# DoubleFloatSpecialFunctions¶

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).