Arbitrary-precision computation of the gamma function

Abstract

We discuss the best methods available for computing the gamma function Γ(z) in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function log Γ(z), the digamma function ψ(z), and derivatives Γ⁽ⁿ⁾(z) and ψ⁽ⁿ⁾(z). Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results.