See Also: Functions List - SpecialFunctions.jl
- By default, all functions are provided by
SpecialFunctions.jl. - ❌: means not implemented
| Function | Description |
|---|
gamma(z) | gamma function |
loggamma(x) | log(gamma(x)) |
logabsgamma(x) | log(abs(gamma(x))) |
digamma(x) | derivative of loggamma |
trigamma(x) | second derivative of loggamma |
polygamma(m,x) | (m+1)-th derivative of the loggamma |
gamma(a,z) | incomplete gamma function |
loggamma(a,z) | log(gamma(a,x)) |
gamma_inc(a,x) | incomplete gamma function |
gamma_inc_inv(a,p,q) | inverse of the incomplete gamma function |
| Function | Description |
|---|
beta(x,y) | beta function |
logbeta(x,y) | log(beta(x,y)) |
logabsbeta(x,y) | log(abs(beta(x,y))) |
beta_inc(a,b,x,y) | incomplete beta function |
beta_inc_inv(a,b,p,q) | inverse of the incomplete beta function |
| Function | Description |
|---|
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 $\operatorname{Ci}(x)$ |
sinhint(x) | ❌ hyperbolic sine integral $\operatorname{Shi}(x)$ |
coshint(x) | ❌ hyperbolic cosine integral $\operatorname{Chi}(x)$ |
Error Functions, Dawson’s and Fresnel Integrals - DLMF
| Function | Description |
|---|
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
| Function | Description |
|---|
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) | scaled Airy Ai function and derivative at z |
airyaiprimex(z) | scaled Airy Ai function and derivative at z |
airybix(z) | scaled Airy Ai function and derivative at z |
airybiprimex(z) | scaled Airy Ai function and derivative at z |
Bessel Functions - DLMF
Struve and Related Functions - DLMF
Parabolic Cylinder Functions - DLMF
Confluent Hypergeometric Functions - DLMF
- MartinMikkelsen/FewSpecialFunctions.jl Few special functions in Julia. Includes Clausen function, Coulomb wave functions, Debye function, Fresnel functions, Struve function, Hypergeometric functions, Confluent hypergeometric functions, Fermi-Dirac
Legendre and Related Functions - DLMF
Hypergeometric Function - DLMF
Elliptic Integrals - DLMF
| Function | Description |
|---|
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
| Function | Description |
|---|
eta(x) | Dirichlet eta function at x |
zeta(x) | Riemann zeta function at x |
Mathieu Functions and Hill’s Equation - DLMF
missing
Lambert W-Function - DLMF