# DoubleFloatSpecialFunctionsΒΆ

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

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`.

digamma: Complex DoubleFloat -> Complex DoubleFloat

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

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

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