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.
翻译：我们讨论在任意精确算术中计算伽马函数$\Gamma(z)$(z)美元的最佳方法。我们处理不同的案例:理性的、代数的、真实的或复杂的论据;大或小的论据;低或高的精确度;有或没有预先计算。方法还包括对伽马函数$(log)\Gamma(z)$(z)$(psi)(z)$(z)$(z)$(z)$(g)$(n)}(z)$(z)和$\psi ⁇ (n)}(z)$(z)$(z)$(z)$(z)。我们除了试图总结目前的情况外,还提出一些新的公式、估计、界限和算法改进,并讨论执行结果。
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