FunctionalSpecialFunction(R, F)ΒΆ

combfunc.spad line 443 [edit on github]

Provides some special functions over an integral domain.

abs: F -> F

abs(f) returns the absolute value operator applied to f.

airyAi: F -> F

airyAi(x) returns the Airy Ai function applied to x.

airyAiPrime: F -> F

airyAiPrime(x) returns the derivative of Airy Ai function applied to x.

airyBi: F -> F

airyBi(x) returns the Airy Bi function applied to x.

airyBiPrime: F -> F

airyBiPrime(x) returns the derivative of Airy Bi function applied to x.

angerJ: (F, F) -> F

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

belong?: BasicOperator -> Boolean

belong?(op) returns true if op is a special function operator.

besselI: (F, F) -> F

besselI(x, y) returns the Bessel I function applied to x and y.

besselJ: (F, F) -> F

besselJ(x, y) returns the Bessel J function applied to x and y.

besselK: (F, F) -> F

besselK(x, y) returns the Bessel K function applied to x and y.

besselY: (F, F) -> F

besselY(x, y) returns the Bessel Y function applied to x and y.

Beta: (F, F) -> F

Beta(x, y) returns the beta function applied to x and y.

Beta: (F, F, F) -> F

Beta(x, a, b) is incomplete Beta function applied to x, a and b.

ceiling: F -> F

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

charlierC: (F, F, F) -> F

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

coerce_Q: Fraction Integer -> F

coerce_Q(x) should be local but conditional

conjugate: F -> F

conjugate(f) returns the conjugate value operator applied to f.

digamma: F -> F

digamma(x) returns the digamma function applied to x.

diracDelta: F -> F

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

ellipticE: (F, F) -> F

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).

ellipticE: F -> F

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).

ellipticF: (F, F) -> F

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: F -> F

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: (F, F, F) -> F

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: F -> F

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

fractionPart: F -> F

fractionPart(x) returns the fractional part of x.

Gamma: (F, F) -> F

Gamma(a, x) returns the incomplete Gamma function applied to a and x.

Gamma: F -> F

Gamma(f) returns the formal Gamma function applied to f.

hahn_p: (F, F, F, F, F) -> F

hahn_p(n, a, b, bar_a, bar_b, z) is the continuous Hahn polynomial.

hahnQ: (F, F, F, F, F) -> F

hahnQ(n, a, b, N, z) is the Hahn polynomial.

hahnR: (F, F, F, F, F) -> F

hahnR(n, c, d, N, z) is the dual Hahn polynomial.

hahnS: (F, F, F, F, F) -> F

hahnS(n, a, b, c, z) is the continuous dual Hahn polynomial.

hankelH1: (F, F) -> F

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

hankelH2: (F, F) -> F

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

hermiteH: (F, F) -> F

hermiteH(n, z) is the Hermite polynomial.

hypergeometricF: (List F, List F, F) -> F

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

iAiryAi: F -> F

iAiryAi(x) should be local but conditional.

iAiryAiPrime: F -> F

iAiryAiPrime(x) should be local but conditional.

iAiryBi: F -> F

iAiryBi(x) should be local but conditional.

iAiryBiPrime: F -> F

iAiryBiPrime(x) should be local but conditional.

iiabs: F -> F

iiabs(x) should be local but conditional.

iiAiryAi: F -> F

iiAiryAi(x) should be local but conditional.

iiAiryAiPrime: F -> F

iiAiryAiPrime(x) should be local but conditional.

iiAiryBi: F -> F

iiAiryBi(x) should be local but conditional.

iiAiryBiPrime: F -> F

iiAiryBiPrime(x) should be local but conditional.

iiBesselI: List F -> F

iiBesselI(x) should be local but conditional.

iiBesselJ: List F -> F

iiBesselJ(x) should be local but conditional.

iiBesselK: List F -> F

iiBesselK(x) should be local but conditional.

iiBesselY: List F -> F

iiBesselY(x) should be local but conditional.

iiBeta: List F -> F

iiBeta(x) should be local but conditional.

iiconjugate: F -> F

iiconjugate(x) should be local but conditional.

iidigamma: F -> F

iidigamma(x) should be local but conditional.

iiGamma: F -> F

iiGamma(x) should be local but conditional.

iiHypergeometricF: List F -> F

iiHypergeometricF(l) should be local but conditional.

iipolygamma: List F -> F

iipolygamma(x) should be local but conditional.

iiPolylog: (F, F) -> F

iiPolylog(x, s) should be local but conditional.

iLambertW: F -> F

iLambertW(x) should be local but conditional.

jacobiCn: (F, F) -> F

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: (F, F) -> F

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: (F, F, F, F) -> F

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

jacobiSn: (F, F) -> F

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

jacobiTheta: (F, F) -> F

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

jacobiZeta: (F, F) -> F

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: (F, F) -> F

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: (F, F) -> F

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: (F, F) -> F

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: (F, F) -> F

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.

krawtchoukK: (F, F, F, F) -> F

krawtchoukK(n, p, N, z) is the Krawtchouk polynomial.

kummerM: (F, F, F) -> F

kummerM(a, b, z) is the Kummer M function.

kummerU: (F, F, F) -> F

kummerU(a, b, z) is the Kummer U function.

laguerreL: (F, F, F) -> F

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

lambertW: F -> F

lambertW(x) is the Lambert W function at x.

legendreP: (F, F, F) -> F

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

legendreQ: (F, F, F) -> F

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

lerchPhi: (F, F, F) -> F

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

lommelS1: (F, F, F) -> F

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

lommelS2: (F, F, F) -> F

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

meijerG: (List F, List F, List F, List F, F) -> F

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

meixnerM: (F, F, F, F) -> F

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

meixnerP: (F, F, F, F) -> F

meixnerP(n, phi, lambda, z) is the Meixner-Pollaczek polynomial.

operator: BasicOperator -> BasicOperator

operator(op) returns a copy of op with the domain-dependent properties appropriate for F; error if op is not a special function operator.

polygamma: (F, F) -> F

polygamma(x, y) returns the polygamma function applied to x and y.

polylog: (F, F) -> F

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

racahR: (F, F, F, F, F, F) -> F

racahR(n, a, b, c, d, z) is the Racah polynomial.

retract_Q: F -> Union(Fraction Integer, failed)

retract_Q(x) should be local but conditional

riemannZeta: F -> F

riemannZeta(z) is the Riemann Zeta function.

sign: F -> F

sign(x) returns the sign of x.

struveH: (F, F) -> F

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

struveL: (F, F) -> F

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: F -> F

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

weberE: (F, F) -> F

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

weierstrassP: (F, F, F) -> F

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

weierstrassPInverse: (F, F, F) -> F

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

weierstrassPPrime: (F, F, F) -> F

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

weierstrassSigma: (F, F, F) -> F

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

weierstrassZeta: (F, F, F) -> F

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

whittakerM: (F, F, F) -> F

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

whittakerW: (F, F, F) -> F

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

wilsonW: (F, F, F, F, F, F) -> F

wilsonW(n, a, b, c, d, z) is the Wilson polynomial.