Gamma and polygamma functions

Since the SpecialFunctions module does not expose methods for its gamma, digamma, trigamma, and polygamma functions for Complex{BigFloat} arguments (or BigFloat arguments in some cases), the BarnesDoubleGamma package imports these methods from the ArbNumerics package instead, and exposes these functions. This means that users of this package can compute gamma functions in arbitrary precision:

julia> using BarnesDoubleGamma

julia> gamma(0.5)
1.772453850905516

julia> gamma(big"0.5")
1.772453850905516027298167483341145182797549456122387128213807789852911284591025

julia> trigamma(big"0.5" + 0.1im)
4.4780986332966090515538950704365981683158900719789372422121210172045798838399969294524 - 1.561594164192146024275706100493907132241385845177793426420827577586671268403448im

julia> gamma(0.5+0.1im)
1.697617826382886 - 0.33284283907262135im

The package also exports a regularised digamma function digamma_reg

BarnesDoubleGamma.digamma_reg — Method

digamma_reg(z)

Digamma function $ψ(z)$ regularised at negative integers thanks to the formula

\[ψ(1-z) - ψ(z) = π \operatorname{cot}(πz)\]

Examples

julia> digamma_reg(-1)
0.4227843350984672

source