Expansion of a class of functions into an integral involving associated Legendre functions

Nanigopal Mandal,B N Mandal

doi:10.1155/s0161171294000438

Abstract

A theorem for expansion of a class of functions into an integral involving associated Legendre functions is obtained in this paper. This is a somewhat general integral expansion formula for a functionf(x)defined in(x1,x2)where-1<x1<x2<1, which is perhaps useful in solving certain boundary value problems of mathematical physics and of elasticity involving conical boundaries.

Highlights

Integral transforms are often used to solve the problems of mathematical physics involving linear partial differential equations and other problems
We present the main result of this paper in the form of the following theorem

Summary

INTRODUCTION

Integral transforms are often used to solve the problems of mathematical physics involving linear partial differential equations and other problems. Integral expansions involving spherical functions of a class of functions are known as Mehler-Fok type transforms. In these transform formulae, the subscript of the Legendre functions appear as the integration variable while its superscript is either zero or a fixed integer (see Sneddon [10]). There is another class of integral transforms involving associated Legendre functions somewhat related to the Mehler-Fok transforms, in which the superscript of the associated Legendre. Mandal and Guha Roy [8] used a similar technique to establish another Mehler-Fok type integral transform formula involving. (1) The function f(z) is piecewise continuous and has a bounded variation in the open interval (Zl, z2)

Then we have

Now introduce the new variables