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.

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  ▪  FoxH  ▪  AppellF1  ▪  BilateralHypergeometricPFQ  ▪  ...

Elliptic Integrals »

EllipticK  ▪  EllipticF  ▪  EllipticE  ▪  EllipticPi  ▪  CarlsonRF  ▪  CarlsonRK  ▪  ...

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  ▪  ...

Heun Functions »

HeunG  ▪  HeunC  ▪  HeunB  ▪  HeunD  ▪  HeunT  ▪  ...

Coulomb Functions

CoulombF  ▪  CoulombG  ▪  CoulombH1  ▪  CoulombH2

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