Ring-shaped potential and a class of relevant integrals involved universal associated Legendre polynomials with complicated arguments

TL;DR

This paper demonstrates that solutions to certain angular differential equations can be expressed using universal associated Legendre polynomials, and introduces a method to evaluate complex integrals involving these functions and Bessel functions.

Contribution

It introduces a novel approach to express solutions with Legendre polynomials and derives analytical integrals with complicated arguments, potentially applicable to other special functions.

Findings

Derived analytical expressions for integrals with complicated arguments

Expressed polar angular solutions using universal associated Legendre polynomials

Extended method applicability to Bessel functions with complex arguments

Abstract

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument. The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument.