Special Functions

Two decades of intense R&D at Wolfram Research have given the Wolfram Language by far the world's broadest and deepest coverage of special functions—and greatly expanded the whole domain of practical closed-form solutions. Often using original results and methods, all special functions in the Wolfram Language support arbitrary-precision evaluation for all complex values of parameters, arbitrary series expansion even at branch points, and an immense web of exact relations, transformations, and simplifications.

(OrangeLink[Gamma, Beta, etc. », paclet:guide/GammaFunctionsAndRelatedFunctions])

(OrangeLink[Error Functions, Exponential Integrals, etc. », paclet:guide/ErrorAndExponentialIntegralFunctions])

Orthogonal Polynomials

(OrangeLink[Bessel-Related Functions », paclet:guide/BesselRelatedFunctions])

Legendre-Related Functions

(OrangeLink[Hypergeometric Functions », paclet:guide/HypergeometricFunctions])

(OrangeLink[Elliptic Integrals », paclet:guide/EllipticIntegrals])

(OrangeLink[Elliptic Functions », paclet:guide/EllipticFunctions])

Modular Forms

(OrangeLink[Zeta Functions & Polylogarithms », paclet:guide/ZetaFunctionsAndPolylogarithms])

(OrangeLink[Mathieu Functions », paclet:guide/MathieuAndRelatedFunctions])

(OrangeLink[Spheroidal Functions », paclet:guide/SpheroidalAndRelatedFunctions])

(OrangeLink[q Functions », paclet:guide/QFunctions])

Fractional Calculus Functions

(OrangeLink[Inverse Functions », paclet:guide/InverseFunctions])

General Solution Functions

(RefLink[N, paclet:ref/N]) — numerical evaluation to any precision

(RefLink[FunctionExpand, paclet:ref/FunctionExpand]) — expand in terms of simpler functions

(RefLink[FullSimplify, paclet:ref/FullSimplify]) — apply full symbolic simplification

(RefLink[Derivative, paclet:ref/Derivative]) ((RefLink[', paclet:ref/Derivative])) — symbolic and numerical derivatives for arguments and parameters

(RefLink[FindRoot, paclet:ref/FindRoot]) — find numerical zeros of functions