# DoubleFloatSpecialFunctionsΒΆ

This package provides special functions for double precision real and complex floating point.

airyAi: Complex DoubleFloat -> Complex DoubleFloat

airyAi(x) is the Airy function Ai(x). This function satisfies the differential equation: Ai''(x) - x * Ai(x) = 0.

airyAi: DoubleFloat -> DoubleFloat

airyAi(x) is the Airy function Ai(x). This function satisfies the differential equation: Ai''(x) - x * Ai(x) = 0.

airyBi: Complex DoubleFloat -> Complex DoubleFloat

airyBi(x) is the Airy function Bi(x). This function satisfies the differential equation: Bi''(x) - x * Bi(x) = 0.

airyBi: DoubleFloat -> DoubleFloat

airyBi(x) is the Airy function Bi(x). This function satisfies the differential equation: Bi''(x) - x * Bi(x) = 0.

besselI: (Complex DoubleFloat, Complex DoubleFloat) -> Complex DoubleFloat

besselI(v, x) is the modified Bessel function of the first kind, I(v, x). This function satisfies the differential equation: x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0.

besselI: (DoubleFloat, DoubleFloat) -> DoubleFloat

besselI(v, x) is the modified Bessel function of the first kind, I(v, x). This function satisfies the differential equation: x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0.

besselJ: (Complex DoubleFloat, Complex DoubleFloat) -> Complex DoubleFloat

besselJ(v, x) is the Bessel function of the first kind, J(v, x). This function satisfies the differential equation: x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0.

besselJ: (DoubleFloat, DoubleFloat) -> DoubleFloat

besselJ(v, x) is the Bessel function of the first kind, J(v, x). This function satisfies the differential equation: x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0.

besselK: (Complex DoubleFloat, Complex DoubleFloat) -> Complex DoubleFloat

besselK(v, x) is the modified Bessel function of the second kind, K(v, x). This function satisfies the differential equation: x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0. Note: The default implementation uses the relation K(v, x) = \%pi/2*(I(-v, x) - I(v, x))/sin(v*\%pi) so is not valid for integer values of v.

besselK: (DoubleFloat, DoubleFloat) -> DoubleFloat

besselK(v, x) is the modified Bessel function of the second kind, K(v, x). This function satisfies the differential equation: x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0. Note: The default implementation uses the relation K(v, x) = \%pi/2*(I(-v, x) - I(v, x))/sin(v*\%pi). so is not valid for integer values of v.

besselY: (Complex DoubleFloat, Complex DoubleFloat) -> Complex DoubleFloat

besselY(v, x) is the Bessel function of the second kind, Y(v, x). This function satisfies the differential equation: x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0. Note: The default implementation uses the relation Y(v, x) = (J(v, x) cos(v*\%pi) - J(-v, x))/sin(v*\%pi) so is not valid for integer values of v.

besselY: (DoubleFloat, DoubleFloat) -> DoubleFloat

besselY(v, x) is the Bessel function of the second kind, Y(v, x). This function satisfies the differential equation: x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0. Note: The default implementation uses the relation Y(v, x) = (J(v, x) cos(v*\%pi) - J(-v, x))/sin(v*\%pi) so is not valid for integer values of v.

Beta: (Complex DoubleFloat, Complex DoubleFloat) -> Complex DoubleFloat

Beta(x, y) is the Euler beta function, B(x, y), defined by Beta(x, y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1). This is related to Gamma(x) by Beta(x, y) = Gamma(x)*Gamma(y) / Gamma(x + y).

Beta: (DoubleFloat, DoubleFloat) -> DoubleFloat

Beta(x, y) is the Euler beta function, B(x, y), defined by Beta(x, y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1). This is related to Gamma(x) by Beta(x, y) = Gamma(x)*Gamma(y) / Gamma(x + y).

digamma: Complex DoubleFloat -> Complex DoubleFloat

digamma(x) is the function, psi(x), defined by psi(x) = Gamma'(x)/Gamma(x).

digamma: DoubleFloat -> DoubleFloat

digamma(x) is the function, psi(x), defined by psi(x) = Gamma'(x)/Gamma(x).

Gamma: Complex DoubleFloat -> Complex DoubleFloat

Gamma(x) is the Euler gamma function, Gamma(x), defined by Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..\%infinity).

Gamma: DoubleFloat -> DoubleFloat

Gamma(x) is the Euler gamma function, Gamma(x), defined by Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..\%infinity).

hypergeometric0F1: (Complex DoubleFloat, Complex DoubleFloat) -> Complex DoubleFloat

hypergeometric0F1(c, z) is the hypergeometric function 0F1(; c; z).

hypergeometric0F1: (DoubleFloat, DoubleFloat) -> DoubleFloat

hypergeometric0F1(c, z) is the hypergeometric function 0F1(; c; z).

logGamma: Complex DoubleFloat -> Complex DoubleFloat

logGamma(x) is the natural log of Gamma(x). This can often be computed even if Gamma(x) cannot.

logGamma: DoubleFloat -> DoubleFloat

logGamma(x) is the natural log of Gamma(x). This can often be computed even if Gamma(x) cannot.

polygamma: (NonNegativeInteger, Complex DoubleFloat) -> Complex DoubleFloat

polygamma(n, x) is the n-th derivative of digamma(x).

polygamma: (NonNegativeInteger, DoubleFloat) -> DoubleFloat

polygamma(n, x) is the n-th derivative of digamma(x).