Functions
Here the Special Functions are listed according to the structure of NIST Digital Library of Mathematical Functions.
Gamma Function
Function | Description |
---|---|
gamma(z) | gamma function $\Gamma(z)$ |
digamma(x) | digamma function (i.e. the derivative of lgamma 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 lgamma function at x ) |
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)) |
beta_inc(a,b,x,y) | incomplete beta function ratio Ix(a,b) and Iy(a,b) (i.e evaluates Ix(a,b) and Iy(a,b) 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 |
loggamma(x) | accurate log(gamma(x)) for large x |
logabsgamma(x) | accurate log(abs(gamma(x))) for large x |
lgamma(x) | accurate log(gamma(x)) for large x |
logfactorial(x) | accurate log(factorial(x)) for large x ; same as lgamma(x+1) for x > 1 , zero otherwise |
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(beta(x,y))) for large x or y |
Trigonometric Integrals
Function | Description |
---|---|
sinint(x) | sine integral $Si(x)$ |
cosint(x) | cosine integral $Ci(x)$ |
Error Functions, Dawson’s and Fresnel Integrals
Function | Description |
---|---|
erf(x) | error function at $x$ |
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 $x$ |
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 $x$ |
Airy and Related Functions
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) , airyaiprimex(z) , airybix(z) , airybiprimex(z) | scaled Airy Ai function and k th derivatives at z |
Bessel Functions
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 |
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 |
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 |
Elliptic Integrals
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
Function | Description |
---|---|
eta(x) | Dirichlet eta function at x |
zeta(x) | Riemann zeta function at x |