Here the Special Functions are listed according to the structure of NIST Digital Library of Mathematical Functions (DLMF).
Gamma Function - DLMF
| Function | Description |
|---|
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 $I_x(a,b)$ 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) | Inverse of the incomplete beta function (i.e evaluates $x$ given $I_{x}(a, b) = p$) |
Exponential and Trigonometric Integrals - DLMF
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
Bessel Functions - DLMF
| Function | Description |
|---|
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 - DLMF
Zeta and Related Functions - DLMF