On evaluation of the Heun functions

TL;DR

This paper develops and tests new algorithms for the numerical evaluation of Heun functions, addressing limitations of existing software like Maple, and provides efficient, accurate methods based on power series and analytic continuation.

Contribution

It introduces alternative algorithms for computing Heun functions numerically, overcoming the limitations of current software and improving accuracy and efficiency.

Findings

Proposed algorithms avoid numerical integration of the differential equation.

Numerical tests demonstrate improved accuracy and efficiency.

Algorithms are based on power series expansions and analytic continuation.

Abstract

In the paper we deal with the Heun functions --- solutions of the Heun equation, which is the most general Fuchsian equation of second order with four regular singular points. Despite the increasing interest to the equation and numerous applications of the functions in a wide variety of physical problems, it is only Maple amidst known software packages which is able to evaluate the Heun functions numerically. But the Maple routine is known to be imperfect: even at regular points it may return infinities or end up with no result. Improving the situation is difficult because the code is not publicly available. The purpose of the work is to suggest and develop alternative algorithms for numerical evaluation of the Heun functions. A procedure based on power series expansions and analytic continuation is suggested which allows us to avoid numerical integration of the differential equation and to achieve reasonable efficiency and accuracy. Results of numerical tests are given.