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| using SpecialFunctions, Documenter | ||
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| makedocs( | ||
| modules = [SpecialFunctions], | ||
| clean = false, | ||
| format = :html, | ||
| sitename = "SpecialFunctions.jl", | ||
| authors = "Jeff Bezanson, Stefan Karpinski, Viral B. Shah, et al.", | ||
| pages = [ | ||
| "Home" => "index.md", | ||
| "Functions" => "special.md", | ||
| modules = [SpecialFunctions], | ||
| sitename = "SpecialFunctions.jl", | ||
| authors = "Jeff Bezanson, Stefan Karpinski, Viral B. Shah, et al.", | ||
| pages = [ | ||
| "Home" => "index.md", | ||
| "Overview" => "functions_overview.md", | ||
| "List" => "functions_list.md", | ||
| ], | ||
| ) | ||
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| deploydocs( | ||
| repo = "github.com/JuliaMath/SpecialFunctions.jl.git", | ||
| target = "build", | ||
| repo = "github.com/JuliaMath/SpecialFunctions.jl.git", | ||
| target = "build", | ||
| ) |
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| # Functions | ||
| Here the *Special Functions* are listed according to the structure of [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/). | ||
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| ## Error Functions, Dawson’s and Fresnel Integrals | ||
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| | Function | Description | | ||
| |:-------- |:----------- | | ||
| | [`erf(x)`](@ref SpecialFunctions.erf) | [error function](https://en.wikipedia.org/wiki/Error_function) at ``x`` | | ||
| | [`erfc(x)`](@ref SpecialFunctions.erfc) | complementary error function, i.e. the accurate version of ``1-\operatorname{erf}(x)`` for large ``x`` | | ||
| | [`erfcinv(x)`](@ref SpecialFunctions.erfcinv) | inverse function to [`erfc()`](@ref SpecialFunctions.erfc) | | ||
| | [`erfcx(x)`](@ref SpecialFunctions.erfcx) | scaled complementary error function, i.e. accurate ``e^{x^2} \operatorname{erfc}(x)`` for large ``x`` | | ||
| | [`erfi(x)`](@ref SpecialFunctions.erfi) | imaginary error function defined as ``-i \operatorname{erf}(ix)`` | | ||
| | [`erfinv(x)`](@ref SpecialFunctions.erfinv) | inverse function to [`erf()`](@ref SpecialFunctions.erf) | | ||
| | [`dawson(x)`](@ref SpecialFunctions.dawson) | 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`` | | ||
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| ## Elliptic Integrals | ||
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| | Function | Description | | ||
| |:-------- |:----------- | | ||
| | [`ellipk(m)`](@ref SpecialFunctions.ellipk) | [complete elliptic integral of 1st kind](https://en.wikipedia.org/wiki/Elliptic_integral#Notational_variants) ``K(m)`` | | ||
| | [`ellipe(m)`](@ref SpecialFunctions.ellipe) | [complete elliptic integral of 2nd kind](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind) ``E(m)`` | | ||
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| ## Trigonometric Integrals | ||
| | Function | Description | | ||
| |:-------- |:----------- | | ||
| | [`sinint(x)`](@ref SpecialFunctions.sinint) | [sine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral) ``Si(x)`` | | ||
| | [`cosint(x)`](@ref SpecialFunctions.cosint) | [cosine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral) ``Ci(x)`` | | ||
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| # WIP not grouped | ||
| | Function | Description | | ||
| |:------------------------------------------------------------- |:--------------------------------------------------------------------------------------------------------------------------------------------------------------- | | ||
| | [`digamma(x)`](@ref SpecialFunctions.digamma) | [digamma function](https://en.wikipedia.org/wiki/Digamma_function) (i.e. the derivative of `lgamma` at `x`) | | ||
| | [`invdigamma(x)`](@ref SpecialFunctions.invdigamma) | [invdigamma function](http://bariskurt.com/calculating-the-inverse-of-digamma-function/) (i.e. inverse of `digamma` function at `x` using fixed-point iteration algorithm) | | ||
| | [`trigamma(x)`](@ref SpecialFunctions.trigamma) | [trigamma function](https://en.wikipedia.org/wiki/Trigamma_function) (i.e the logarithmic second derivative of `gamma` at `x`) | | ||
| | [`polygamma(m,x)`](@ref SpecialFunctions.polygamma) | [polygamma function](https://en.wikipedia.org/wiki/Polygamma_function) (i.e the (m+1)-th derivative of the `lgamma` function at `x`) | | ||
| | [`gamma_inc(a,x,IND)`](@ref SpecialFunctions.gamma_inc) | [incomplete gamma function ratio P(a,x) and Q(a,x)](https://en.wikipedia.org/wiki/Incomplete_gamma_function) (i.e evaluates P(a,x) and Q(a,x)for accuracy specified by IND and returns tuple (p,q)) | | ||
| | [`beta_inc(a,b,x,y)`](@ref SpecialFunctions.beta_inc) | [incomplete beta function ratio Ix(a,b) and Iy(a,b)](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) (i.e evaluates Ix(a,b) and Iy(a,b) and returns tuple (p,q)) | | ||
| | [`gamma_inc_inv(a,p,q)`](@ref SpecialFunctions.gamma_inc_inv) | [inverse of incomplete gamma function ratio P(a,x) and Q(a,x)](https://en.wikipedia.org/wiki/Incomplete_gamma_function) (i.e evaluates x given P(a,x)=p and Q(a,x)=q | | ||
| | [`eta(x)`](@ref SpecialFunctions.eta) | [Dirichlet eta function](https://en.wikipedia.org/wiki/Dirichlet_eta_function) at `x` | | ||
| | [`zeta(x)`](@ref SpecialFunctions.zeta) | [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function) at `x` | | ||
| | [`airyai(z)`](@ref SpecialFunctions.airyai) | [Airy Ai function](https://en.wikipedia.org/wiki/Airy_function) at `z` | | ||
| | [`airyaiprime(z)`](@ref SpecialFunctions.airyaiprime) | derivative of the Airy Ai function at `z` | | ||
| | [`airybi(z)`](@ref SpecialFunctions.airybi) | [Airy Bi function](https://en.wikipedia.org/wiki/Airy_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 `k`th derivatives at `z` | | ||
| | [`besselj(nu,z)`](@ref SpecialFunctions.besselj) | [Bessel function](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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` | | ||
| | [`gamma(x)`](@ref SpecialFunctions.gamma) | [gamma function](https://en.wikipedia.org/wiki/Gamma_function) at `x` | | ||
| | [`loggamma(x)`](@ref SpecialFunctions.loggamma) | accurate `log(gamma(x))` for large `x` | | ||
| | [`logabsgamma(x)`](@ref SpecialFunctions.logabsgamma) | accurate `log(abs(gamma(x)))` for large `x` | | ||
| | [`lgamma(x)`](@ref SpecialFunctions.lgamma) | accurate `log(gamma(x))` for large `x` | | ||
| | [`logfactorial(x)`](@ref SpecialFunctions.logfactorial) | accurate `log(factorial(x))` for large `x`; same as `lgamma(x+1)` for `x > 1`, zero otherwise | | ||
| | [`beta(x,y)`](@ref SpecialFunctions.beta) | [beta function](https://en.wikipedia.org/wiki/Beta_function) at `x,y` | | ||
| | [`logbeta(x,y)`](@ref SpecialFunctions.logbeta) | accurate `log(beta(x,y))` for large `x` or `y` | | ||
| | [`logabsbeta(x,y)`](@ref SpecialFunctions.logabsbeta) | accurate `log(abs(beta(x,y)))` for large `x` or `y` | | ||
| | [`logabsbinomial(x,y)`](@ref SpecialFunctions.logabsbinomial) | accurate `log(abs(beta(x,y)))` for large `x` or `y` | |
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