Finite precision computation of the polylogarithm function for complex parameters and arguments

The polylogarithm function is a member 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 treatment lavished on its siblings. Only a handful of formal publications consider its evaluation, and most focus on specific domains or presume the use of arbitrary-precision arithmetic. 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 values and test it thoroughly in comparison to the arbitrary-precision implementation in Mathematica. Our 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.