-
Functions
- AiryAi
- AiryAiPrime
- AppellF1
- BesselI
- BesselJ
- BesselJZero
- BesselK
- BesselY
- Beta
- BilateralHypergeometricPFQ
- CarlsonRF
- CarlsonRK
- ChebyshevT
- ChebyshevU
- CoulombF
- CoulombG
- CoulombH1
- CoulombH2
- DedekindEta
- Derivative
- DifferenceRoot
- DifferentialRoot
- EllipticE
- EllipticF
- EllipticK
- EllipticPi
- EllipticTheta
- Erf
- Erfc
- ExpIntegralE
- ExpIntegralEi
- FindRoot
- FoxH
- FresnelS
- FullSimplify
- FunctionExpand
- Gamma
- GegenbauerC
- HankelH1
- HermiteH
- HeunB
- HeunC
- HeunD
- HeunG
- HeunT
- Hypergeometric2F1
- HypergeometricPFQ
- HypergeometricU
- InverseEllipticNomeQ
- InverseErf
- InverseGammaRegularized
- InverseJacobiSN
- InverseWeierstrassP
- JacobiP
- JacobiSN
- KelvinBer
- KleinInvariantJ
- LaguerreL
- LegendreP
- LegendreQ
- LerchPhi
- LogGamma
- LogIntegral
- MathieuC
- MathieuCharacteristicA
- MathieuS
- MathieuSPrime
- MeijerG
- MittagLefflerE
- ModularLambda
- N
- Pochhammer
- PolyGamma
- PolyLog
- ProductLog
- QFactorial
- QHypergeometricPFQ
- QPochhammer
- RiemannSiegelZ
- Root
- SiegelTheta
- SinIntegral
- SphericalBesselJ
- SphericalHarmonicY
- SpheroidalEigenvalue
- SpheroidalPS
- SpheroidalQS
- SpheroidalS1
- StruveH
- WeierstrassP
- WignerD
- ZernikeR
- Zeta
- ZetaZero
- Related Guides
- Tech Notes
-
-
Functions
- AiryAi
- AiryAiPrime
- AppellF1
- BesselI
- BesselJ
- BesselJZero
- BesselK
- BesselY
- Beta
- BilateralHypergeometricPFQ
- CarlsonRF
- CarlsonRK
- ChebyshevT
- ChebyshevU
- CoulombF
- CoulombG
- CoulombH1
- CoulombH2
- DedekindEta
- Derivative
- DifferenceRoot
- DifferentialRoot
- EllipticE
- EllipticF
- EllipticK
- EllipticPi
- EllipticTheta
- Erf
- Erfc
- ExpIntegralE
- ExpIntegralEi
- FindRoot
- FoxH
- FresnelS
- FullSimplify
- FunctionExpand
- Gamma
- GegenbauerC
- HankelH1
- HermiteH
- HeunB
- HeunC
- HeunD
- HeunG
- HeunT
- Hypergeometric2F1
- HypergeometricPFQ
- HypergeometricU
- InverseEllipticNomeQ
- InverseErf
- InverseGammaRegularized
- InverseJacobiSN
- InverseWeierstrassP
- JacobiP
- JacobiSN
- KelvinBer
- KleinInvariantJ
- LaguerreL
- LegendreP
- LegendreQ
- LerchPhi
- LogGamma
- LogIntegral
- MathieuC
- MathieuCharacteristicA
- MathieuS
- MathieuSPrime
- MeijerG
- MittagLefflerE
- ModularLambda
- N
- Pochhammer
- PolyGamma
- PolyLog
- ProductLog
- QFactorial
- QHypergeometricPFQ
- QPochhammer
- RiemannSiegelZ
- Root
- SiegelTheta
- SinIntegral
- SphericalBesselJ
- SphericalHarmonicY
- SpheroidalEigenvalue
- SpheroidalPS
- SpheroidalQS
- SpheroidalS1
- StruveH
- WeierstrassP
- WignerD
- ZernikeR
- Zeta
- ZetaZero
- Related Guides
- Tech Notes
-
Functions
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.
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