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arXiv · 2109.08392

Arbitrary-precision computation of the gamma function

Fredrik Johansson
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Abstract

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.

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Fredrik Johansson. 2021-09-17. Arbitrary-precision computation of the gamma function. https://arxiv.org/abs/2109.08392

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