From 2673cebd83402989addabd4191ca1b100b0045d4 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Tue, 7 May 2024 01:46:53 +0000 Subject: [PATCH] build based on e08ff8d --- dev/functions_list/index.html | 36 +++++++++++++++---------------- dev/functions_overview/index.html | 2 +- dev/index.html | 2 +- dev/search/index.html | 2 +- 4 files changed, 21 insertions(+), 21 deletions(-) diff --git a/dev/functions_list/index.html b/dev/functions_list/index.html index 9b3d7876..70a87f12 100644 --- a/dev/functions_list/index.html +++ b/dev/functions_list/index.html @@ -1,28 +1,28 @@ -Reference · SpecialFunctions.jl
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Functions

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, Wikipedia

See also: besselh

source
SpecialFunctions.beta — Method
beta(x, y)

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

source
SpecialFunctions.beta_inc — Method
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

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

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

External links: DLMF, Wikipedia

See also: beta_inc_inv

source
SpecialFunctions.beta_inc_cont_fraction — Method
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, Wikipedia

See also: beta_inc

Implementation

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

source
SpecialFunctions.beta_inc_inv — Method
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

source
SpecialFunctions.beta_inc_power_series — Method
beta_inc_power_series(a, b, x, epps)

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

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

External links: DLMF, Wikipedia

See also: beta_inc

Implementation

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

source
SpecialFunctions.coeff1 — Method

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

source
SpecialFunctions.coeff2 — Method

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

source
SpecialFunctions.coeff3 — Method

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

source
SpecialFunctions.cosint — Function
cosint(x)

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

\[\operatorname{Ci}(x) +Reference · SpecialFunctions.jl

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Functions

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, Wikipedia

See also: besselh

source
SpecialFunctions.beta — Method
beta(x, y)

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

source
SpecialFunctions.beta_inc — Method
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

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

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

External links: DLMF, Wikipedia

See also: beta_inc_inv

source
SpecialFunctions.beta_inc_cont_fraction — Method
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, Wikipedia

See also: beta_inc

Implementation

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

source
SpecialFunctions.beta_inc_inv — Method
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

source
SpecialFunctions.beta_inc_power_series — Method
beta_inc_power_series(a, b, x, epps)

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

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

External links: DLMF, Wikipedia

See also: beta_inc

Implementation

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

source
SpecialFunctions.coeff1 — Method

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

source
SpecialFunctions.coeff2 — Method

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

source
SpecialFunctions.coeff3 — Method

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

source
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, 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: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

source
SpecialFunctions.dawson — Function
dawson(x)

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

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

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

External links: DLMF, 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: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

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, Wikipedia.

See also: erfi(x).

Implementation by

  • Float32/Float64: C standard math library libm.
source
SpecialFunctions.ellipe — Method
ellipe(m)

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

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

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

External links: DLMF, Wikipedia.

See also: erfi(x).

Implementation by

  • Float32/Float64: C standard math library libm.
source
SpecialFunctions.ellipe — Method
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, 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.ellipk — Method
ellipk(m)

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

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

External links: DLMF, 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.ellipk — Method
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, 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.erf — Function
erf(x)

Compute the error function of $x$, defined by

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

erf(x, y)

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

External links: DLMF, 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
source
SpecialFunctions.erfc — Function
erfc(x)

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

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

External links: DLMF, 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.erf — Function
erf(x)

Compute the error function of $x$, defined by

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

erf(x, y)

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

External links: DLMF, 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
source
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, Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: C library for multiple-precision floating-point MPFR
source
SpecialFunctions.erfcinv — Method
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.

source
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, Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: C library for multiple-precision floating-point MPFR
source
SpecialFunctions.erfcinv — Method
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.

source
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, 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.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, 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.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.
source
SpecialFunctions.erfinv — Method
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.expint — Function
expint(z)
-expint(ν, z)

Computes the exponential integral $\operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{t^\nu} dt$. If $\nu$ is not specified, $\nu=1$ is used. Arbitrary complex $\nu$ and $z$ are supported.

External links: DLMF, Wikipedia

source
SpecialFunctions.expinti — Method
expinti(x::Real)

Computes the exponential integral function $\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} dt$, which is equivalent to $-\Re[\operatorname{E}_1(-x)]$ where $\operatorname{E}_1$ is the expint function.

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

Computes the scaled exponential integral $\exp(z) \operatorname{E}_\nu(z) = e^z \int_1^\infty \frac{e^{-zt}}{t^\nu} dt$. If $\nu$ is not specified, $\nu=1$ is used. Arbitrary complex $\nu$ and $z$ are supported.

See also: expint(ν, z)

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.gamma — Method
gamma(a,x)

Returns the upper incomplete gamma function

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

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, Wikipedia

source
SpecialFunctions.gamma — Method
gamma(z)

Compute the gamma function for complex $z$, defined by

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

External links: Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
source
SpecialFunctions.erfinv — Method
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.expint — Function
expint(z)
+expint(ν, z)

Computes the exponential integral $\operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{t^\nu} dt$. If $\nu$ is not specified, $\nu=1$ is used. Arbitrary complex $\nu$ and $z$ are supported.

External links: DLMF, Wikipedia

source
SpecialFunctions.expinti — Method
expinti(x::Real)

Computes the exponential integral function $\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} dt$, which is equivalent to $-\Re[\operatorname{E}_1(-x)]$ where $\operatorname{E}_1$ is the expint function.

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

Computes the scaled exponential integral $\exp(z) \operatorname{E}_\nu(z) = e^z \int_1^\infty \frac{e^{-zt}}{t^\nu} dt$. If $\nu$ is not specified, $\nu=1$ is used. Arbitrary complex $\nu$ and $z$ are supported.

See also: expint(ν, z)

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.gamma — Method
gamma(a,x)

Returns the upper incomplete gamma function

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

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, Wikipedia

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SpecialFunctions.gamma — Method
gamma(z)

Compute the gamma function for complex $z$, defined by

\[\Gamma(z) := \begin{cases} n! @@ -48,11 +48,11 @@ 24 julia> gamma(4+1) == prod(1:4) == factorial(4) -true

External links: DLMF, 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
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:

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

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

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

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, Wikipedia

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

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SpecialFunctions.gamma_inc_asym — Method
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

See also: gamma_inc(a,x,ind)

source
SpecialFunctions.gamma_inc_cf — Method
gamma_inc_cf(a, x, ind)

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

\[R(a,x) * \frac{1}{1-\frac{z}{a+1+\frac{z}{a+2-\frac{(a+1)z}{a+3+\frac{2z}{a+4-\frac{(a+2)z}{a+5+\frac{3z}{a+6-\dots}}}}}}}\]

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

External links: DLMF

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_inv_alarge — Method
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:

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

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.

source
SpecialFunctions.gamma_inc_inv_psmall — Method
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 $x = r + \sum_{k=2}^{\infty}c_{k}r^{k}$.

source
SpecialFunctions.gamma_inc_inv_qsmall — Method
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 $b = 1-a$, $L = \ln{x_0}$.

source
SpecialFunctions.gamma_inc_minimax — Method
gamma_inc_minimax(a,x,z)

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

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

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 .

External links: DLMF

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_taylor_x — Method
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 — Method
gamma_inc_temme(a, x, z)

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

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

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

See also: gamma_inc(a,x,ind)

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SpecialFunctions.gamma_inc_temme_1 — Method
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 $\sigma <= e_{0}/\sqrt{a}$.

External links: DLMF

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SpecialFunctions.gammax — Method
gammax(x)

\[\operatorname{gammax}(x) = \begin{cases}e^{\operatorname{stirling}(x)}\quad\quad\quad \text{if} \quad x>0,\\ +true

External links: DLMF, 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
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:

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

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

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

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, Wikipedia

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

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SpecialFunctions.gamma_inc_asym — Method
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

See also: gamma_inc(a,x,ind)

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

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

\[R(a,x) * \frac{1}{1-\frac{z}{a+1+\frac{z}{a+2-\frac{(a+1)z}{a+3+\frac{2z}{a+4-\frac{(a+2)z}{a+5+\frac{3z}{a+6-\dots}}}}}}}\]

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

External links: DLMF

See also: gamma_inc(a,x,ind)

source
SpecialFunctions.gamma_inc_inv_alarge — Method
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:

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

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.

source
SpecialFunctions.gamma_inc_inv_psmall — Method
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 $x = r + \sum_{k=2}^{\infty}c_{k}r^{k}$.

source
SpecialFunctions.gamma_inc_inv_qsmall — Method
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 $b = 1-a$, $L = \ln{x_0}$.

source
SpecialFunctions.gamma_inc_minimax — Method
gamma_inc_minimax(a,x,z)

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

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

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 .

External links: DLMF

See also: gamma_inc(a,x,ind)

source
SpecialFunctions.gamma_inc_taylor_x — Method
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)

source
SpecialFunctions.gamma_inc_temme — Method
gamma_inc_temme(a, x, z)

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

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

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

See also: gamma_inc(a,x,ind)

source
SpecialFunctions.gamma_inc_temme_1 — Method
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 $\sigma <= e_{0}/\sqrt{a}$.

External links: DLMF

source
SpecialFunctions.gammax — Method
gammax(x)

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

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SpecialFunctions.jinc — Method
jinc(x)

Bessel function of the first kind divided by x. Following convention: $\operatorname{jinc}{x} = \frac{2 \cdot J_1{\pi x}}{\pi x}$. Sometimes known as sombrero or besinc function.

External links: Wikipedia

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SpecialFunctions.logabsbinomial — Method
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))).

source
SpecialFunctions.logerf — Method
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.logerfc — Method
logerfc(x)

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

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

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

External links: Wikipedia.

See also: erfcx(x).

Implementation

Based on the erfc(x) and erfcx(x) functions. Currently only implemented for Float32, Float64, and BigFloat.

source
SpecialFunctions.logerfcx — Method
logerfcx(x)

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

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

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

External links: Wikipedia.

See also: erfcx(x).

Implementation

Based on the erfc(x) and erfcx(x) functions. Currently only implemented for Float32, Float64, and BigFloat.

source
SpecialFunctions.loggamma — Method
loggamma(a,x)

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

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

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.loggamma — Method
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 $2\pi i$ (i.e. it may employ a different branch cut).

See also logabsgamma for real x.

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

Compute CDF of noncentral F distribution given by:

\[F(x, v1, v2; lambda) = I_{v1*x/(v1*x + v2)}(v1/2, v2/2; \lambda)\]

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

Wikipedia: https://en.wikipedia.org/wiki/Noncentral_F-distribution

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SpecialFunctions.ncbeta — Method
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 $\lambda >= 54$ : modification in Chattamvelli(1997) in ncbeta_poisson(a,b,lambda,x) by using both forward and backward recurrences.

source
SpecialFunctions.ncbeta_poisson — Method
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 — Method
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 https://dlmf.nist.gov/8.17.21.

source
SpecialFunctions.rgamma1pm1 — Method

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.jinc — Method
jinc(x)

Bessel function of the first kind divided by x. Following convention: $\operatorname{jinc}{x} = \frac{2 \cdot J_1{\pi x}}{\pi x}$. Sometimes known as sombrero or besinc function.

External links: Wikipedia

source
SpecialFunctions.logabsbinomial — Method
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))).

source
SpecialFunctions.logerf — Method
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).

source
SpecialFunctions.logerfc — Method
logerfc(x)

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

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

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

External links: Wikipedia.

See also: erfcx(x).

Implementation

Based on the erfc(x) and erfcx(x) functions. Currently only implemented for Float32, Float64, and BigFloat.

source
SpecialFunctions.logerfcx — Method
logerfcx(x)

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

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

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

External links: Wikipedia.

See also: erfcx(x).

Implementation

Based on the erfc(x) and erfcx(x) functions. Currently only implemented for Float32, Float64, and BigFloat.

source
SpecialFunctions.loggamma — Method
loggamma(a,x)

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

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

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).

source
SpecialFunctions.loggamma — Method
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 $2\pi i$ (i.e. it may employ a different branch cut).

See also logabsgamma for real x.

source
SpecialFunctions.ncF — Method
ncF(x,v1,v2,lambda)

Compute CDF of noncentral F distribution given by:

\[F(x, v1, v2; lambda) = I_{v1*x/(v1*x + v2)}(v1/2, v2/2; \lambda)\]

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

Wikipedia: https://en.wikipedia.org/wiki/Noncentral_F-distribution

source
SpecialFunctions.ncbeta — Method
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 $\lambda >= 54$ : modification in Chattamvelli(1997) in ncbeta_poisson(a,b,lambda,x) by using both forward and backward recurrences.

source
SpecialFunctions.ncbeta_poisson — Method
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.

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

source
SpecialFunctions.ncbeta_tail — Method
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 https://dlmf.nist.gov/8.17.21.

source
SpecialFunctions.rgamma1pm1 — Method

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).

source
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, 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.sphericalbesselj — Method
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.sphericalbessely — Method
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.zeta — Method
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
+x \in \mathbb{R} \,.\]

External links: DLMF, 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: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

source
SpecialFunctions.sphericalbesselj — Method
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.sphericalbessely — Method
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.zeta — Method
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
diff --git a/dev/functions_overview/index.html b/dev/functions_overview/index.html index b9bd057f..a211f16e 100644 --- a/dev/functions_overview/index.html +++ b/dev/functions_overview/index.html @@ -1,2 +1,2 @@ -Overview · SpecialFunctions.jl
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Functions

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

Gamma Function

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

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

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 $\operatorname{erfcx}(-ix)$
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 kth derivatives at z

Bessel Functions

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

FunctionDescription
ellipk(m)complete elliptic integral of 1st kind $K(m)$
ellipe(m)complete elliptic integral of 2nd kind $E(m)$
FunctionDescription
eta(x)Dirichlet eta function at x
zeta(x)Riemann zeta function at x
+Overview · SpecialFunctions.jl
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Functions

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

Gamma Function

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

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

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 $\operatorname{erfcx}(-ix)$
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 kth derivatives at z

Bessel Functions

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

FunctionDescription
ellipk(m)complete elliptic integral of 1st kind $K(m)$
ellipe(m)complete elliptic integral of 2nd kind $E(m)$
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 9bf3538d..47e7aa0d 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · SpecialFunctions.jl
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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:

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:

Pkg.add("SpecialFunctions")
diff --git a/dev/search/index.html b/dev/search/index.html index cd0b7148..6d4fddba 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · SpecialFunctions.jl

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