Arbitrary-precision computation of the gamma function

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