On regularization of the Heun functions

TL;DR

This paper introduces a regularization method for Heun functions, addressing singularities at integer parameter values to ensure smooth dependence on parameters, which enhances their evaluation and applicability in physics.

Contribution

It proposes a novel regularization technique for Heun functions near integer parameter values, improving their analytical properties and computational stability.

Findings

Regularized Heun functions depend smoothly on parameters near integers.

The method reduces singular behavior at critical parameter values.

Enhanced evaluation of Heun functions in physical applications.

Abstract

In the paper we consider the Heun functions, which are solutions of the equation introduced by Karl Heun in 1889. The Heun functions generalize many known special functions and appear in many fields of modern physics. Evaluation of the functions was described in [1]. It is based on local power series solutions near the origin, derived by the Frobenius method, and analytic continuation to the whole complex plane with branch cuts. However, exceptional cases can occur at integer values of an exponent-related parameter $\gamma$ of the equation, when one of the two local solutions should include a logarithmic term. This also means singular behavior of the Heun functions as $\gamma$ approaches the integer values. Here we suggest a method of regularization and redefine the Heun functions in some vicinities of the integer values of $\gamma$, where the new functions depend smoothly on the parameter.