Lerch超越的数值方法及任意精度计算
Numerical methods and arbitrary-precision computation of the Lerch transcendent
摘要：
我们研究使用的欧拉-麦克劳林公式和新派生的均匀渐近展开的数值评估的Lerch超越$Φ(z, s, a)$为$z, s, a \in \mathbb{C}$任意精度。这些扩展的详细分析是伴随着严格的误差界。一个完整的方案，计算的参数和参数的大小值与算法的细节一起描述，以实现高性能。所描述的算法已经过广泛的测试，在不同的制度的参数，并与当前国家的最先进的代码。基于本文所述算法的$Φ(z, s, a)$的开源实现是可用的。
英文摘要：
We examine the use of the Euler-Maclaurin formula and new derived uniform asymptotic expansions for the numerical evaluation of the Lerch transcendent $Φ(z, s, a)$ for $z, s, a \in \mathbb{C}$ to arbitrary precision. A detailed analysis of these expansions is accompanied by rigorous error bounds. A complete scheme of computation for large and small values of the parameters and argument is described along with algorithmic details to achieve high performance. The described algorithm has been extensively tested in different regimes of the parameters and compared with current state-of-the-art codes. An open-source implementation of $Φ(z, s, a)$ based on the algorithms described in this paper is available.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
