SpecialFunctionCategoryΒΆ

trigcat.spad line 235

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).
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).
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).
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.
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).
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.