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HIGHER MATHEMATICAL FUNCTIONS

Combinatorial polynomials

Orthogonal polynomials

Gamma and related functions

Zeta and related functions

Exponential Integral and related functions

Error function and related functions

Bessel functions

Legendre and related functions

Confluent hypergeometric functions

Hypergeometric functions and generalizations

ProductLog function

Elliptic integrals

Elliptic functions

Mathieu and related functions


Combinatorial Polynomials

Fibonacci[n, x] Fibonacci polynomial
BernoulliB[n, x] Bernoulli polynomial
EulerE[n, x] Euler polynomial


Orthogonal Polynomials
LegendreP[n, x] Legendre polynomial
LegendreP[n, m, x] associated Legendre polynomial
GegenbauerC[n, m, x] Gegenbauer polynomial
GegenbauerC[n, x] renormalized Gegenbauer polynomial
SphericalHarmonicY[l, m, theta, phi] spherical harmonic
ChebyshevT[n, x] Chebyshev polynomial of the first kind
ChebyshevU[n, x] Chebyshev polynomial of the second kind
HermiteH[n, x] Hermite polynomial
LaguerreL[n, x] Laguerre polynomial
LaguerreL[n, a, x] associated Laguerre polynomial
JacobiP[n, a, b, x] Jacobi polynomial


Gamma and Related Functions
Beta[a, b] beta function
Beta[x, a, b] incomplete beta function
BetaRegularized[x, a, b] regularized incomplete beta function
Gamma[x] gamma function
Gamma[a, x] incomplete gamma function
Gamma[a, x0, x1] generalized incomplete gamma function
GammaRegularized[a, x] regularized incomplete gamma function
InverseBetaRegularized[x, a, b] inverse regularized beta function
InverseGammaRegularized[a, x] inverse regularized gamma function
Pochhammer[a, n] Pochhammer symbol
PolyGamma[x] digamma function
PolyGamma[n, x] nth derivative of the digamma function


Zeta and Related Functions
LerchPhi[x, s, a] Lerch's transcendent
PolyLog[n, x] polylogarithm function
PolyLog[n, p, x] Nielsen generalized polylogarithm function
RiemannSiegelTheta[x] Riemann-Siegel theta function
RiemannSiegelZ[x] Riemann-Siegel Z function
Zeta[x] Riemann zeta function
Zeta[x, a] generalized Riemann zeta function


Exponential Integral and Related Functions
CosIntegral[x] cosine integral
CoshIntegral[x] hyperbolic cosine integral
ExpIntegralE[n, x] exponential integral E
ExpIntegralEi[x] exponential integral Ei
LogIntegral[x] logarithmic integral
SinIntegral[x] sine integral
SinhIntegral[x] hyperbolic sine integral


Error Function and Related Functions
Erf[x] error function
Erf[x0, x1] generalized error function
Erfc[x] complementary error function
Erfi[x] imaginary error function
FresnelC[x], FresnelS[x] Fresnel integrals
InverseErf[x] inverse error function
InverseErfc[x] inverse complementary error function


Bessel Functions
AiryAi[x], AiryBi[x] Airy functions
AiryAiPrime[x], AiryBiPrime[x] derivatives of Airy functions
BesselJ[n, x] Bessel function of the first kind
BesselY[n, x] Bessel function of the second kind
BesselI[n, x] modified Bessel function of the first kind
BesselK[n, x] modified Bessel function of the second kind
StruveH[n, x] Struve function
StruveL[n, x] modified Struve function


Legendre and Related Functions
LegendreP[n, x] Legendre function of the first kind
LegendreP[n, m, x] associated Legendre function of the first kind
LegendreQ[n, x] Legendre function of the second kind
LegendreQ[n, m, x] associated Legendre function of the second kind
LegendreP[n, m, a, x], LegendreQ[n, m, a, x] Legendre function of type a


Confluent Hypergeometric Functions
Hypergeometric0F1[a, x] hypergeometric function
Hypergeometric0F1Regularized[a, x] regularized hypergeometric function
Hypergeometric1F1[a, b, x] Kummer confluent hypergeometric function
Hypergeometric1F1Regularized[a, b, x] regularized confluent hypergeometric function
HypergeometricU[a, b, x] confluent hypergeometric function of the second kind


Hypergeometric Functions and Generalizations
Hypergeometric2F1[a, b, c, x] hypergeometric function
Hypergeometric2F1Regularized[a, b, c, x] regularized hypergeometric function
HypergeometricPFQ[{a1, ... , ap}, {b1, ... , bq}, x] generalized hypergeometric function
HypergeometricPFQRegularized[{a1, ... , ap}, {b1, ... , bq}, x] regularized generalized hypergeometric function
MeijerG[{{a1, ... , an}, {an+1, ... , ap}}, {{b1, ... , bm}, {bm+1, ... , bq}}, x] Meijer G-function
AppelF1[a, b1, b2, c, x, y] Appell hypergeometric function of two variables


ProductLog Function
ProductLog[x] Lambert W-function
ProductLog[k, x] multiple values of the Lambert W-function


Elliptic Integrals
EllipticK[x] complete elliptic integral of the first kind
EllipticF[x, m] elliptic integral of the first kind
EllipticE[x] complete elliptic integral of the second kind
EllipticE[x, m] elliptic integral of the second kind
EllipticPi[n, m] complete elliptic integral of the third kind
EllipticPi[n, x, m] elliptic integral of the third kind
JacobiZeta[x, m] Jacobi zeta function


Elliptic Functions
JacobiAmplitude[x, m] Jacobi amplitude function
JacobiCD[x, m], JacobiCN[x, m], JacobiCS[x, m], JacobiDC[x, m], JacobiDN[x, m], JacobiDS[x, m], JacobiNC[x, m], JacobiND[x, m], JacobiNS[x, m], JacobiSC[x, m], JacobiSD[x, m], JacobiSN[x, m] Jacobi elliptic functions
InverseJacobiCD[x, m], InverseJacobiCN[x, m], InverseJacobiCS[x, m], InverseJacobiDC[x, m], InverseJacobiDN[x, m], InverseJacobiDS[x, m], InverseJacobiNC[x, m], InverseJacobiND[x, m], InverseJacobiNS[x, m], InverseJacobiSC[x, m], InverseJacobiSD[x, m], InverseJacobiSN[x, m] inverse Jacobi elliptic functions
EllipticTheta[a, x, q] theta function
EllipticThetaPrime[a, x, q] derivative of theta function
WeierstrassP[x, {g2, g3}] Weierstrass elliptic function
WeierstrassPPrime[x, {g2, g3}] derivative of Weierstrass elliptic function
InverseWeierstrassP[x, {g2, g3}] inverse Weierstrass elliptic function
WeierstrassSigma[x, {g2, g3}] Weierstrass sigma function
WeierstrassZeta[z, {g2, g3}] Weierstrass zeta function


Mathieu and Related Functions
MathieuC[a, q, x] even Mathieu function
MathieuS[b, q, x] odd Mathieu function
MathieuCPrime[a, q, x] derivative of even Mathieu function
MathieuSPrime[b, q, x] derivative of odd Mathieu function
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