A New Integral Equation for the Spheroidal equations in case of m equal 1

TL;DR

This paper derives a new integral equation for spheroidal functions with m=1, explicitly relating eigenvalues from differential and integral forms, offering a novel approach to eigenvalue problems.

Contribution

The paper introduces a new integral equation for m=1 spheroidal functions and explicitly relates the two types of eigenvalues, filling a gap in existing literature.

Findings

Explicit relation between eigenvalues of differential and integral equations.

Provides a new integral equation specific to m=1 spheroidal functions.

Includes an example illustrating the eigenvalue problem.

Abstract

The spheroidal wave functions are investigated in the case m=1. The integral equation is obtained for them. For the two kinds of eigenvalues in the differential and corresponding integral equations, the relation between them are given explicitly. Though there are already some integral equations for the spheroidal equations, the relation between their two kinds of eigenvalues is not known till now. This is the great advantage of our integral equation, which will provide useful information through the study of the integral equation. Also an example is given for the special case, which shows another way to study the eigenvalue problem.