# SpecialFunctionCategoryΒΆ

Category for the other special functions.

abs: % -> %

abs(x) returns the absolute value of x.

airyAi: % -> %

airyAi(x) is the Airy function Ai(x).

airyAiPrime: % -> %

airyAiPrime(x) is the derivative of the Airy function Ai(x).

airyBi: % -> %

airyBi(x) is the Airy function Bi(x).

airyBiPrime: % -> %

airyBiPrime is the derivative of the Airy function Bi(x).

angerJ: (%, %) -> %

angerJ(v, z) is the Anger J function.

besselI: (%, %) -> %

besselI(v, z) is the modified Bessel function of the first kind.

besselJ: (%, %) -> %

besselJ(v, z) is the Bessel function of the first kind.

besselK: (%, %) -> %

besselK(v, z) is the modified Bessel function of the second kind.

besselY: (%, %) -> %

besselY(v, z) is the Bessel function of the second kind.

Beta: (%, %) -> %

Beta(x, y) is Gamma(x) * Gamma(y)/Gamma(x+y).

Beta: (%, %, %) -> %

Beta(x, a, b) is the incomplete Beta function.

ceiling: % -> %

ceiling(x) returns the smallest integer above or equal x.

charlierC: (%, %, %) -> %

charlierC(n, a, z) is the Charlier polynomial

conjugate: % -> %

conjugate(x) returns the conjugate of x.

digamma: % -> %

digamma(x) is the logarithmic derivative of Gamma(x) (often written psi(x) in the literature).

diracDelta: % -> %

diracDelta(x) is unit mass at zeros of x.

ellipticE: % -> %

ellipticE(m) is the complete elliptic integral of the second kind: ellipticE(m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t = 0..1).

ellipticE: (%, %) -> %

ellipticE(z, m) is the incomplete elliptic integral of the second kind: ellipticE(z, m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t = 0..z).

ellipticF: (%, %) -> %

ellipticF(z, m) is the incomplete elliptic integral of the first kind : ellipticF(z, m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)), t = 0..z).

ellipticK: % -> %

ellipticK(m) is the complete elliptic integral of the first kind: ellipticK(m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)), t = 0..1).

ellipticPi: (%, %, %) -> %

ellipticPi(z, n, m) is the incomplete elliptic integral of the third kind: ellipticPi(z, n, m) = integrate(1/((1-n*t^2)*sqrt((1-t^2)*(1-m*t^2))), t = 0..z).

floor: % -> %

floor(x) returns the largest integer below or equal x.

fractionPart: % -> %

fractionPart(x) returns the fractional part of x. Note: fractionPart(x) = x - floor(x).

Gamma: % -> %

Gamma(x) is the Euler Gamma function.

Gamma: (%, %) -> %

Gamma(a, x) is the incomplete Gamma function.

hankelH1: (%, %) -> %

hankelH1(v, z) is first Hankel function (Bessel function of the third kind).

hankelH2: (%, %) -> %

hankelH2(v, z) is the second Hankel function (Bessel function of the third kind).

hermiteH: (%, %) -> %

hermiteH(n, z) is the Hermite polynomial

hypergeometricF: (List %, List %, %) -> % if % has RetractableTo Integer

hypergeometricF(la, lb, z) is the generalized hypergeometric function.

jacobiCn: (%, %) -> %

jacobiCn(z, m) is the Jacobi elliptic cn function, defined by jacobiCn(z, m)^2 + jacobiSn(z, m)^2 = 1 and jacobiCn(0, m) = 1.

jacobiDn: (%, %) -> %

jacobiDn(z, m) is the Jacobi elliptic dn function, defined by jacobiDn(z, m)^2 + m*jacobiSn(z, m)^2 = 1 and jacobiDn(0, m) = 1.

jacobiP: (%, %, %, %) -> %

jacobiP(n, a, b, z) is the Jacobi polynomial

jacobiSn: (%, %) -> %

jacobiSn(z, m) is the Jacobi elliptic sn function, defined by the formula jacobiSn(ellipticF(z, m), m) = z.

jacobiTheta: (%, %) -> %

jacobiTheta(z, m) is the Jacobi Theta function in Jacobi notation.

jacobiZeta: (%, %) -> %

jacobiZeta(z, m) is the Jacobi elliptic zeta function, defined by D(jacobiZeta(z, m), z) = jacobiDn(z, m)^2 - ellipticE(m)/ellipticK(m) and jacobiZeta(0, m) = 0.

kelvinBei: (%, %) -> %

kelvinBei(v, z) is the Kelvin bei function defined by equality kelvinBei(v, z) = imag(besselJ(v, exp(3*\%pi*\%i/4)*z)) for z and v real.

kelvinBer: (%, %) -> %

kelvinBer(v, z) is the Kelvin ber function defined by equality kelvinBer(v, z) = real(besselJ(v, exp(3*\%pi*\%i/4)*z)) for z and v real.

kelvinKei: (%, %) -> %

kelvinKei(v, z) is the Kelvin kei function defined by equality kelvinKei(v, z) = imag(exp(-v*\%pi*\%i/2)*besselK(v, exp(\%pi*\%i/4)*z)) for z and v real.

kelvinKer: (%, %) -> %

kelvinKer(v, z) is the Kelvin kei function defined by equality kelvinKer(v, z) = real(exp(-v*\%pi*\%i/2)*besselK(v, exp(\%pi*\%i/4)*z)) for z and v real.

kummerM: (%, %, %) -> %

kummerM(mu, nu, z) is the Kummer M function.

kummerU: (%, %, %) -> %

kummerU(mu, nu, z) is the Kummer U function.

laguerreL: (%, %, %) -> %

laguerreL(n, a, z) is the Laguerre polynomial

lambertW: % -> %

lambertW(z) = w is the principial branch of the solution to the equation we^w = z.

legendreP: (%, %, %) -> %

legendreP(nu, mu, z) is the Legendre P function.

legendreQ: (%, %, %) -> %

legendreQ(nu, mu, z) is the Legendre Q function.

lerchPhi: (%, %, %) -> %

lerchPhi(z, s, a) is the Lerch Phi function.

lommelS1: (%, %, %) -> %

lommelS1(mu, nu, z) is the Lommel s function.

lommelS2: (%, %, %) -> %

lommelS2(mu, nu, z) is the Lommel S function.

meijerG: (List %, List %, List %, List %, %) -> % if % has RetractableTo Integer

meijerG(la, lb, lc, ld, z) is the meijerG function.

meixnerM: (%, %, %, %) -> %

meixnerM(n, b, c, z) is the Meixner polynomial

polygamma: (%, %) -> %

polygamma(k, x) is the k-th derivative of digamma(x), (often written psi(k, x) in the literature).

polylog: (%, %) -> %

polylog(s, x) is the polylogarithm of order s at x.

riemannZeta: % -> %

riemannZeta(z) is the Riemann Zeta function.

sign: % -> %

sign(x) returns the sign of x.

struveH: (%, %) -> %

struveH(v, z) is the Struve H function.

struveL: (%, %) -> %

struveL(v, z) is the Struve L function defined by the formula struveL(v, z) = -\%i^exp(-v*\%pi*\%i/2)*struveH(v, \%i*z).

unitStep: % -> %

unitStep(x) is 0 for x less than 0, 1 for x bigger or equal 0.

weberE: (%, %) -> %

weberE(v, z) is the Weber E function.

weierstrassP: (%, %, %) -> %

weierstrassP(g2, g3, z) is the Weierstrass P function.

weierstrassPInverse: (%, %, %) -> %

weierstrassPInverse(g2, g3, z) is the inverse of Weierstrass P function, defined by the formula weierstrassP(g2, g3, weierstrassPInverse(g2, g3, z)) = z.

weierstrassPPrime: (%, %, %) -> %

weierstrassPPrime(g2, g3, z) is the derivative of Weierstrass P function.

weierstrassSigma: (%, %, %) -> %

weierstrassSigma(g2, g3, z) is the Weierstrass Sigma function.

weierstrassZeta: (%, %, %) -> %

weierstrassZeta(g2, g3, z) is the Weierstrass Zeta function.

whittakerM: (%, %, %) -> %

whittakerM(k, m, z) is the Whittaker M function.

whittakerW: (%, %, %) -> %

whittakerW(k, m, z) is the Whittaker W function.