Special Functions
Topic Overview 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
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