Special Functions

59 built-in function(s) in this category.

• AiryAi — AiryAi[z] (Stable)
• AiryBi — AiryBi[z] (Stable)
• BarnesG — BarnesG[z] (Stable)
• BernoulliB — BernoulliB[n] (Stable)
• BesselI — BesselI[n, z] (Stable)
• BesselJ — BesselJ[n, z] (Stable)
• BesselK — BesselK[n, z] (Stable)
• BesselY — BesselY[n, z] (Stable)
• Beta — Beta[a, b] (Stable)
• ChebyshevT — ChebyshevT[n, x] (Stable)
• ChebyshevU — ChebyshevU[n, x] (Stable)
• CosIntegral — CosIntegral[z] (Stable)
• CoshIntegral — CoshIntegral[z] (Stable)
• DirichletL — DirichletL[k, j, s] (Stable)
• EllipticE — EllipticE[m] is the complete elliptic integral of the second kind, Integrate[Sqrt[1 - m Sin[t]^2], {t, 0, Pi/2}], and EllipticE[phi, m] the incomplete one, with upper limit phi. The parameter argument is m = k^2, not the modulus k. EllipticE[0] is Pi/2 and EllipticE[1] is 1. EllipticE[phi, 1] is Sin[phi] only for |phi| <= Pi/2, since E(phi|1) is the integral of Abs[Cos[t]]: EllipticE[2, 1] is 2 - Sin[2], not Sin[2]. (Stable)
• EllipticF — EllipticF[phi, m] is the incomplete elliptic integral of the first kind, Integrate[1/Sqrt[1 - m Sin[t]^2], {t, 0, phi}]. The second argument is the PARAMETER m = k^2, not the modulus k. EllipticF[phi, 0] is phi and EllipticF[Pi/2, m] is EllipticK[m]; phi may be complex and of any size (the quasi-period is applied). (Stable)
• EllipticK — EllipticK[m] is the complete elliptic integral of the first kind, Integrate[1/Sqrt[1 - m Sin[t]^2], {t, 0, Pi/2}]. The argument is the PARAMETER m = k^2, not the modulus k. EllipticK[0] is Pi/2 and EllipticK[1] is ComplexInfinity; exact arguments otherwise stay symbolic and inexact ones evaluate numerically at their precision. (Stable)
• EllipticPi — EllipticPi[n, m] is the complete elliptic integral of the third kind, Integrate[1/((1 - n Sin[t]^2) Sqrt[1 - m Sin[t]^2]), {t, 0, Pi/2}], and EllipticPi[n, phi, m] the incomplete one, with upper limit phi. The parameter argument is m = k^2, not the modulus k. EllipticPi[0, m] is EllipticK[m] and EllipticPi[0, phi, m] is EllipticF[phi, m]; EllipticPi[1, m] is ComplexInfinity. Where the path crosses the pole at Sin[t]^2 == 1/n the value is complex, not a real principal value: EllipticPi[3/2, 1/2] is -0.4567203134529101 - 2.720699046351328 I. (Stable)
• Erf — Erf[z] (Stable)
• Erfc — Erfc[z] (Stable)
• Erfi — Erfi[z] (Stable)
• EulerE — EulerE[n] (Stable)
• ExpIntegralE — ExpIntegralE[n, z] (Stable)
• ExpIntegralEi — ExpIntegralEi[z] (Stable)
• FresnelC — FresnelC[z] (Stable)
• FresnelS — FresnelS[z] (Stable)
• FunctionExpand — FunctionExpand[expr] (Stable)
• Gamma — Gamma[z] (Stable)
• HarmonicNumber — HarmonicNumber[n] (Stable)
• HermiteH — HermiteH[n, x] (Stable)
• HurwitzZeta — HurwitzZeta[s, a] (Stable)
• Hyperfactorial — Hyperfactorial[n] (Stable)
• Hypergeometric0F1 — Hypergeometric0F1[b, z] (Stable)
• Hypergeometric1F1 — Hypergeometric1F1[a, b, z] (Stable)
• Hypergeometric2F1 — Hypergeometric2F1[a, b, c, z] (Stable)
• HypergeometricPFQ — HypergeometricPFQ[{a1, ...}, {b1, ...}, z] (Stable)
• HypergeometricU — HypergeometricU[a, b, z] (Stable)
• InverseErf — InverseErf[s] (Stable)
• InverseErfc — InverseErfc[s] (Stable)
• LaguerreL — LaguerreL[n, x] (Stable)
• LegendreP — LegendreP[n, x] (Stable)
• LegendreQ — LegendreQ[n, x] (Stable)
• LerchPhi — LerchPhi[z, s, a] (Stable)
• LogGamma — LogGamma[z] (Stable)
• LogIntegral — LogIntegral[z] (Stable)
• MeijerG — MeijerG[{{a1, ..., an}, {an+1, ..., ap}}, {{b1, ..., bm}, {bm+1, ..., bq}}, z] (Stable)
• Pochhammer — Pochhammer[a, n] (Stable)
• PolyGamma — PolyGamma[z] (Stable)
• PolyLog — PolyLog[n, z] (Stable)
• ProductLog — ProductLog[z] (Stable)
• QPochhammer — QPochhammer[a, q, n] (Stable)
• SinIntegral — SinIntegral[z] (Stable)
• Sinc — Sinc[z] (Stable)
• SinhIntegral — SinhIntegral[z] (Stable)
• StieltjesGamma — StieltjesGamma[n] (Stable)
• StruveH — StruveH[nu, z] (Stable)
• WhittakerM — WhittakerM[k, m, z] (Stable)
• WhittakerW — WhittakerW[k, m, z] (Stable)
• Zeta — Zeta[s] (Stable)