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 functionsand 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.

ReferenceReference

Gamma, Beta, etc. »

Gamma  ▪  Pochhammer  ▪  Beta  ▪  PolyGamma  ▪  LogGamma  ▪  ...

Error Functions, Exponential Integrals, etc. »

Erf  ▪  Erfc  ▪  ExpIntegralE  ▪  ExpIntegralEi  ▪  LogIntegral  ▪  FresnelS  ▪  SinIntegral  ▪  ...

Orthogonal Polynomials

LegendreP  ▪  HermiteH  ▪  LaguerreL  ▪  JacobiP  ▪  GegenbauerC  ▪  ChebyshevT  ▪  ChebyshevU  ▪  ZernikeR  ▪  SphericalHarmonicY  ▪  WignerD

Bessel-Related Functions »

BesselJ  ▪  BesselY  ▪  BesselI  ▪  BesselK  ▪  AiryAi  ▪  AiryAiPrime  ▪  SphericalBesselJ  ▪  KelvinBer  ▪  HankelH1  ▪  StruveH  ▪  ...

Legendre-Related Functions

LegendreP  ▪  LegendreQ  ▪  SpheroidalPS  ▪  SpheroidalQS

Hypergeometric Functions »

Hypergeometric2F1  ▪  HypergeometricPFQ  ▪  HypergeometricU  ▪  MeijerG  ▪  AppellF1  ▪  ...

Elliptic Integrals »

EllipticK  ▪  EllipticF  ▪  EllipticE  ▪  EllipticPi  ▪  ...

Elliptic Functions »

JacobiSN  ▪  InverseJacobiSN  ▪  WeierstrassP  ▪  EllipticTheta  ▪  ...

Modular Forms

DedekindEta  ▪  KleinInvariantJ  ▪  ModularLambda  ▪  SiegelTheta

Zeta Functions & Polylogarithms »

Zeta  ▪  PolyLog  ▪  LerchPhi  ▪  RiemannSiegelZ  ▪  ...

Mathieu Functions »

MathieuS  ▪  MathieuSPrime  ▪  MathieuC  ▪  MathieuCharacteristicA  ▪  ...

Spheroidal Functions »

SpheroidalPS  ▪  SpheroidalS1  ▪  SpheroidalEigenvalue  ▪  ...

q Functions »

QFactorial  ▪  QPochhammer  ▪  QHypergeometricPFQ  ▪  ...

Fractional Calculus Functions

MittagLefflerE

Inverse Functions »

ProductLog  ▪  InverseErf  ▪  InverseGammaRegularized  ▪  InverseEllipticNomeQ  ▪  InverseWeierstrassP  ▪  BesselJZero  ▪  ZetaZero  ▪  ...

General Solution Functions

Root  ▪  DifferentialRoot  ▪  DifferenceRoot

N numerical evaluation to any precision

FunctionExpand expand in terms of simpler functions

FullSimplify apply full symbolic simplification

Derivative (') symbolic and numerical derivatives for arguments and parameters

FindRoot find numerical zeros of functions