Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory

TL;DR

This paper introduces explicit elementary integral series representations for all types of Heun functions, including an application to black-hole perturbation theory, providing convergent solutions across the entire domain.

Contribution

It presents the first explicit elementary integral series representations for all Heun functions, solving a long-standing open problem and applying it to black-hole perturbation equations.

Findings

Explicit integral representations for all Heun functions.

Application to the Teukolsky radial equation for black holes.

Convergent solutions from horizon to infinity.

Abstract

Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide explicit integral representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly-confluent and triconfluent, with integrals involving only rational functions and exponential integrands. All the series are illustrated with concrete examples of use. These results stem from the technique of path-sums, which we use to evaluate the path-ordered exponential of a variable matrix chosen specifically to yield Heun functions. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity.