https://link.springer.com/article/10.1007/BF02142806.

Note: the second zero of $\text{Ci}(x)$ has a typo that is fixed: $r_1 = 3.38418 0422\mathbf{8} 51186 42639 78511 46402$ in the article, but is in fact: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

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Error Functions, Dawson’s and Fresnel Integrals

SpecialFunctions.erf — Function
erf(x)

Compute the error function of $x$, defined by

\[\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp(-t^2) \; \mathrm{d}t -\quad \text{for} \quad x \in \mathbb{C} \, .\]

External links: DLMF 7.2.1, Wikipedia.

See also: erfc(x), erfcx(x), erfi(x), dawson(x), erfinv(x), erfcinv(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: C library for multiple-precision floating-point MPFR
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SpecialFunctions.erf — Method
erf(x, y)

Accurate version of erf(y) - erf(x) (for real arguments only).

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SpecialFunctions.erfc — Function
erfc(x)

Compute the complementary error function of $x$, defined by

\[\operatorname{erfc}(x) +x > 0 \,,\]

where $\gamma$ is the Euler-Mascheroni constant.

External links: DLMF 6.2.13, Wikipedia.

See also: sinint(x).

Implementation

Using the rational approximants tabulated in:

A.J. MacLeod, "Rational approximations, software and test methods for sine and cosine integrals", Numer. Algor. 12, pp. 259–272 (1996). https://doi.org/10.1007/BF02142806, https://link.springer.com/article/10.1007/BF02142806.

Note: the second zero of $\text{Ci}(x)$ has a typo that is fixed: $r_1 = 3.38418 0422\mathbf{8} 51186 42639 78511 46402$ in the article, but is in fact: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

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Error Functions, Dawson’s and Fresnel Integrals

SpecialFunctions.erf — Function
erf(x)

Compute the error function of $x$, defined by

\[\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp(-t^2) \; \mathrm{d}t +\quad \text{for} \quad x \in \mathbb{C} \, .\]

External links: DLMF 7.2.1, Wikipedia.

See also: erfc(x), erfcx(x), erfi(x), dawson(x), erfinv(x), erfcinv(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: C library for multiple-precision floating-point MPFR
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SpecialFunctions.erf — Method
erf(x, y)

Accurate version of erf(y) - erf(x) (for real arguments only).

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SpecialFunctions.erfc — Function
erfc(x)

Compute the complementary error function of $x$, defined by

\[\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty \exp(-t^2) \; \mathrm{d}t -\quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of 1-erf(x) for large $x$.

External links: DLMF 7.2.2, Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: C library for multiple-precision floating-point MPFR
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SpecialFunctions.logerf — Function
logerf(x, y)

Compute the natural logarithm of two-argument error function. This is an accurate version of log(erf(x, y)), which works for large x, y.

External links: Wikipedia.

See also: erf(x,y).

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SpecialFunctions.erfcinv — Function
erfcinv(x)

Compute the inverse error complementary function of a real $x$, that is

\[\operatorname{erfcinv}(x) = \operatorname{erfc}^{-1}(x) -\quad \text{for} \quad x \in \mathbb{R} \, .\]

External links: Wikipedia.

See also: erfc(x).

Implementation

Using the rational approximants tabulated in:

J. M. Blair, C. A. Edwards, and J. H. Johnson, "Rational Chebyshev approximations for the inverse of the error function", Math. Comp. 30, pp. 827–830 (1976). https://doi.org/10.1090/S0025-5718-1976-0421040-7, http://www.jstor.org/stable/2005402

combined with Newton iterations for BigFloat.

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SpecialFunctions.erfcx — Function
erfcx(x)

Compute the scaled complementary error function of $x$, defined by

\[\operatorname{erfcx}(x) +\quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of 1-erf(x) for large $x$.

External links: DLMF 7.2.2, Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: C library for multiple-precision floating-point MPFR
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SpecialFunctions.logerf — Function
logerf(x, y)

Compute the natural logarithm of two-argument error function. This is an accurate version of log(erf(x, y)), which works for large x, y.

External links: Wikipedia.

See also: erf(x,y).

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SpecialFunctions.erfcinv — Function
erfcinv(x)

Compute the inverse error complementary function of a real $x$, that is

\[\operatorname{erfcinv}(x) = \operatorname{erfc}^{-1}(x) +\quad \text{for} \quad x \in \mathbb{R} \, .\]

External links: Wikipedia.

See also: erfc(x).

Implementation

Using the rational approximants tabulated in:

J. M. Blair, C. A. Edwards, and J. H. Johnson, "Rational Chebyshev approximations for the inverse of the error function", Math. Comp. 30, pp. 827–830 (1976). https://doi.org/10.1090/S0025-5718-1976-0421040-7, http://www.jstor.org/stable/2005402

combined with Newton iterations for BigFloat.

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SpecialFunctions.erfcx — Function
erfcx(x)

Compute the scaled complementary error function of $x$, defined by

\[\operatorname{erfcx}(x) = e^{x^2} \operatorname{erfc}(x) -\quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of $e^{x^2} \operatorname{erfc}(x)$ for large $x$. Note also that $\operatorname{erfcx}(-ix)$ computes the Faddeeva function w(x).

External links: DLMF 7.2.3, Wikipedia.

See also: erfc(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: MPFR has an open TODO item for this function until then, we use DLMF 7.12.1 for the tail.
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SpecialFunctions.logerfc — Function
logerfc(x)

Compute the natural logarithm of the complementary error function of $x$, that is

\[\operatorname{logerfc}(x) = \ln(\operatorname{erfc}(x)) -\quad \text{for} \quad x \in \mathbb{R} \, .\]

This is the accurate version of $\ln(\operatorname{erfc}(x))$ for large $x$.

External links: Wikipedia.

See also: erfcx(x).

Implementation

Based on the erfc(x) and erfcx(x) functions.

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SpecialFunctions.logerfcx — Function
logerfcx(x)

Compute the natural logarithm of the scaled complementary error function of $x$, that is

\[\operatorname{logerfcx}(x) = \ln(\operatorname{erfcx}(x)) -\quad \text{for} \quad x \in \mathbb{R} \, .\]

This is the accurate version of $\ln(\operatorname{erfcx}(x))$ for large and negative $x$.

External links: Wikipedia.

See also: erfcx(x).

Implementation

Based on the erfc(x) and erfcx(x) functions.

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SpecialFunctions.erfi — Function
erfi(x)

Compute the imaginary error function of $x$, defined by

\[\operatorname{erfi}(x) +\quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of $e^{x^2} \operatorname{erfc}(x)$ for large $x$. Note also that $\operatorname{erfcx}(-ix)$ computes the Faddeeva function w(x).

External links: DLMF 7.2.3, Wikipedia.

See also: erfc(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: MPFR has an open TODO item for this function until then, we use DLMF 7.12.1 for the tail.
source
SpecialFunctions.logerfc — Function
logerfc(x)

Compute the natural logarithm of the complementary error function of $x$, that is

\[\operatorname{logerfc}(x) = \ln(\operatorname{erfc}(x)) +\quad \text{for} \quad x \in \mathbb{R} \, .\]

This is the accurate version of $\ln(\operatorname{erfc}(x))$ for large $x$.

External links: Wikipedia.

See also: erfcx(x).

Implementation

Based on the erfc(x) and erfcx(x) functions.

source
SpecialFunctions.logerfcx — Function
logerfcx(x)

Compute the natural logarithm of the scaled complementary error function of $x$, that is

\[\operatorname{logerfcx}(x) = \ln(\operatorname{erfcx}(x)) +\quad \text{for} \quad x \in \mathbb{R} \, .\]

This is the accurate version of $\ln(\operatorname{erfcx}(x))$ for large and negative $x$.

External links: Wikipedia.

See also: erfcx(x).

Implementation

Based on the erfc(x) and erfcx(x) functions.

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SpecialFunctions.erfi — Function
erfi(x)

Compute the imaginary error function of $x$, defined by

\[\operatorname{erfi}(x) = -i \operatorname{erf}(ix) -\quad \text{for} \quad x \in \mathbb{C} \, .\]

External links: Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
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SpecialFunctions.erfinv — Function
erfinv(x)

Compute the inverse error function of a real $x$, that is

\[\operatorname{erfinv}(x) = \operatorname{erf}^{-1}(x) -\quad \text{for} \quad x \in \mathbb{R} \, .\]

External links: Wikipedia.

See also: erf(x).

Implementation

Using the rational approximants tabulated in:

J. M. Blair, C. A. Edwards, and J. H. Johnson, "Rational Chebyshev approximations for the inverse of the error function", Math. Comp. 30, pp. 827–830 (1976). https://doi.org/10.1090/S0025-5718-1976-0421040-7, http://www.jstor.org/stable/2005402

combined with Newton iterations for BigFloat.

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SpecialFunctions.dawson — Function
dawson(x)

Compute the Dawson function (scaled imaginary error function) of $x$, defined by

\[\operatorname{dawson}(x) +\quad \text{for} \quad x \in \mathbb{C} \, .\]

External links: Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
source
SpecialFunctions.erfinv — Function
erfinv(x)

Compute the inverse error function of a real $x$, that is

\[\operatorname{erfinv}(x) = \operatorname{erf}^{-1}(x) +\quad \text{for} \quad x \in \mathbb{R} \, .\]

External links: Wikipedia.

See also: erf(x).

Implementation

Using the rational approximants tabulated in:

J. M. Blair, C. A. Edwards, and J. H. Johnson, "Rational Chebyshev approximations for the inverse of the error function", Math. Comp. 30, pp. 827–830 (1976). https://doi.org/10.1090/S0025-5718-1976-0421040-7, http://www.jstor.org/stable/2005402

combined with Newton iterations for BigFloat.

source
SpecialFunctions.dawson — Function
dawson(x)

Compute the Dawson function (scaled imaginary error function) of $x$, defined by

\[\operatorname{dawson}(x) = \frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x) -\quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of $\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)$ for large $x$.

External links: DLMF 7.2.5, Wikipedia.

See also: erfi(x).

Implementation by

  • Float32/Float64: C standard math library libm.
source
SpecialFunctions.faddeeva — Function
faddeeva(z)

Compute the Faddeeva function of complex z, defined by $e^{-z^2} \operatorname{erfc}(-iz)$. Note that this function, also named w (original Faddeeva package) or wofz (Scilab package), is equivalent to $\operatorname{erfcx}(-iz)$.

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SpecialFunctions.airyai — Function
airyai(x)

Airy function of the first kind $\operatorname{Ai}(x)$.

External links: DLMF 9.2, Wikipedia

See also: airyaix, airyaiprime, airybi

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SpecialFunctions.airyaiprime — Function
airyaiprime(x)

Derivative of the Airy function of the first kind $\operatorname{Ai}'(x)$.

External links: DLMF 9.2, Wikipedia

See also: airyaiprimex, airyai, airybi

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SpecialFunctions.airybi — Function
airybi(x)

Airy function of the second kind $\operatorname{Bi}(x)$.

External links: DLMF 8.2, Wikipedia

See also: airybix, airybiprime, airyai

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SpecialFunctions.airybiprime — Function
airybiprime(x)

Derivative of the Airy function of the second kind $\operatorname{Bi}'(x)$.

External links: DLMF 9.2, Wikipedia

See also: airybiprimex, airybi, airyai

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SpecialFunctions.airyaix — Function
airyaix(x)

Scaled Airy function of the first kind $\operatorname{Ai}(x) e^{\frac{2}{3} x \sqrt{x}}$. Throws DomainError for negative Real arguments.

External links: DLMF 9.2, Wikipedia

See also: airyai, airyaiprime, airybi

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SpecialFunctions.airyaiprimex — Function
airyaiprimex(x)

Scaled derivative of the Airy function of the first kind $\operatorname{Ai}'(x) e^{\frac{2}{3} x \sqrt{x}}$. Throws DomainError for negative Real arguments.

External links: DLMF 9.2, Wikipedia

See also: airyaiprime, airyai, airybi

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SpecialFunctions.airybix — Function
airybix(x)

Scaled Airy function of the second kind $\operatorname{Bi}(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$.

External links: DLMF 9.2, Wikipedia

See also: airybi, airybiprime, airyai

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SpecialFunctions.airybiprimex — Function
airybiprimex(x)

Scaled derivative of the Airy function of the second kind $\operatorname{Bi}'(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$.

External links: DLMF 9.2, Wikipedia

See also: airybiprime, airybi, airyai

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Bessel Functions

SpecialFunctions.besselj — Function
besselj(nu, x)

Bessel function of the first kind of order nu, $J_\nu(x)$.

External links: DLMF 10.2.2, Wikipedia

See also: besseljx(nu,x), besseli(nu,x), besselk(nu,x)

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SpecialFunctions.besselj0 — Function
besselj0(x)

Bessel function of the first kind of order 0, $J_0(x)$.

External links: DLMF 10.2.2, Wikipedia

See also: besselj(nu,x)

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SpecialFunctions.besselj1 — Function
besselj1(x)

Bessel function of the first kind of order 1, $J_1(x)$.

External links: DLMF 10.2.2, Wikipedia

See also: besselj(nu,x)

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SpecialFunctions.besseljx — Function
besseljx(nu, x)

Scaled Bessel function of the first kind of order nu, $J_\nu(x) e^{- | \operatorname{Im}(x) |}$.

External links: DLMF 10.2.2, Wikipedia

See also: besselj(nu,x), besseli(nu,x), besselk(nu,x)

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SpecialFunctions.sphericalbesselj — Function
sphericalbesselj(nu, x)

Spherical bessel function of the first kind at order nu, $j_ν(x)$. This is the non-singular solution to the radial part of the Helmholz equation in spherical coordinates.

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SpecialFunctions.bessely — Function
bessely(nu, x)

Bessel function of the second kind of order nu, $Y_\nu(x)$.

External links: DLMF 10.2.3, Wikipedia

See also besselyx(nu,x) for a scaled variant.

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SpecialFunctions.bessely0 — Function
bessely0(x)

Bessel function of the second kind of order 0, $Y_0(x)$.

External links: DLMF 10.2.3, Wikipedia

See also: bessely(nu,x)

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SpecialFunctions.bessely1 — Function
bessely1(x)

Bessel function of the second kind of order 1, $Y_1(x)$.

External links: DLMF 10.2.3, Wikipedia

See also: bessely(nu,x)

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SpecialFunctions.besselyx — Function
besselyx(nu, x)

Scaled Bessel function of the second kind of order nu, $Y_\nu(x) e^{- | \operatorname{Im}(x) |}$.

External links: DLMF 10.2.3, Wikipedia

See also bessely(nu,x)

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SpecialFunctions.sphericalbessely — Function
sphericalbessely(nu, x)

Spherical bessel function of the second kind at order nu, $y_ν(x)$. This is the singular solution to the radial part of the Helmholz equation in spherical coordinates. Sometimes known as a spherical Neumann function.

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SpecialFunctions.besselh — Function
besselh(nu, [k=1,] x)

Bessel function of the third kind of order nu (the Hankel function). k is either 1 or 2, selecting hankelh1 or hankelh2, respectively. k defaults to 1 if it is omitted.

External links: DLMF 10.2.5 and DLMF 10.2.6, Wikipedia

See also: besselhx for an exponentially scaled variant.

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SpecialFunctions.besselhx — Function
besselhx(nu, [k=1,] z)

Compute the scaled Hankel function $\exp(∓iz) H_ν^{(k)}(z)$, where $k$ is 1 or 2, $H_ν^{(k)}(z)$ is besselh(nu, k, z), and $∓$ is $-$ for $k=1$ and $+$ for $k=2$. k defaults to 1 if it is omitted.

The reason for this function is that $H_ν^{(k)}(z)$ is asymptotically proportional to $\exp(∓iz)/\sqrt{z}$ for large $|z|$, and so the besselh function is susceptible to overflow or underflow when z has a large imaginary part. The besselhx function cancels this exponential factor (analytically), so it avoids these problems.

External links: DLMF 10.2, Wikipedia

See also: besselh

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SpecialFunctions.hankelh1 — Function
hankelh1(nu, x)

Bessel function of the third kind of order nu, $H^{(1)}_\nu(x)$.

External links: DLMF 10.2.5, Wikipedia

See also: hankelh1x

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SpecialFunctions.hankelh1x — Function
hankelh1x(nu, x)

Scaled Bessel function of the third kind of order nu, $H^{(1)}_\nu(x) e^{-x i}$.

External links: DLMF 10.2.5, Wikipedia

See also: hankelh1

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SpecialFunctions.hankelh2 — Function
hankelh2(nu, x)

Bessel function of the third kind of order nu, $H^{(2)}_\nu(x)$.

External links: DLMF 10.2.6, Wikipedia

See also: hankelh2x(nu,x)

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SpecialFunctions.hankelh2x — Function
hankelh2x(nu, x)

Scaled Bessel function of the third kind of order nu, $H^{(2)}_\nu(x) e^{x i}$.

External links: DLMF 10.2.6, Wikipedia

See also: hankelh2(nu,x)

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SpecialFunctions.besseli — Function
besseli(nu, x)

Modified Bessel function of the first kind of order nu, $I_\nu(x)$.

External links: DLMF 10.25.2, Wikipedia

See also: besselix(nu,x), besselj(nu,x), besselk(nu,x)

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SpecialFunctions.besselix — Function
besselix(nu, x)

Scaled modified Bessel function of the first kind of order nu, $I_\nu(x) e^{- | \operatorname{Re}(x) |}$.

External links: DLMF 10.25.2, Wikipedia

See also: besseli(nu,x), besselj(nu,x), besselk(nu,x)

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SpecialFunctions.besselk — Function
besselk(nu, x)

Modified Bessel function of the second kind of order nu, $K_\nu(x)$.

External links: DLMF 10.25.3, Wikipedia

See also: besselkx(nu,x), besseli(nu,x), besselj(nu,x)

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SpecialFunctions.besselkx — Function
besselkx(nu, x)

Scaled modified Bessel function of the second kind of order nu, $K_\nu(x) e^x$.

External links: DLMF 10.25.3, Wikipedia

See also: besselk(nu,x), besseli(nu,x), besselj(nu,x)

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SpecialFunctions.jinc — Function
jinc(x)

Bessel function of the first kind divided by x. Following convention:

\[\operatorname{jinc}{x} = \frac{2 J_1({\pi x})}{\pi x}.\]

Sometimes known as sombrero or besinc function.

External links: Wikipedia

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Elliptic Integrals

SpecialFunctions.ellipk — Function
ellipk(m)

Computes Complete Elliptic Integral of 1st kind $K(m)$ for parameter $m$ given by

\[\operatorname{ellipk}(m) +\quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of $\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)$ for large $x$.

External links: DLMF 7.2.5, Wikipedia.

See also: erfi(x).

Implementation by

  • Float32/Float64: C standard math library libm.
source
SpecialFunctions.faddeeva — Function
faddeeva(z)

Compute the Faddeeva function of complex z, defined by $e^{-z^2} \operatorname{erfc}(-iz)$. Note that this function, also named w (original Faddeeva package) or wofz (Scilab package), is equivalent to $\operatorname{erfcx}(-iz)$.

source
SpecialFunctions.airyai — Function
airyai(x)

Airy function of the first kind $\operatorname{Ai}(x)$.

External links: DLMF 9.2, Wikipedia

See also: airyaix, airyaiprime, airybi

source
SpecialFunctions.airyaiprime — Function
airyaiprime(x)

Derivative of the Airy function of the first kind $\operatorname{Ai}'(x)$.

External links: DLMF 9.2, Wikipedia

See also: airyaiprimex, airyai, airybi

source
SpecialFunctions.airybi — Function
airybi(x)

Airy function of the second kind $\operatorname{Bi}(x)$.

External links: DLMF 8.2, Wikipedia

See also: airybix, airybiprime, airyai

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SpecialFunctions.airybiprime — Function
airybiprime(x)

Derivative of the Airy function of the second kind $\operatorname{Bi}'(x)$.

External links: DLMF 9.2, Wikipedia

See also: airybiprimex, airybi, airyai

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SpecialFunctions.airyaix — Function
airyaix(x)

Scaled Airy function of the first kind $\operatorname{Ai}(x) e^{\frac{2}{3} x \sqrt{x}}$. Throws DomainError for negative Real arguments.

External links: DLMF 9.2, Wikipedia

See also: airyai, airyaiprime, airybi

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SpecialFunctions.airyaiprimex — Function
airyaiprimex(x)

Scaled derivative of the Airy function of the first kind $\operatorname{Ai}'(x) e^{\frac{2}{3} x \sqrt{x}}$. Throws DomainError for negative Real arguments.

External links: DLMF 9.2, Wikipedia

See also: airyaiprime, airyai, airybi

source
SpecialFunctions.airybix — Function
airybix(x)

Scaled Airy function of the second kind $\operatorname{Bi}(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$.

External links: DLMF 9.2, Wikipedia

See also: airybi, airybiprime, airyai

source
SpecialFunctions.airybiprimex — Function
airybiprimex(x)

Scaled derivative of the Airy function of the second kind $\operatorname{Bi}'(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$.

External links: DLMF 9.2, Wikipedia

See also: airybiprime, airybi, airyai

source

Bessel Functions

SpecialFunctions.besselj — Function
besselj(nu, x)

Bessel function of the first kind of order nu, $J_\nu(x)$.

External links: DLMF 10.2.2, Wikipedia

See also: besseljx(nu,x), besseli(nu,x), besselk(nu,x)

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SpecialFunctions.besselj0 — Function
besselj0(x)

Bessel function of the first kind of order 0, $J_0(x)$.

External links: DLMF 10.2.2, Wikipedia

See also: besselj(nu,x)

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SpecialFunctions.besselj1 — Function
besselj1(x)

Bessel function of the first kind of order 1, $J_1(x)$.

External links: DLMF 10.2.2, Wikipedia

See also: besselj(nu,x)

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SpecialFunctions.besseljx — Function
besseljx(nu, x)

Scaled Bessel function of the first kind of order nu, $J_\nu(x) e^{- | \operatorname{Im}(x) |}$.

External links: DLMF 10.2.2, Wikipedia

See also: besselj(nu,x), besseli(nu,x), besselk(nu,x)

source
SpecialFunctions.sphericalbesselj — Function
sphericalbesselj(nu, x)

Spherical bessel function of the first kind at order nu, $j_ν(x)$. This is the non-singular solution to the radial part of the Helmholz equation in spherical coordinates.

source
SpecialFunctions.bessely — Function
bessely(nu, x)

Bessel function of the second kind of order nu, $Y_\nu(x)$.

External links: DLMF 10.2.3, Wikipedia

See also besselyx(nu,x) for a scaled variant.

source
SpecialFunctions.bessely0 — Function
bessely0(x)

Bessel function of the second kind of order 0, $Y_0(x)$.

External links: DLMF 10.2.3, Wikipedia

See also: bessely(nu,x)

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SpecialFunctions.bessely1 — Function
bessely1(x)

Bessel function of the second kind of order 1, $Y_1(x)$.

External links: DLMF 10.2.3, Wikipedia

See also: bessely(nu,x)

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SpecialFunctions.besselyx — Function
besselyx(nu, x)

Scaled Bessel function of the second kind of order nu, $Y_\nu(x) e^{- | \operatorname{Im}(x) |}$.

External links: DLMF 10.2.3, Wikipedia

See also bessely(nu,x)

source
SpecialFunctions.sphericalbessely — Function
sphericalbessely(nu, x)

Spherical bessel function of the second kind at order nu, $y_ν(x)$. This is the singular solution to the radial part of the Helmholz equation in spherical coordinates. Sometimes known as a spherical Neumann function.

source
SpecialFunctions.besselh — Function
besselh(nu, [k=1,] x)

Bessel function of the third kind of order nu (the Hankel function). k is either 1 or 2, selecting hankelh1 or hankelh2, respectively. k defaults to 1 if it is omitted.

External links: DLMF 10.2.5 and DLMF 10.2.6, Wikipedia

See also: besselhx for an exponentially scaled variant.

source
SpecialFunctions.besselhx — Function
besselhx(nu, [k=1,] z)

Compute the scaled Hankel function $\exp(∓iz) H_ν^{(k)}(z)$, where $k$ is 1 or 2, $H_ν^{(k)}(z)$ is besselh(nu, k, z), and $∓$ is $-$ for $k=1$ and $+$ for $k=2$. k defaults to 1 if it is omitted.

The reason for this function is that $H_ν^{(k)}(z)$ is asymptotically proportional to $\exp(∓iz)/\sqrt{z}$ for large $|z|$, and so the besselh function is susceptible to overflow or underflow when z has a large imaginary part. The besselhx function cancels this exponential factor (analytically), so it avoids these problems.

External links: DLMF 10.2, Wikipedia

See also: besselh

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SpecialFunctions.hankelh1 — Function
hankelh1(nu, x)

Bessel function of the third kind of order nu, $H^{(1)}_\nu(x)$.

External links: DLMF 10.2.5, Wikipedia

See also: hankelh1x

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SpecialFunctions.hankelh1x — Function
hankelh1x(nu, x)

Scaled Bessel function of the third kind of order nu, $H^{(1)}_\nu(x) e^{-x i}$.

External links: DLMF 10.2.5, Wikipedia

See also: hankelh1

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SpecialFunctions.hankelh2 — Function
hankelh2(nu, x)

Bessel function of the third kind of order nu, $H^{(2)}_\nu(x)$.

External links: DLMF 10.2.6, Wikipedia

See also: hankelh2x(nu,x)

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SpecialFunctions.hankelh2x — Function
hankelh2x(nu, x)

Scaled Bessel function of the third kind of order nu, $H^{(2)}_\nu(x) e^{x i}$.

External links: DLMF 10.2.6, Wikipedia

See also: hankelh2(nu,x)

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SpecialFunctions.besseli — Function
besseli(nu, x)

Modified Bessel function of the first kind of order nu, $I_\nu(x)$.

External links: DLMF 10.25.2, Wikipedia

See also: besselix(nu,x), besselj(nu,x), besselk(nu,x)

source
SpecialFunctions.besselix — Function
besselix(nu, x)

Scaled modified Bessel function of the first kind of order nu, $I_\nu(x) e^{- | \operatorname{Re}(x) |}$.

External links: DLMF 10.25.2, Wikipedia

See also: besseli(nu,x), besselj(nu,x), besselk(nu,x)

source
SpecialFunctions.besselk — Function
besselk(nu, x)

Modified Bessel function of the second kind of order nu, $K_\nu(x)$.

External links: DLMF 10.25.3, Wikipedia

See also: besselkx(nu,x), besseli(nu,x), besselj(nu,x)

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SpecialFunctions.besselkx — Function
besselkx(nu, x)

Scaled modified Bessel function of the second kind of order nu, $K_\nu(x) e^x$.

External links: DLMF 10.25.3, Wikipedia

See also: besselk(nu,x), besseli(nu,x), besselj(nu,x)

source
SpecialFunctions.jinc — Function
jinc(x)

Bessel function of the first kind divided by x. Following convention:

\[\operatorname{jinc}{x} = \frac{2 J_1({\pi x})}{\pi x}.\]

Sometimes known as sombrero or besinc function.

External links: Wikipedia

source

Elliptic Integrals

SpecialFunctions.ellipk — Function
ellipk(m)

Computes Complete Elliptic Integral of 1st kind $K(m)$ for parameter $m$ given by

\[\operatorname{ellipk}(m) = K(m) = \int_0^{ \frac{\pi}{2} } \frac{1}{\sqrt{1 - m \sin^2 \theta}} \, \mathrm{d}\theta -\quad \text{for} \quad m \in \left( -\infty, 1 \right] \, .\]

External links: DLMF 19.2.8, Wikipedia.

See also: ellipe(m).

Arguments

  • m: parameter $m$, restricted to the domain $(-\infty,1]$, is related to the elliptic modulus $k$ by $k^2=m$ and to the modular angle $\alpha$ by $k = \sin \alpha$.

Implementation

Using piecewise approximation polynomial as given in

'Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions', Fukushima, Toshio. (2014). F09-FastEI. Celest Mech Dyn Astr, DOI 10.1007/s10569-009-9228-z, https://pdfs.semanticscholar.org/8112/c1f56e833476b61fc54d41e194c962fbe647.pdf

For $m<0$, followed by

Fukushima, Toshio. (2014). 'Precise, compact, and fast computation of complete elliptic integrals by piecewise minimax rational function approximation'. Journal of Computational and Applied Mathematics. 282. DOI 10.13140/2.1.1946.6245., https://www.researchgate.net/publication/267330394

As suggested in this paper, the domain is restricted to $(-\infty,1]$.

source
SpecialFunctions.ellipe — Function
ellipe(m)

Computes Complete Elliptic Integral of 2nd kind $E(m)$ for parameter $m$ given by

\[\operatorname{ellipe}(m) +\quad \text{for} \quad m \in \left( -\infty, 1 \right] \, .\]

External links: DLMF 19.2.8, Wikipedia.

See also: ellipe(m).

Arguments

  • m: parameter $m$, restricted to the domain $(-\infty,1]$, is related to the elliptic modulus $k$ by $k^2=m$ and to the modular angle $\alpha$ by $k = \sin \alpha$.

Implementation

Using piecewise approximation polynomial as given in

'Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions', Fukushima, Toshio. (2014). F09-FastEI. Celest Mech Dyn Astr, DOI 10.1007/s10569-009-9228-z, https://pdfs.semanticscholar.org/8112/c1f56e833476b61fc54d41e194c962fbe647.pdf

For $m<0$, followed by

Fukushima, Toshio. (2014). 'Precise, compact, and fast computation of complete elliptic integrals by piecewise minimax rational function approximation'. Journal of Computational and Applied Mathematics. 282. DOI 10.13140/2.1.1946.6245., https://www.researchgate.net/publication/267330394

As suggested in this paper, the domain is restricted to $(-\infty,1]$.

source
SpecialFunctions.ellipe — Function
ellipe(m)

Computes Complete Elliptic Integral of 2nd kind $E(m)$ for parameter $m$ given by

\[\operatorname{ellipe}(m) = E(m) = \int_0^{ \frac{\pi}{2} } \sqrt{1 - m \sin^2 \theta} \, \mathrm{d}\theta -\quad \text{for} \quad m \in \left( -\infty, 1 \right] .\]

External links: DLMF 19.2.8, Wikipedia.

See also: ellipk(m).

Arguments

  • m: parameter $m$, restricted to the domain $(-\infty,1]$, is related to the elliptic modulus $k$ by $k^2=m$ and to the modular angle $\alpha$ by $k=\sin \alpha$.

Implementation

Using piecewise approximation polynomial as given in

'Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions', Fukushima, Toshio. (2014). F09-FastEI. Celest Mech Dyn Astr, DOI 10.1007/s10569-009-9228-z, https://pdfs.semanticscholar.org/8112/c1f56e833476b61fc54d41e194c962fbe647.pdf

For $m<0$, followed by

Fukushima, Toshio. (2014). 'Precise, compact, and fast computation of complete elliptic integrals by piecewise minimax rational function approximation'. Journal of Computational and Applied Mathematics. 282. DOI 10.13140/2.1.1946.6245., https://www.researchgate.net/publication/267330394

As suggested in this paper, the domain is restricted to $(-\infty,1]$.

source
SpecialFunctions.eta — Function
eta(s)

Dirichlet eta function

\[\eta(s) = \sum_{n=1}^\infty (-1)^{n-1} / n^{s}.\]

source
SpecialFunctions.zeta — Function
zeta(s, z)

Generalized zeta function defined by

\[\zeta(s, z) = \sum_{k=0}^\infty \frac{1}{((k+z)^2)^{s/2}},\]

where any term with $k+z = 0$ is excluded. For $\Re z > 0$, this definition is equivalent to the Hurwitz zeta function $\sum_{k=0}^\infty (k+z)^{-s}$.

The Riemann zeta function is recovered as $\zeta(s) = \zeta(s,1)$.

External links: Riemann zeta function, Hurwitz zeta function

source
zeta(s)

Riemann zeta function

\[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}\quad\text{for}\quad s\in\mathbb{C}.\]

External links: Wikipedia

source
+\quad \text{for} \quad m \in \left( -\infty, 1 \right] .\]

External links: DLMF 19.2.8, Wikipedia.

See also: ellipk(m).

Arguments

Implementation

Using piecewise approximation polynomial as given in

'Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions', Fukushima, Toshio. (2014). F09-FastEI. Celest Mech Dyn Astr, DOI 10.1007/s10569-009-9228-z, https://pdfs.semanticscholar.org/8112/c1f56e833476b61fc54d41e194c962fbe647.pdf

For $m<0$, followed by

Fukushima, Toshio. (2014). 'Precise, compact, and fast computation of complete elliptic integrals by piecewise minimax rational function approximation'. Journal of Computational and Applied Mathematics. 282. DOI 10.13140/2.1.1946.6245., https://www.researchgate.net/publication/267330394

As suggested in this paper, the domain is restricted to $(-\infty,1]$.

source
SpecialFunctions.eta — Function
eta(s)

Dirichlet eta function

\[\eta(s) = \sum_{n=1}^\infty (-1)^{n-1} / n^{s}.\]

source
SpecialFunctions.zeta — Function
zeta(s, z)

Generalized zeta function defined by

\[\zeta(s, z) = \sum_{k=0}^\infty \frac{1}{((k+z)^2)^{s/2}},\]

where any term with $k+z = 0$ is excluded. For $\Re z > 0$, this definition is equivalent to the Hurwitz zeta function $\sum_{k=0}^\infty (k+z)^{-s}$.

The Riemann zeta function is recovered as $\zeta(s) = \zeta(s,1)$.

External links: Riemann zeta function, Hurwitz zeta function

source
zeta(s)

Riemann zeta function

\[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}\quad\text{for}\quad s\in\mathbb{C}.\]

External links: Wikipedia

source
diff --git a/dev/functions_overview/index.html b/dev/functions_overview/index.html index 43aa76cd..8b329f36 100644 --- a/dev/functions_overview/index.html +++ b/dev/functions_overview/index.html @@ -1,2 +1,2 @@ -Overview · SpecialFunctions.jl
Edit on GitHub

Functions

Here the Special Functions are listed according to the structure of NIST Digital Library of Mathematical Functions (DLMF).

Gamma Function

Gamma Function - DLMF

FunctionDescription
gamma(z)gamma function $\Gamma(z)$
loggamma(x)accurate log(gamma(x)) for large x
logabsgamma(x)accurate log(abs(gamma(x))) for large x
logfactorial(x)accurate log(factorial(x)) for large x; same as loggamma(x+1) for x > 1, zero otherwise
digamma(x)digamma function (i.e. the derivative of loggamma at x)
invdigamma(x)invdigamma function (i.e. inverse of digamma function at x using fixed-point iteration algorithm)
trigamma(x)trigamma function (i.e the logarithmic second derivative of gamma at x)
polygamma(m,x)polygamma function (i.e the (m+1)-th derivative of the loggamma function at x)
gamma(a,z)upper incomplete gamma function $\Gamma(a,z)$
loggamma(a,z)accurate log(gamma(a,x)) for large arguments
gamma_inc(a,x,IND)incomplete gamma function ratio P(a,x) and Q(a,x) (i.e evaluates P(a,x) and Q(a,x) for accuracy specified by IND and returns tuple (p,q))
gamma_inc_inv(a,p,q)inverse of incomplete gamma function ratio P(a,x) and Q(a,x) (i.e evaluates x given P(a,x)=p and Q(a,x)=q)
beta(x,y)beta function at x,y
logbeta(x,y)accurate log(beta(x,y)) for large x or y
logabsbeta(x,y)accurate log(abs(beta(x,y))) for large x or y
logabsbinomial(x,y)accurate log(abs(binomial(n,k))) for large n and k near n/2
beta_inc(a,b,x,y)incomplete beta function ratio Ix(a,b) and Iy(a,b) (i.e evaluates Ix(a,b) and Iy(a,b) and returns tuple (p,q))
beta_inc_inv(a,b,p,q)Inverse of the incomplete beta function (i.e evaluates x given $I_{x}(a, b) = p$)

Exponential and Trigonometric Integrals

Exponential and Trigonometric Integrals - DLMF

FunctionDescription
expint(ν, z)exponential integral $\operatorname{E}_\nu(z)$
expinti(x)exponential integral $\operatorname{Ei}(x)$
expintx(x)scaled exponential integral $e^z \operatorname{E}_\nu(z)$
sinint(x)sine integral $\operatorname{Si}(x)$
cosint(x)cosine integral $\operatorname{Ci}(x)$

Error Functions, Dawson’s and Fresnel Integrals

Error Functions, Dawson’s and Fresnel Integrals - DLMF

FunctionDescription
erf(x)error function at $x$
erf(x,y)accurate version of $\operatorname{erf}(y) - \operatorname{erf}(x)$
erfc(x)complementary error function, i.e. the accurate version of $1-\operatorname{erf}(x)$ for large $x$
erfcinv(x)inverse function to erfc()
erfcx(x)scaled complementary error function, i.e. accurate $e^{x^2} \operatorname{erfc}(x)$ for large $x$
logerfc(x)log of the complementary error function, i.e. accurate $\operatorname{ln}(\operatorname{erfc}(x))$ for large $x$
logerfcx(x)log of the scaled complementary error function, i.e. accurate $\operatorname{ln}(\operatorname{erfcx}(x))$ for large negative $x$
erfi(x)imaginary error function defined as $-i \operatorname{erf}(ix)$
erfinv(x)inverse function to erf()
dawson(x)scaled imaginary error function, a.k.a. Dawson function, i.e. accurate $\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)$ for large $x$
faddeeva(x)Faddeeva function, equivalent to $\operatorname{erfcx}(-ix)$

Airy and Related Functions - DLMF

FunctionDescription
airyai(z)Airy Ai function at z
airyaiprime(z)derivative of the Airy Ai function at z
airybi(z)Airy Bi function at z
airybiprime(z)derivative of the Airy Bi function at z
airyaix(z), airyaiprimex(z), airybix(z), airybiprimex(z)scaled Airy Ai function and derivative at z

Bessel Functions

Bessel Functions - DLMF

FunctionDescription
besselj(nu,z)Bessel function of the first kind of order nu at z
besselj0(z)besselj(0,z)
besselj1(z)besselj(1,z)
besseljx(nu,z)scaled Bessel function of the first kind of order nu at z
sphericalbesselj(nu,z)Spherical Bessel function of the first kind of order nu at z
bessely(nu,z)Bessel function of the second kind of order nu at z
bessely0(z)bessely(0,z)
bessely1(z)bessely(1,z)
besselyx(nu,z)scaled Bessel function of the second kind of order nu at z
sphericalbessely(nu,z)Spherical Bessel function of the second kind of order nu at z
besselh(nu,k,z)Bessel function of the third kind (a.k.a. Hankel function) of order nu at z; k must be either 1 or 2
hankelh1(nu,z)besselh(nu, 1, z)
hankelh1x(nu,z)scaled besselh(nu, 1, z)
hankelh2(nu,z)besselh(nu, 2, z)
hankelh2x(nu,z)scaled besselh(nu, 2, z)
besseli(nu,z)modified Bessel function of the first kind of order nu at z
besselix(nu,z)scaled modified Bessel function of the first kind of order nu at z
besselk(nu,z)modified Bessel function of the second kind of order nu at z
besselkx(nu,z)scaled modified Bessel function of the second kind of order nu at z
jinc(x)scaled Bessel function of the first kind divided by x. A.k.a. sombrero or besinc

Elliptic Integrals

Elliptic Integrals - DLMF

FunctionDescription
ellipk(m)complete elliptic integral of 1st kind $K(m)$
ellipe(m)complete elliptic integral of 2nd kind $E(m)$

Zeta and Related Functions - DLMF

FunctionDescription
eta(x)Dirichlet eta function at x
zeta(x)Riemann zeta function at x
+Overview · SpecialFunctions.jl
Edit on GitHub

Functions

Here the Special Functions are listed according to the structure of NIST Digital Library of Mathematical Functions (DLMF).

Gamma Function

Gamma Function - DLMF

FunctionDescription
gamma(z)gamma function $\Gamma(z)$
loggamma(x)accurate log(gamma(x)) for large x
logabsgamma(x)accurate log(abs(gamma(x))) for large x
logfactorial(x)accurate log(factorial(x)) for large x; same as loggamma(x+1) for x > 1, zero otherwise
digamma(x)digamma function (i.e. the derivative of loggamma at x)
invdigamma(x)invdigamma function (i.e. inverse of digamma function at x using fixed-point iteration algorithm)
trigamma(x)trigamma function (i.e the logarithmic second derivative of gamma at x)
polygamma(m,x)polygamma function (i.e the (m+1)-th derivative of the loggamma function at x)
gamma(a,z)upper incomplete gamma function $\Gamma(a,z)$
loggamma(a,z)accurate log(gamma(a,x)) for large arguments
gamma_inc(a,x,IND)incomplete gamma function ratio P(a,x) and Q(a,x) (i.e evaluates P(a,x) and Q(a,x) for accuracy specified by IND and returns tuple (p,q))
gamma_inc_inv(a,p,q)inverse of incomplete gamma function ratio P(a,x) and Q(a,x) (i.e evaluates x given P(a,x)=p and Q(a,x)=q)
beta(x,y)beta function at x,y
logbeta(x,y)accurate log(beta(x,y)) for large x or y
logabsbeta(x,y)accurate log(abs(beta(x,y))) for large x or y
logabsbinomial(x,y)accurate log(abs(binomial(n,k))) for large n and k near n/2
beta_inc(a,b,x,y)incomplete beta function ratio Ix(a,b) and Iy(a,b) (i.e evaluates Ix(a,b) and Iy(a,b) and returns tuple (p,q))
beta_inc_inv(a,b,p,q)Inverse of the incomplete beta function (i.e evaluates x given $I_{x}(a, b) = p$)

Exponential and Trigonometric Integrals

Exponential and Trigonometric Integrals - DLMF

FunctionDescription
expint(ν, z)exponential integral $\operatorname{E}_\nu(z)$
expinti(x)exponential integral $\operatorname{Ei}(x)$
expintx(x)scaled exponential integral $e^z \operatorname{E}_\nu(z)$
sinint(x)sine integral $\operatorname{Si}(x)$
cosint(x)cosine integral $\operatorname{Ci}(x)$

Error Functions, Dawson’s and Fresnel Integrals

Error Functions, Dawson’s and Fresnel Integrals - DLMF

FunctionDescription
erf(x)error function at $x$
erf(x,y)accurate version of $\operatorname{erf}(y) - \operatorname{erf}(x)$
erfc(x)complementary error function, i.e. the accurate version of $1-\operatorname{erf}(x)$ for large $x$
erfcinv(x)inverse function to erfc()
erfcx(x)scaled complementary error function, i.e. accurate $e^{x^2} \operatorname{erfc}(x)$ for large $x$
logerfc(x)log of the complementary error function, i.e. accurate $\operatorname{ln}(\operatorname{erfc}(x))$ for large $x$
logerfcx(x)log of the scaled complementary error function, i.e. accurate $\operatorname{ln}(\operatorname{erfcx}(x))$ for large negative $x$
erfi(x)imaginary error function defined as $-i \operatorname{erf}(ix)$
erfinv(x)inverse function to erf()
dawson(x)scaled imaginary error function, a.k.a. Dawson function, i.e. accurate $\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)$ for large $x$
faddeeva(x)Faddeeva function, equivalent to $\operatorname{erfcx}(-ix)$

Airy and Related Functions - DLMF

FunctionDescription
airyai(z)Airy Ai function at z
airyaiprime(z)derivative of the Airy Ai function at z
airybi(z)Airy Bi function at z
airybiprime(z)derivative of the Airy Bi function at z
airyaix(z), airyaiprimex(z), airybix(z), airybiprimex(z)scaled Airy Ai function and derivative at z

Bessel Functions

Bessel Functions - DLMF

FunctionDescription
besselj(nu,z)Bessel function of the first kind of order nu at z
besselj0(z)besselj(0,z)
besselj1(z)besselj(1,z)
besseljx(nu,z)scaled Bessel function of the first kind of order nu at z
sphericalbesselj(nu,z)Spherical Bessel function of the first kind of order nu at z
bessely(nu,z)Bessel function of the second kind of order nu at z
bessely0(z)bessely(0,z)
bessely1(z)bessely(1,z)
besselyx(nu,z)scaled Bessel function of the second kind of order nu at z
sphericalbessely(nu,z)Spherical Bessel function of the second kind of order nu at z
besselh(nu,k,z)Bessel function of the third kind (a.k.a. Hankel function) of order nu at z; k must be either 1 or 2
hankelh1(nu,z)besselh(nu, 1, z)
hankelh1x(nu,z)scaled besselh(nu, 1, z)
hankelh2(nu,z)besselh(nu, 2, z)
hankelh2x(nu,z)scaled besselh(nu, 2, z)
besseli(nu,z)modified Bessel function of the first kind of order nu at z
besselix(nu,z)scaled modified Bessel function of the first kind of order nu at z
besselk(nu,z)modified Bessel function of the second kind of order nu at z
besselkx(nu,z)scaled modified Bessel function of the second kind of order nu at z
jinc(x)scaled Bessel function of the first kind divided by x. A.k.a. sombrero or besinc

Elliptic Integrals

Elliptic Integrals - DLMF

FunctionDescription
ellipk(m)complete elliptic integral of 1st kind $K(m)$
ellipe(m)complete elliptic integral of 2nd kind $E(m)$

Zeta and Related Functions - DLMF

FunctionDescription
eta(x)Dirichlet eta function at x
zeta(x)Riemann zeta function at x
diff --git a/dev/index.html b/dev/index.html index 7e203a1a..e363fee8 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · SpecialFunctions.jl
Edit on GitHub

SpecialFunctions.jl

SpecialFunctions.jl provides a comprehensive collection of special functions based on the OpenSpecFun and OpenLibm libraries.

Special mathematical functions in Julia, include Bessel, Hankel, Airy, error, Dawson, exponential (or sine and cosine) integrals, eta, zeta, digamma, inverse digamma, trigamma, and polygamma functions.

Installation

The latest version of the package is available for Julia versions 1.3 and up. To install it, run the following at the Julia REPL:

import Pkg; Pkg.add("SpecialFunctions")
+Home · SpecialFunctions.jl
Edit on GitHub

SpecialFunctions.jl

SpecialFunctions.jl provides a comprehensive collection of special functions based on the OpenSpecFun and OpenLibm libraries.

Special mathematical functions in Julia, include Bessel, Hankel, Airy, error, Dawson, exponential (or sine and cosine) integrals, eta, zeta, digamma, inverse digamma, trigamma, and polygamma functions.

Installation

The latest version of the package is available for Julia versions 1.3 and up. To install it, run the following at the Julia REPL:

import Pkg; Pkg.add("SpecialFunctions")
diff --git a/dev/search/index.html b/dev/search/index.html index 6f1d9b4b..b7d40ba7 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · SpecialFunctions.jl

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