# FunctionalSpecialFunction(R, F)¶

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) is true if op is a special function operator.
besselI: (F, F) -> F
besselI(x, y) returns the besseli function applied to x and y
besselJ: (F, F) -> F
besselJ(x, y) returns the besselj function applied to x and y
besselK: (F, F) -> F
besselK(x, y) returns the besselk function applied to x and y
besselY: (F, F) -> F
besselY(x, y) returns the bessely function applied to x and y
Beta: (F, F) -> F
Beta(x, y) returns the beta function applied to x and y
charlierC: (F, F, F) -> F
charlierC(n, a, z) is the Charlier polynomial
digamma: F -> F
digamma(x) returns the digamma function applied to 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)
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
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.
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
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
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
riemannZeta: F -> F
riemannZeta(z) is the Riemann Zeta function
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)
weberE: (F, F) -> F
weberE(v, z) is the Weber E function
weierstrassP: (F, F, F) -> F
weierstrassP(g2, g3, x)
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)
weierstrassSigma: (F, F, F) -> F
weierstrassSigma(g2, g3, x)
weierstrassZeta: (F, F, F) -> F
weierstrassZeta(g2, g3, x)
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