Computations of general Heun functions from their integral series representations

TL;DR

This paper introduces a numerical method for computing general Heun functions using integral series representations, providing Python codes that outperform existing software in efficiency, especially for large datasets.

Contribution

It develops and releases new Python implementations of Heun functions based on integral series, improving computational efficiency and accuracy over existing tools.

Findings

The Python codes compare favorably with Mathematica's HeunG.

Performance is better at large numbers of evaluation points.

The method offers a reliable alternative for large-scale computations.

Abstract

We present a numerical implementation of the recently developed unconditionally convergent representation of general Heun functions as integral series. We produce two codes in Python available for download, one of which is especially aimed at reproducing the output of Mathematica's HeunG function. We show that the present code compares favorably with Mathematica's HeunG and with an Octave/Matlab code of Motygin, in particular when the Heun function is to be evaluated at a large number of points if less accuracy is sufficient. We suggest further improvements concerning the accuracy and discuss the issue of singularities.