Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments

Wei Li,Chang-Yuan Chen,Shi-Hai Dong

doi:10.1155/2017/7374256

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, that is,∫-11Pl′m′xt-1/1+t2-2xtPk′m′(x)/(1+t2-2tx)(l′+1)/2dx, wheret∈(0,1). 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∫0∞Jn(αx2+z2)/(x2+z2)nx2m+1dx.

Highlights

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials
It is well known that the associated Legendre polynomials play an important role in the central fields when one solves the physical problems in the spherical coordinates
The choice of positive or negative b depends on b ≥ −m2

Summary

Introduction

It is well known that the associated Legendre polynomials play an important role in the central fields when one solves the physical problems in the spherical coordinates. 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, that is, ∫−11 Plm󸀠 󸀠 ((xt − 1)/√1 + t2 − 2xt)(Pkm󸀠󸀠 (x)/(1 + t2 − 2tx)(l󸀠+1)/2)dx, where t ∈ (0, 1).

Results

Conclusion