Normalization Integrals of Orthogonal Heun Functions

TL;DR

This paper derives an explicit formula for the normalization integrals of orthogonal Heun functions, enabling efficient computation of orthonormal bases without numerical integration, based on local power-series solutions.

Contribution

It introduces a simple limiting procedure to evaluate normalization integrals of orthogonal Heun functions explicitly in terms of local solutions and derivatives.

Findings

Provides an explicit formula for normalization integrals.

Enables efficient orthonormal basis construction using Heun functions.

Reduces reliance on numerical integration.

Abstract

A formula for evaluating the quadratic normalization integrals of orthogonal Heun functions over the real interval 0 <= x <= 1 is derived using a simple limiting procedure based upon the associated differential equation. The resulting expression gives the value of the normalization integral explicitly in terms of the local power-series solutions about x=0 and x=1 and their derivatives. This provides an extremely efficient alternative to numerical integration for the development of an orthonormal basis using Heun functions, because all of the required information is available as a by-product of the search for the eigenvalues of the differential equation. 02.30.Gp; 02.30.Hq; 02.70.-c; 02.60.Jh