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Special Functions implementation survey (2025-12-14)
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| Class | SubClass | Function Name | NO | SpecialFunctions.jl | Mathematica | scipy | Desc | DLMF | MathWorld | |
|---|---|---|---|---|---|---|---|---|---|---|
| Gamma Functions | Factorial Function | n! | 1 | factorial(n::Integer) | n! | factorial(n) | factorial of non-negative integer n | |||
| Gamma Functions | Factorial Function | n!! | 2 | n!! | factorial2(n) | Double factorial | ||||
| Gamma Functions | Factorial Function | FactorialK(n, k) | 3 | factorialk(n, k) | Multifactorial of n of order k | |||||
| Gamma Functions | Factorial Function | Binomial(n, m) | 4 | binomial(n::Integer, k::Integer) | Binomial[n, m] | binom(x, y) | binomial coefficient | |||
| Gamma Functions | Factorial Function | n$ | 5 | BarnesG[z] | Superfactorial, equivalent to the integral values of the Barnes G-function | https://mathworld.wolfram.com/Superfactorial.html | ||||
| Gamma Functions | Factorial Function | H(n) | 6 | Hyperfactorial[n] | Hyperfactorial | https://mathworld.wolfram.com/Hyperfactorial.html | ||||
| Gamma Functions | Factorial Function | Multinomial(n1, n2, …) | 7 | Multinomial[n1, n2, ...] | multinomial coefficient | |||||
| Gamma Functions | Gamma Function | Γ(z) | 8 | gamma(z) | Gamma[z] | gamma(z) | gamma function | https://mathworld.wolfram.com/GammaFunction.html | ||
| Gamma Functions | Gamma Function | 1/Γ(z) | 9 | rgamma(z) | Reciprocal of the gamma function | |||||
| Gamma Functions | Gamma Function | ln Γ(z) | 10 | loggamma(z) | LogGamma[z] | loggamma(z) | log gamma function | |||
| Gamma Functions | Gamma Function | ln |Γ(z)| | 11 | logabsgamma(x) | gammaln(x) | log abs gamma function | ||||
| Gamma Functions | Gamma Function | ψ(z) | 12 | digamma(x) | PolyGamma[z] | psi(z); digamma(z) | psi function, digamma function | https://mathworld.wolfram.com/DigammaFunction.html | ||
| Gamma Functions | Gamma Function | ψ'(z) | 13 | trigamma(x) | trigamma function | |||||
| Gamma Functions | Gamma Function | ψ⁽ⁿ⁾(z) | 14 | polygamma(m, x) | PolyGamma[n, z] | polygamma(n, x) | polygamma functions | |||
| Gamma Functions | Gamma Function | Γd(a) | 15 | multigammaln(a, d) | log of multivariate gamma, generalized gamma | |||||
| Gamma Functions | Gamma Function | G(n) | 16 | BarnesG[z] | Barnes G-function | https://mathworld.wolfram.com/BarnesG-Function.html | ||||
| Gamma Functions | Gamma Function | ln G(n) | 17 | LogBarnesG[z] | logarithm of the Barnes G-function | |||||
| Gamma Functions | Gamma Function | K(n) | 18 | K-Function, | https://mathworld.wolfram.com/K-Function.html | |||||
| Gamma Functions | Incomplete Gamma Function | γ(a, z) | 19 | Gamma[a, 0, z] | (lower, [0, z]) incomplete gamma function | https://dlmf.nist.gov/8.2.1 | https://mathworld.wolfram.com/IncompleteGammaFunction.html | |||
| Gamma Functions | Incomplete Gamma Function | Γ(a, z) | 20 | gamma_inc(a, x, IND=0) | Gamma[a, z] | (upper, [z, Inf]) incomplete gamma function | https://dlmf.nist.gov/8.2.2 | https://mathworld.wolfram.com/IncompleteGammaFunction.html | ||
| Gamma Functions | Incomplete Gamma Function | γ*(a, z) | 21 | Tricomi’s incomplete gamma function | https://dlmf.nist.gov/8.2.6 | |||||
| Gamma Functions | Incomplete Gamma Function | P(a, z) | 22 | GammaRegularized[a, 0, z] | gammainc(a, x) | γ(a, z)/Γ(z), Normalized lower incomplete gamma function | https://dlmf.nist.gov/8.2.4 | https://mathworld.wolfram.com/RegularizedGammaFunction.html | ||
| Gamma Functions | Incomplete Gamma Function | Q(a, z) | 23 | GammaRegularized[a, z] | gammaincc(a, x) | Γ(a, z)/Γ(z), Normalized upper incomplete gamma function | https://dlmf.nist.gov/8.2.4 | https://mathworld.wolfram.com/RegularizedGammaFunction.html | ||
| Gamma Functions | Incomplete Gamma Function | P(a, y)_inv | 24 | gammaincinv(a, y) | Inverse to the regularized lower incomplete gamma function | |||||
| Gamma Functions | Incomplete Gamma Function | Q(a, y)_inv | 25 | gammainccinv(a, y) | Inverse of the regularized upper incomplete gamma function | |||||
| Gamma Functions | Pochhammer Function | poch(z, n) | 26 | Pochhammer[a, n] | poch(z, m) | Pochhammer’s symbol (or shifted factorial) | ||||
| Gamma Functions | Pochhammer Function | poch1(z, n) | 27 | ≡ (poch(z, n) - 1)/z | ||||||
| Gamma Functions | Beta Function | B(a, b) | 28 | beta(x, y) | Beta[a,b] | beta(a, b) | Beta function | |||
| Gamma Functions | Beta Function | Bₓ(a, b) | 29 | Beta[z,a,b] | betainc(a, b, x) | incomplete beta function | ||||
| Gamma Functions | Beta Function | Iₓ(a, b) | 30 | beta_inc(a, b, x) | betaincc(a, b, x) | regularized incomplete beta function | ||||
| Gamma Functions | Beta Function | 1-Iₓ(a, b) | 31 | betaincc(a, b, x) | Complement of the regularized incomplete beta function | |||||
| Gamma Functions | Beta Function | ln B(a, b) | 32 | logbeta(x, y) | log beta function | |||||
| Gamma Functions | Beta Function | ln |B(a, b)| | 33 | logabsbeta(x, y) | betaln(a, b) | log abs beta function | ||||
| Gamma Functions | Beta Function | beta_inc_inv(a, b, y) | 34 | beta_inc_inv(a,b, p,q) | Inverse of the regularized incomplete beta function | |||||
| Exponential and Trigonometric Integrals | Exponential Integral | E₁(z) | 35 | expint(z::Complex) | ExpIntegralE[1, z] | exp1(z) | (principal value of) exponential integral | |||
| Exponential and Trigonometric Integrals | Exponential Integral | Eν(z) | 36 | expint(ν::Complex, z::Complex) | ExpIntegralE[n,z] | expn(n, x) | generalized exponential integral | |||
| Exponential and Trigonometric Integrals | Exponential Integral | eᶻEν(z) | 37 | expintx(ν::Complex, z::Complex) | scaled (generalized) exponential integral | |||||
| Exponential and Trigonometric Integrals | Exponential Integral | Ei(x) | 38 | expinti(x::Real) | ExpIntegralEi[z] | expi(x) | exponential integral | |||
| Exponential and Trigonometric Integrals | Exponential Integral | Li(x), {x>1} | 39 | logint(x::Real) | LogIntegral[z] | logarithmic integral | ||||
| Exponential and Trigonometric Integrals | Trigonometric Integral | Si(z) | 40 | sinint(z) | SinIntegral[z] | sici(x) | sine integral function | |||
| Exponential and Trigonometric Integrals | Trigonometric Integral | Ci(z) | 41 | cosint(z) | CosIntegral[z] | sici(x) | cosine integral function | |||
| Exponential and Trigonometric Integrals | Trigonometric Integral | Shi(z) | 42 | sinhint(z) | SinhIntegral[z] | shichi(x) | hyperbolic sine integral function | |||
| Exponential and Trigonometric Integrals | Trigonometric Integral | Chi(z) | 43 | coshint(z) | CoshIntegral[z] | shichi(x) | hyperbolic cosine integral function | |||
| Error Functions | Error Function | erf(z) | 44 | erf(x) | Erf[z] | error function | ||||
| Error Functions | Error Function | erf(z0, z1) | 45 | erf(x,y) | Erf[z0,z1] | generalized error function `erf(z1) - erf(z0)` | ||||
| Error Functions | Error Function | erfc(z) | 46 | erfc(x) | Erfc[z] | complementary error function | ||||
| Error Functions | Error Function | erfcx(z) | 47 | erfcx(x) | Scaled complementary error function | |||||
| Error Functions | Error Function | ω(z) | 48 | faddeeva(x) | Faddeeva function | |||||
| Error Functions | Error Function | erfi(z) | 49 | erfi(x) | Erfi[z] | Imaginary error function | ||||
| Error Functions | Error Function | erf_inv(z) | 50 | erfinv(x) | InverseErf[s] | inverse error function | ||||
| Error Functions | Error Function | erfc_inv(z) | 51 | erfcinv(x) | InverseErfc[s] | inverse complementary error function | ||||
| Error Functions | Dawson Integral | dawson(z) | 52 | dawson(x) | Dawson integral | |||||
| Error Functions | Fresnel Integral | C(z) | 53 | FewSpecialFunctions.FresnelC(x) | FresnelC[z] | Fresnel integral C(z) | ||||
| Error Functions | Fresnel Integral | S(z) | 54 | FewSpecialFunctions.FresnelS(x) | FresnelS[z] | Fresnel integral S(z) | ||||
| Error Functions | Fresnel Integral | f(z) | 55 | FresnelF[z] | Fresnel auxiliary function f(z) | |||||
| Error Functions | Fresnel Integral | g(z) | 56 | FresnelG[z] | Fresnel auxiliary function g(z) | |||||
| Error Functions | Fresnel Integral | ℱ(z) | 57 | Fresnel Integrals ℱ(z) | ||||||
| Error Functions | Fresnel Integral | G(z) | 58 | Goodwin–Staton Integral | ||||||
| Error Functions | Fresnel Integral | U(x, t) | 59 | Voigt Functions U | ||||||
| Error Functions | Fresnel Integral | V(x, t) | 60 | Voigt Functions V | ||||||
| Airy Functions | Airy Function | Ai(z) | 61 | airyai(z) | AiryAi[z] | airy(z) | Airy Ai function | |||
| Airy Functions | Airy Function | Aix(z) | 62 | airyaix(z) | airye(z) | scaled Airy Ai function | ||||
| Airy Functions | Airy Function | Bi(z) | 63 | airybi(z) | AiryBi[z] | airy(z) | Airy Bi function | |||
| Airy Functions | Airy Function | Bix(z) | 64 | airybix(z) | airye(z) | scaled Airy Bi function | ||||
| Airy Functions | Airy Function | Ai'(z) | 65 | airyaiprime(z) | AiryAiPrime[z] | airy(z) | derivative of Airy Ai function | |||
| Airy Functions | Airy Function | Aix'(z) | 66 | airyaiprimex(z) | airye(z) | scaled derivative of Airy Ai function | ||||
| Airy Functions | Airy Function | Bi'(z) | 67 | airybiprime(z) | AiryBiPrime[z] | airy(z) | derivative of Airy Bi function | |||
| Airy Functions | Airy Function | Bix'(z) | 68 | airybiprimex(z) | airye(z) | scaled derivative of Airy Bi function | ||||
| Airy Functions | Zeros of Airy Function | AiZeros(nt) | 69 | AiryAiZero[k] | ai_zeros(nt) | Compute nt zeros and values of the Airy function Ai and its derivative | ||||
| Airy Functions | Zeros of Airy Function | BiZeros(nt) | 70 | AiryBiZero[k] | bi_zeros(nt) | Compute nt zeros and values of the Airy function Bi and its derivative | ||||
| Airy Functions | Integral of Airy Function | Ai_int(z) | 71 | itairy(x) | Integral of Airy Ai function | |||||
| Airy Functions | Integral of Airy Function | Bi_int(z) | 72 | itairy(x) | Integral of Airy Bi function | |||||
| Airy Functions | Scorer Function | Gi(z) | 73 | ScorerGi[z] | Scorer function Gi(z) | |||||
| Airy Functions | Scorer Function | Gi'(z) | 74 | ScorerGiPrime[z] | derivative of the Scorer function Gi(z) | |||||
| Airy Functions | Scorer Function | Hi(z) | 75 | ScorerHi[z] | Scorer function Hi(z) | |||||
| Airy Functions | Scorer Function | Hi'(z) | 76 | ScorerHiPrime[z] | derivative of the Scorer function Hi(z) | |||||
| Bessel Functions | Bessel Function | Jν(x) | 77 | besselj(nu,z) | BesselJ[n,z] | jv(v, z[, out]) | ||||
| Bessel Functions | Bessel Function | J0(x) | 78 | besselj0(z) | ||||||
| Bessel Functions | Bessel Function | J1(x) | 79 | besselj1(z) | ||||||
| Bessel Functions | Bessel Function | Jν(x)*exp(?) | 80 | besseljx(nu,z) | jve(v, z[, out]) | |||||
| Bessel Functions | Bessel Function | Yν(x) | 81 | bessely(nu,z) | BesselY[n,z] | yv(v, z[, out]) | ||||
| Bessel Functions | Bessel Function | Y0(x) | 82 | bessely0(z) | ||||||
| Bessel Functions | Bessel Function | Y1(x) | 83 | bessely1(z) | ||||||
| Bessel Functions | Bessel Function | Yν(x)*exp(?) | 84 | besselyx(nu,z) | yve(v, z[, out]) | |||||
| Bessel Functions | Modified Bessel Function | Iν(x) | 85 | besseli(nu,z) | BesselI[n,z] | iv(v, z[, out]) | ||||
| Bessel Functions | Modified Bessel Function | I0(x) | 86 | besseli0(z) | ||||||
| Bessel Functions | Modified Bessel Function | I1(x) | 87 | besseli1(z) | ||||||
| Bessel Functions | Modified Bessel Function | Iν(x)*exp(-x) | 88 | besselix(nu,z) | ive(v, z[, out]) | |||||
| Bessel Functions | Modified Bessel Function | Kν(x) | 89 | besselk(nu,z) | BesselK[n,z] | kv(v, z[, out]) | ||||
| Bessel Functions | Modified Bessel Function | K0(x) | 90 | besselk0(z) | ||||||
| Bessel Functions | Modified Bessel Function | K1(x) | 91 | besselk1(z) | ||||||
| Bessel Functions | Modified Bessel Function | Kν(x)*exp(x) | 92 | besselkx(nu,z) | kve(v, z[, out]) | |||||
| Bessel Functions | Hankel Function | Hkν(z) | 93 | besselh(nu,k,z) | Bessel function of third kind (Hankel function) | |||||
| Bessel Functions | Hankel Function | H1v(z) | 94 | hankelh1(nu,z) | HankelH1[n,z] | hankel1(v, z[, out]) | Hankel function of the first kind | |||
| Bessel Functions | Hankel Function | H1vx(z) | 95 | hankelh1x(nu,z) | hankel1e(v, z[, out]) | Exponentially scaled Hankel function of the first kind | ||||
| Bessel Functions | Hankel Function | H2v(z) | 96 | hankelh2(nu,z) | HankelH2[n,z] | hankel2(v, z[, out]) | Hankel function of the second kind | |||
| Bessel Functions | Hankel Function | H2vx(z) | 97 | hankelh2x(nu,z) | hankel2e(v, z[, out]) | Exponentially scaled Hankel function of the second kind | ||||
| Bessel Functions | Hankel Function | H1v‘(z) | 98 | derivatives of Hankel function of the first kind | ||||||
| Bessel Functions | Hankel Function | H2v’(z) | 99 | derivatives of Hankel function of the second kind | ||||||
| Bessel Functions | Spherical Bessel Function | jν(x) | 100 | sphericalbesselj(ν, x) | spherical_jn(n, z[, derivative]) | |||||
| Bessel Functions | Spherical Bessel Function | yν(x) | 101 | sphericalbessely(ν, x) | spherical_yn(n, z[, derivative]) | |||||
| Bessel Functions | Spherical Bessel Function | iν(x) | 102 | Bessels.sphericalbesseli(ν, x) | spherical_in(n, z[, derivative]) | |||||
| Bessel Functions | Spherical Bessel Function | kν(x) | 103 | Bessels.sphericalbesselk(ν, x) | spherical_kn(n, z[, derivative]) | |||||
| Bessel Functions | Kelvin Function | kelvin(x) | 104 | kelvin(x[, out]) | Kelvin functions as complex numbers | |||||
| Bessel Functions | Kelvin Function | kelvin_zeros(nt) | 105 | kelvin_zeros(nt) | Zero of Kelvin Function | |||||
| Bessel Functions | Kelvin Function | ber(x) | 106 | KelvinBer[n,z] | ber(x) | Kelvin Function of the First Kind bei | ||||
| Bessel Functions | Kelvin Function | bei(x) | 107 | KelvinBei[n,z] | bei(x) | Kelvin Function of the First Kind ber | ||||
| Bessel Functions | Kelvin Function | ker(x) | 108 | KelvinKer[n,z] | ker(x) | Kelvin Function of the Second Kind ker | ||||
| Bessel Functions | Kelvin Function | kei(x) | 109 | KelvinKei[n,z] | kei(x) | Kelvin Function of the Second Kind kei | ||||
| Bessel Functions | Kelvin Function | ber'(x) | 110 | berp(x) | Derivative of the Kelvin Function of the First Kind ber | |||||
| Bessel Functions | Kelvin Function | bei'(x) | 111 | beip(x) | Derivative of the Kelvin Function of the First Kind bei | |||||
| Bessel Functions | Kelvin Function | ker'(x) | 112 | kerp(x) | Derivative of the Kelvin Function of the Second Kind ker | |||||
| Bessel Functions | Kelvin Function | kei'(x) | 113 | keip(x) | Derivative of the Kelvin Function of the Second Kind kei | |||||
| Struve Functions | Struve Function | Hν(z) | 114 | Struve.jl | StruveH[n,z] | Struve function | https://dlmf.nist.gov/11.2.1 | |||
| Struve Functions | Struve Function | Lν(z) | 115 | StruveL[n,z] | Modified Struve function | https://dlmf.nist.gov/11.2.2 | ||||
| Struve Functions | Struve Function | ∫Hν(z) | 116 | Integrals of Struve function | ||||||
| Struve Functions | Struve Function | ∫Lν(z) | 117 | Integrals of Modified Struve function | ||||||
| Struve Functions | Lommel Function | sμν(z) | 118 | LommelS1[m,n,z] | Lommel function `s` | https://dlmf.nist.gov/11.9.3 | ||||
| Struve Functions | Lommel Function | Sμν(z) | 119 | LommelS2[m,n,z] | Lommel function `S` | https://dlmf.nist.gov/11.9.5 | ||||
| Struve Functions | Lommel Function | tₘₙ(z) | 120 | LommelT1[m,n,z] | Modified Lommel Function `t` | |||||
| Struve Functions | Lommel Function | Tₘₙ(z) | 121 | LommelT2.[m,n,z] | Modified Lommel Function `T` | |||||
| Struve Functions | Anger and Weber Function | Jν(z) | 122 | AngerJ[ν,z] | Anger Function | https://dlmf.nist.gov/11.10.1 | ||||
| Struve Functions | Anger and Weber Function | Eν(z) | 123 | WeberE[ν,z] | Weber Function | https://dlmf.nist.gov/11.10.2 | ||||
| Struve Functions | Anger and Weber Function | Aν(z) | 124 | AngerWeberA[ν,z] | associated Anger-Weber function | https://dlmf.nist.gov/11.10.4 | ||||
| Struve Functions | Anger and Weber Function | ∫Jν(z) | 125 | Integrals of Anger Function | ||||||
| Struve Functions | Anger and Weber Function | ∫Eν(z) | 126 | Integrals of Weber Function | ||||||
| Parabolic Cylinder Functions | Parabolic Cylinder Function | Dν(z) | 127 | ParabolicCylinderD[ν,z] | pbdv(v, x) | Parabolic cylinder function, in Whittaker’s notation Dn | ||||
| Parabolic Cylinder Functions | Parabolic Cylinder Function | V(a, z) | 128 | FewSpecialFunctions.V(a, x) | ParabolicCylinderV[a,z] | pbvv(v, x) | Parabolic cylinder function V | |||
| Parabolic Cylinder Functions | Parabolic Cylinder Function | U(a, z) | 129 | FewSpecialFunctions.U(a, x) | ParabolicCylinderU[a,z] | Parabolic cylinder function U | ||||
| Parabolic Cylinder Functions | Parabolic Cylinder Function | W(a, z) | 130 | pbwa(a, x) | Parabolic cylinder function W | https://dlmf.nist.gov/12.14 | ||||
| Parabolic Cylinder Functions | Parabolic Cylinder Function | Dv'(x) | 131 | pbdv(v, x) | ||||||
| Parabolic Cylinder Functions | Parabolic Cylinder Function | Vv'(z) | 132 | pbvv(v, x) | ||||||
| Hypergeometric Functions | Hypergeometric Function | ₂F₁(a,b,c,x) | 133 | HypergeometricFunctions._₂F₁ | Hypergeometric2F1[a,b,c,z] | hyp2f1(a, b, c, z) | Gauss hypergeometric function 2F1 | https://dlmf.nist.gov/15.2.2 | ||
| Hypergeometric Functions | Confluent Hypergeometric Function | ₀F₁(a,z) | 134 | Hypergeometric0F1[a,z] | hyp0f1(v, z[, out]) | hypergeometric function 0F1 | ||||
| Hypergeometric Functions | Kummer Functions | ₁F₁(a,b,z) | 135 | HypergeometricFunctions._₁F₁ | Hypergeometric1F1[a,b,z] | hyp1f1(a, b, x[, out]) | Kummer confluent hypergeometric function 1F1 | https://dlmf.nist.gov/13.2.2 | ||
| Hypergeometric Functions | Kummer Functions | U(a,b,x) | 136 | HypergeometricFunctions.U(a, b, z) | HypergeometricU[a,b,z] | hyperu(a, b, x[, out]) | confluent hypergeometric function U | https://dlmf.nist.gov/13.2.6 | ||
| Hypergeometric Functions | Kummer Functions | M(a,b,x) | 137 | HypergeometricFunctions.M(a, b, z) | Olver’s confluent hypergeometric function | https://dlmf.nist.gov/13.2.3 | ||||
| Hypergeometric Functions | Whittaker Functions | Mκμ(z) | 138 | WhittakerM[k,m,z] | Whittaker confluent hypergeometric function M | https://dlmf.nist.gov/13.14.2 | ||||
| Hypergeometric Functions | Whittaker Functions | Wκμ(z) | 139 | WhittakerW[k,m,z] | Whittaker confluent hypergeometric function W | https://dlmf.nist.gov/13.14.3 | ||||
| Hypergeometric Functions | Generalized Hypergeometric Function | pFq(A, B, z) | 140 | HypergeometricFunctions.pFq(α, β, z) | HypergeometricPFQ[{a1,…,ap},{b1,…,bq},z] | Generalized Hypergeometric Function | ||||
| Hypergeometric Functions | Generalized Hypergeometric Function | Gmnpq(A, B, z) | 141 | MeijerG[{{a1,…,an},{an+1,…,ap}},{{b1,…,bm},{bm+1,…,bq}},z] | Meijer G-Function | |||||
| Legendre Functions | Legendre Function | Pn(z) | 142 | LegendrePolynomials.Pl(x, l) | LegendreP[n,x] | Legendre functions of the first kind | ||||
| Legendre Functions | Legendre Function | Qn(z) | 143 | LegendreQ[n,z] | Legendre functions of the second kind | |||||
| Legendre Functions | Associated Legendre Function | Pmn(z) | 144 | LegendrePolynomials.Plm(x, l, m) | LegendreP[n,m,x] | associated Legendre functions of the first kind | ||||
| Legendre Functions | Associated Legendre Function | Qmn(z) | 145 | LegendreQ[n,m,z] | associated Legendre functions of the second kind | |||||
| Legendre Functions | Spherical and Spheroidal Harmonics | Yml(θ, ϕ) | 146 | spherical harmonic | ||||||
| q Functions | 147 | |||||||||
| Orthogonal Polynomials | 148 | |||||||||
| Elliptic Integrals | Legendre Integral | F(Φ, m) | 149 | JacobiElliptic.F(φ, m) | EllipticF[ϕ,m] | ellipkinc(phi, m) | (Legendre’s) incomplete elliptic integral of the first kind | |||
| Elliptic Integrals | Legendre Integral | E(Φ, m) | 150 | JacobiElliptic.E(φ, m) | EllipticE[ϕ,m] | ellipeinc(phi, m) | (Legendre’s) incomplete elliptic integral of the second kind | |||
| Elliptic Integrals | Legendre Integral | D(Φ, m) | 151 | incomplete elliptic integral of Legendre’s type | ||||||
| Elliptic Integrals | Legendre Integral | Π(Φ, u, m) | 152 | JacobiElliptic.Pi(n, φ, m) | EllipticPi[n,ϕ,m] | (Legendre’s) incomplete elliptic integral of the third kind | ||||
| Elliptic Integrals | Legendre Integral | K(m) | 153 | ellipk(m) | EllipticK[m] | ellipk(m) | complete elliptic integral of the first kind | |||
| Elliptic Integrals | Legendre Integral | E(m) | 154 | ellipe(m) | EllipticE[m] | ellipe(m) | complete elliptic integral of the second kind | |||
| Elliptic Integrals | Legendre Integral | Π(u, m) | 155 | JacobiElliptic.Pi(n, m) | EllipticPi[n,m] | ≡ Π(π/2, u, m), complete elliptic integral of the third kind | ||||
| Elliptic Integrals | Symmetric Integral | RF(x, y, z) | 156 | EllipticFunctions.CarlsonRF(x, y, z) | CarlsonRF[x, y, z] | elliprf(x, y, z) | symmetric elliptic integral of first kind | |||
| Elliptic Integrals | Symmetric Integral | RG(x, y, z) | 157 | EllipticFunctions.CarlsonRG(x, y, z) | CarlsonRG[x, y, z] | elliprg(x, y, z) | symmetric elliptic integral of second kind | |||
| Elliptic Integrals | Symmetric Integral | RJ(x, y, z, p) | 158 | EllipticFunctions.CarlsonRJ(x, y, z) | CarlsonRJ[x, y, z, rho] | elliprj(x, y, z, p) | symmetric elliptic integral of third kind | |||
| Elliptic Integrals | Symmetric Integral | RD(x, y, z) | 159 | EllipticFunctions.CarlsonRD(x, y, z) | CarlsonRD[x, y, z] | elliprd(x, y, z) | ≡ RJ(x, y, z, z), elliptic integral symmetric in only two variables | |||
| Elliptic Integrals | Symmetric Integral | RC(x, y) | 160 | EllipticFunctions.CarlsonRC(x, y, z) | CarlsonRC[x, y] | elliprc(x, y) | Carlson’s combination of inverse circular and inverse hyperbolic functions | |||
| Elliptic Functions | Theta Functions | θ(n, z, q) | 161 | EllipticFunctions.ljtheta[1-4] | EllipticTheta[a,u,q] | (Jacobi) Theta functions | ||||
| Elliptic Functions | Jacobi Elliptic Function | sn(z, k) | 162 | JacobiElliptic.sn(u, m); EllipticFunctions.jellip | JacobiSN[u, m] | Jacobi Elliptic Function sn | ||||
| Elliptic Functions | Jacobi Elliptic Function | cn(z, k) | 163 | JacobiElliptic.cn(u, m) | JacobiCN[u, m] | Jacobi Elliptic Function cn | ||||
| Elliptic Functions | Jacobi Elliptic Function | dn(z, k) | 164 | JacobiElliptic.dn(u, m) | JacobiDN[u, m] | Jacobi Elliptic Function dn | ||||
| Elliptic Functions | Weierstrass Elliptic Function | WeierstrassP(z) | 165 | EllipticFunctions.wp(z; tau, omega) | WeierstrassP[u, {g2,g3}] | Weierstrass ℘ function | ||||
| Elliptic Functions | Weierstrass Elliptic Function | WeierstrassZeta(z) | 166 | EllipticFunctions.wzeta(z; tau, omega) | WeierstrassZeta[u, {g2,g3}] | Weierstrass zeta function | ||||
| Elliptic Functions | Weierstrass Elliptic Function | WeierstrassSigma(x) | 167 | EllipticFunctions.wsigma(z; tau, omega) | WeierstrassSigma[u, {g2,g3}] | Weierstrass sigma function | ||||
| Zeta Functions | Zeta Function | ζ(s) | 168 | zeta(x) | Zeta[s] | zeta(x) | Riemann zeta function | https://dlmf.nist.gov/25.2.1 | ||
| Zeta Functions | Zeta Function | ζ(s)-1 | 169 | zetac(x) | ||||||
| Zeta Functions | Zeta Function | ζ(s, a) | 170 | Zeta[s,a] | zeta(x[, q]) | generalized Riemann zeta function, Hurwitz zeta function | https://dlmf.nist.gov/25.11.1 | |||
| Zeta Functions | Dilogarithm | Li₂(z) | 171 | PolyLog.li2(z) | Spence’s function, dilogarithm | https://dlmf.nist.gov/25.12.1 | ||||
| Zeta Functions | Dilogarithm | Liₛ(z) | 172 | PolyLog.li(n, z); Polylogarithms.polylog(s, z) | PolyLog[n,z] | polylogarithm | https://dlmf.nist.gov/25.12.10 | |||
| Zeta Functions | Lerchs Transcendent | Φ(z, s, a) | 173 | LerchPhi[z,s,a] | Lerch transcendent | https://dlmf.nist.gov/25.14.1 | ||||
| Zeta Functions | Dirichlet L-function | L(s, χ) | 174 | DirichletL[k,j,s] | Dirichlet L-function | https://dlmf.nist.gov/25.15.1 | ||||
| Zeta Functions | Dirichlet L-function | η(s) | 175 | eta(x) | Dirichlet eta function | https://mathworld.wolfram.com/DirichletEtaFunction.html | ||||
| Number Theory Functions | 176 | |||||||||
| Mathieu Functions | Mathieu Functions | cem(a, q, z) | 177 | MathieuFunctions.ce(m, q, x) | MathieuC[a,q,z] | mathieu_cem(m, q, x) | even, periodic Mathieu functions | |||
| Mathieu Functions | Mathieu Functions | sem(a, q, z) | 178 | MathieuFunctions.se(m, q, x) | MathieuS[b,q,z] | mathieu_sem(m, q, x) | odd, periodic Mathieu functions | |||
| Mathieu Functions | Mathieu Functions | cem‘(a, q, z) | 179 | MathieuCPrime[a,q,z] | mathieu_cem(m, q, x) | z derivatives of even Mathieu functions | ||||
| Mathieu Functions | Mathieu Functions | sem’(b, q, z) | 180 | MathieuSPrime[b,q,z] | mathieu_sem(m, q, x) | z derivatives of odd Mathieu functions | ||||
| Mathieu Functions | Characteristic Value of Mathieu function | mathieu_a(n, q) | 181 | MathieuFunctions.charA(q; k) | MathieuCharacteristicA[r,q] | mathieu_a(m, q) | eigenvalues of even Mathieu functions | |||
| Mathieu Functions | Characteristic Value of Mathieu function | mathieu_b(n, q) | 182 | MathieuFunctions.charB(q, k) | MathieuCharacteristicB[r,q] | mathieu_b(m, q) | eigenvalues of odd Mathieu functions | |||
| Mathieu Functions | Characteristic Value of Mathieu function | mathieu_exp(a, q) | 183 | MathieuCharacteristicExponent[a,q] | characteristic exponent r for Mathieu functions | |||||
| Spheroidal Wave Functions | Spheroidal Wave Function | S1mn(z, γ) | 184 | SpheroidalS1[n,m,γ,z] | Radial Spheroidal Wave Function S1 | |||||
| Spheroidal Wave Functions | Spheroidal Wave Function | S2mn(z, γ) | 185 | SpheroidalS2[n,m,γ,z] | Radial Spheroidal Wave Function S2 | |||||
| Spheroidal Wave Functions | Spheroidal Wave Function | S1'mn(z, γ) | 186 | SpheroidalS1Prime[n,m,γ,z] | z derivatives of Radial Spheroidal Wave Function S1 | |||||
| Spheroidal Wave Functions | Spheroidal Wave Function | S2'mn(z, γ) | 187 | SpheroidalS2Prime[n,m,γ,z] | z derivatives of Radial Spheroidal Wave Function S2 | |||||
| Spheroidal Wave Functions | Spheroidal Wave Function | PSmn(z, γ) | 188 | SpheroidalPS[n,m,γ,z] | Angular Spheroidal Wave Function PS | |||||
| Spheroidal Wave Functions | Spheroidal Wave Function | QSmn(z, γ) | 189 | SpheroidalQS[n,m,γ,z] | Angular Spheroidal Wave Function QS | |||||
| Spheroidal Wave Functions | Spheroidal Wave Function | PS'mn(z, γ) | 190 | SpheroidalPSPrime[n,m,γ,z] | z derivatives of Angular Spheroidal Wave Function PS | |||||
| Spheroidal Wave Functions | Spheroidal Wave Function | QS'mn(z, γ) | 191 | SpheroidalQSPrime[n,m,γ,z] | z derivatives of Angular Spheroidal Wave Function QS | |||||
| Spheroidal Wave Functions | Spheroidal Eigenvalue | λmn(γ) | 192 | SpheroidalEigenvalue[n,m,γ] | Spheroidal Eigenvalue of degree `n` and order `m` | |||||
| Heun Functions | Heun Functions | HeunG | 193 | HeunG[a,q,α,β,γ,δ,z] | ||||||
| Heun Functions | Heun Functions | HeunC | 194 | HeunC[q,α,γ,δ,ϵ,z] | ||||||
| Heun Functions | Heun Functions | HeunD | 195 | HeunD[q,α,γ,δ,ϵ,z] | ||||||
| Heun Functions | Heun Functions | HeunB | 196 | HeunB[q,α,γ,δ,ϵ,z] | ||||||
| Heun Functions | Heun Functions | HeunT | 197 | HeunT[q,α,γ,δ,ϵ,z] | ||||||
| Heun Functions | Lamé Functions | LameC | 198 | LameC[ν,j,z,m] | ||||||
| Heun Functions | Lamé Functions | LameS | 199 | LameS[ν,j,z,m] | ||||||
| Heun Functions | Lamé Functions | LameEigenvalueA | 200 | LameEigenvalueA[ν,j,m] | ||||||
| Heun Functions | Lamé Functions | LameEigenvalueB | 201 | LameEigenvalueB[ν,j,m] | ||||||
| Coulomb Functions | F_L(η,ρ) | 202 | CoulombF[l,η,r] | regular Coulomb wavefunction | ||||||
| Coulomb Functions | G_L(η,ρ) | 203 | CoulombG[l,η,r] | irregular Coulomb wave function | ||||||
| Coulomb Functions | H⁺L(η,ρ) | 204 | CoulombH1[l,η,r] | outgoing irregular Coulomb wavefunction | ||||||
| Coulomb Functions | H⁻L(η,ρ) | 205 | CoulombH2[l,η,r] | incoming irregular Coulomb wavefunction | ||||||
| Coulomb Functions | σL(η) | 206 | Coulomb phase shift | https://dlmf.nist.gov/33.2.10 | ||||||
| Miscellaneous Functions | 207 |
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