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Special Functions

This package provides a comprehensive collection of special functions based on the OpenSpecFun and OpenLibm libraries.

Function Description
[erf(x)](@ref SpecialFunctions.erf) error function at x
[erfc(x)](@ref SpecialFunctions.erfc) complementary error function, i.e. the accurate version of 1-erf(x) for large x
[erfinv(x)](@ref SpecialFunctions.erfinv) inverse function to [erf()](@ref SpecialFunctions.erf)
[erfcinv(x)](@ref SpecialFunctions.erfcinv) inverse function to [erfc()](@ref SpecialFunctions.erfc)
[erfi(x)](@ref SpecialFunctions.erfi) imaginary error function defined as -im * erf(x * im), where im is the imaginary unit
[erfcx(x)](@ref SpecialFunctions.erfcx) scaled complementary error function, i.e. accurate exp(x^2) * erfc(x) for large x
[dawson(x)](@ref SpecialFunctions.dawson) scaled imaginary error function, a.k.a. Dawson function, i.e. accurate exp(-x^2) * erfi(x) * sqrt(pi) / 2 for large x
[sinint(x)](@ref SpecialFunctions.sinint) sine integral at x
[cosint(x)](@ref SpecialFunctions.cosint) cosine integral at x
[digamma(x)](@ref SpecialFunctions.digamma) digamma function (i.e. the derivative of lgamma at x)
[eta(x)](@ref SpecialFunctions.eta) Dirichlet eta function at x
[zeta(x)](@ref SpecialFunctions.zeta) Riemann zeta function at x
[airyai(z)](@ref SpecialFunctions.airyai) Airy Ai function at z
[airyaiprime(z)](@ref SpecialFunctions.airyaiprime) derivative of the Airy Ai function at z
[airybi(z)](@ref SpecialFunctions.airybi) Airy Bi function at z
[airybiprime(z)](@ref SpecialFunctions.airybiprime) derivative of the Airy Bi function at z
[airyaix(z)](@ref SpecialFunctions.airyaix), [airyaiprimex(z)](@ref SpecialFunctions.airyaiprimex), [airybix(z)](@ref SpecialFunctions.airybix), [airybiprimex(z)](@ref SpecialFunctions.airybiprimex) scaled Airy Ai function and kth derivatives at z
[besselj(nu,z)](@ref SpecialFunctions.besselj) Bessel function of the first kind of order nu at z
[besselj0(z)](@ref SpecialFunctions.besselj0) besselj(0,z)
[besselj1(z)](@ref SpecialFunctions.besselj1) besselj(1,z)
[besseljx(nu,z)](@ref SpecialFunctions.besseljx) scaled Bessel function of the first kind of order nu at z
[bessely(nu,z)](@ref SpecialFunctions.bessely) Bessel function of the second kind of order nu at z
[bessely0(z)](@ref SpecialFunctions.bessely0) bessely(0,z)
[bessely1(z)](@ref SpecialFunctions.bessely1) bessely(1,z)
[besselyx(nu,z)](@ref SpecialFunctions.besselyx) scaled Bessel function of the second kind of order nu at z
[besselh(nu,k,z)](@ref SpecialFunctions.besselh) 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)](@ref SpecialFunctions.hankelh1) besselh(nu, 1, z)
[hankelh1x(nu,z)](@ref SpecialFunctions.hankelh1x) scaled besselh(nu, 1, z)
[hankelh2(nu,z)](@ref SpecialFunctions.hankelh2) besselh(nu, 2, z)
[hankelh2x(nu,z)](@ref SpecialFunctions.hankelh2x) scaled besselh(nu, 2, z)
[besseli(nu,z)](@ref SpecialFunctions.besseli) modified Bessel function of the first kind of order nu at z
[besselix(nu,z)](@ref SpecialFunctions.besselix) scaled modified Bessel function of the first kind of order nu at z
[besselk(nu,z)](@ref SpecialFunctions.besselk) modified Bessel function of the second kind of order nu at z
[besselkx(nu,z)](@ref SpecialFunctions.besselkx) scaled modified Bessel function of the second kind of order nu at z
[lambertw(z,k)](@ref SpecialFunctions.lambertw) kth branch of the Lambert W function at z

Installation

The package is available for Julia versions 0.5 and up. To install it, run

Pkg.add("SpecialFunctions")

from the Julia REPL.

Note

Prior to Julia 0.6, most of these functions were available in Julia's Base module. Because of this, the symbols from this package are not exported on Julia 0.5 to avoid name conflicts. In this case, the symbols will need to be explicitly imported or called with the prefix SpecialFunctions. This is not necessary for Julia versions 0.6 and later.

On Julia 0.7, openspecfun is not build as part of Julia. Thus for Julia versions 0.7 and later, installing this package downloads openspecfun. Binaries of openspecfun are available for macOS, Windows, and Linux (glibc >= 2.6). Other systems will need to build the library from source. You can force a build from source by setting an environment variable called JULIA_SPECIALFUNCTIONS_BUILD_SOURCE equal to true before running Pkg.build. This ensures that the library is built locally from source, even if binaries are available. Doing this requires a C compiler (Clang on macOS and FreeBSD, GCC elsewhere) and gfortran. If you always want to build this library from source, consider adding

ENV["JULIA_SPECIALFUNCTIONS_BUILD_SOURCE"] = "true"

to your .juliarc.jl file.