Arbitrary-precision computation of the gamma function

Fredrik Johansson (1, 2)

1 LFANT - Lithe and fast algorithmic number theory
2 CANARI - Analyse cryptographique et arithmétique

We discuss the best methods available for computing the gamma function $\Gamma(z)$ 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 $\log \Gamma(z)$, the digamma function $\psi(z)$, and derivatives $\Gamma^{(n)}(z)$ and $\psi^{(n)}(z)$. Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results.

Domaines: Analyse numérique [cs.NA]; Logiciel mathématique [cs.MS]; Analyse classique [math.CA]

Fredrik Johansson. Arbitrary-precision computation of the gamma function. Maple Transactions, 2023, 3 (1), ⟨10.5206/mt.v3i1.14591⟩. ⟨hal-03346642⟩