The polylogarithm function is one of the constellation of important mathematical functions. It has a long history, and many connections to other special functions and series, and many applications, for instance in statistical physics. However, the practical aspects of its numerical evaluation have not received the type of comprehensive treatments lavished on its siblings. Only a handful of formal publications consider the evaluation of the function, and most focus on a specific domain and/or presume arbitrary precision arithmetic will be used. And very little of the literature contains any formal validation of numerical performance. In this paper we present an algorithm for calculating polylogarithms for both complex parameter and argument and evaluate it thoroughly in comparison to the arbitrary precision implementation in Mathematica. The implementation was created in a new scientific computing language Julia, which is ideal for the purpose, but also allows us to write the code in a simple, natural manner so as to make it easy to port the implementation to other such languages.

中文翻译：

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Julia 中的多对数函数

多对数函数是重要的数学函数之一。它有着悠久的历史，与其他特殊函数和系列的许多联系，以及许多应用，例如在统计物理学中。然而，它的数值评估的实际方面并没有得到对其兄弟姐妹的全面处理。只有少数正式出版物考虑了函数的评估，并且大多数专注于特定领域和/或假定将使用任意精度算法。很少有文献包含对数值性能的任何正式验证。在本文中，我们提出了一种计算复参数和实参的多对数的算法，并与 Mathematica 中的任意精度实现进行比较，对其进行彻底评估。