Here the Special Functions are listed according to the structure of NIST Digital Library of Mathematical Functions (DLMF).
| Function | Description |
|---|---|
[gamma(z)](@ref SpecialFunctions.gamma(::Number)) |
gamma function \Gamma(z) |
[loggamma(x)](@ref SpecialFunctions.loggamma(::Number)) |
accurate log(gamma(x)) for large x |
[logabsgamma(x)](@ref SpecialFunctions.logabsgamma) |
accurate log(abs(gamma(x))) for large x |
[logfactorial(x)](@ref SpecialFunctions.logfactorial) |
accurate log(factorial(x)) for large x; same as loggamma(x+1) for x > 1, zero otherwise |
[digamma(x)](@ref SpecialFunctions.digamma) |
digamma function (i.e. the derivative of loggamma at x) |
[invdigamma(x)](@ref SpecialFunctions.invdigamma) |
invdigamma function (i.e. inverse of digamma function at x using fixed-point iteration algorithm) |
[trigamma(x)](@ref SpecialFunctions.trigamma) |
trigamma function (i.e the logarithmic second derivative of gamma at x) |
[polygamma(m,x)](@ref SpecialFunctions.polygamma) |
polygamma function (i.e the (m+1)-th derivative of the loggamma function at x) |
[gamma(a,z)](@ref SpecialFunctions.gamma(::Number,::Number)) |
upper incomplete gamma function \Gamma(a,z) |
[loggamma(a,z)](@ref SpecialFunctions.loggamma(::Number,::Number)) |
accurate log(gamma(a,x)) for large arguments |
[gamma_inc(a,x,IND)](@ref SpecialFunctions.gamma_inc) |
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)](@ref SpecialFunctions.gamma_inc_inv) |
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)](@ref SpecialFunctions.beta) |
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(binomial(n,k))) for large n and k near n/2 |
[beta_inc(a,b,x,y)](@ref SpecialFunctions.beta_inc) |
incomplete beta function ratio I_x(a,b) and I_y(a,b) (i.e evaluates I_x(a,b) and I_y(a,b) and returns tuple (p,q)) |
[beta_inc_inv(a,b,p,q)](@ref SpecialFunctions.beta_inc_inv) |
Inverse of the incomplete beta function (i.e evaluates x given I_{x}(a, b) = p) |
Exponential and Trigonometric Integrals - DLMF
| Function | Description |
|---|---|
[expint(ν, z)](@ref SpecialFunctions.expint) |
exponential integral \operatorname{E}_\nu(z) |
[expinti(x)](@ref SpecialFunctions.expinti) |
exponential integral \operatorname{Ei}(x) |
[expintx(x)](@ref SpecialFunctions.expintx) |
scaled exponential integral e^z \operatorname{E}_\nu(z) |
[sinint(x)](@ref SpecialFunctions.sinint) |
sine integral \operatorname{Si}(x) |
[cosint(x)](@ref SpecialFunctions.cosint) |
cosine integral \operatorname{Ci}(x) |
Error Functions, Dawson’s and Fresnel Integrals - DLMF
| Function | Description |
|---|---|
[erf(x)](@ref SpecialFunctions.erf) |
error function at x |
[erf(x,y)](@ref SpecialFunctions.erf) |
accurate version of \operatorname{erf}(y) - \operatorname{erf}(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 |
[logerfc(x)](@ref SpecialFunctions.logerfc) |
log of the complementary error function, i.e. accurate \operatorname{ln}(\operatorname{erfc}(x)) for large x |
[logerfcx(x)](@ref SpecialFunctions.logerfcx) |
log of the scaled complementary error function, i.e. accurate \operatorname{ln}(\operatorname{erfcx}(x)) for large negative 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 |
[faddeeva(x)](@ref SpecialFunctions.faddeeva) |
Faddeeva function, equivalent to \operatorname{erfcx}(-ix) |
Airy and Related Functions - DLMF
| Function | Description |
|---|---|
[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 derivative at z |
| Function | Description |
|---|---|
[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 |
[sphericalbesselj(nu,z)](@ref SpecialFunctions.sphericalbesselj) |
Spherical 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 |
[sphericalbessely(nu,z)](@ref SpecialFunctions.sphericalbessely) |
Spherical 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 |
[jinc(x)](@ref SpecialFunctions.jinc) |
scaled Bessel function of the first kind divided by x. A.k.a. sombrero or besinc |
| Function | Description |
|---|---|
[ellipk(m)](@ref SpecialFunctions.ellipk) |
complete elliptic integral of 1st kind K(m) |
[ellipe(m)](@ref SpecialFunctions.ellipe) |
complete elliptic integral of 2nd kind E(m) |
[ellipke(m)](@ref SpecialFunctions.ellipke) |
complete elliptic integrals K(m) and E(m) simultaneously |
Zeta and Related Functions - DLMF
| Function | Description |
|---|---|
[eta(x)](@ref SpecialFunctions.eta) |
Dirichlet eta function at x |
[zeta(x)](@ref SpecialFunctions.zeta) |
Riemann zeta function at x |