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The spheroidal wave functions are investigated in the case $m = 1$. The integral equation is obtained for them. There are two kinds of eigenvalues in the differential and corresponding integral equations, and the relation between them is given explicitly. 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.

The spheroidal wave equations are extension of the ordinary spherical equations. There are many fields where spheroidal functions play important roles just as the spherical functions do. So far, in comparison to simpler spherical special functions (the associated Lengdre’s functions), their properties are still difficult to study. The equations for them are

with −1 < x < 1 and the natural conditions that Θ is finite at the boundaries x = ±1. This is a kind of the singular Sturm-Liouville eigenvalue problem. To satisfy the boundaries condition, the parameter E can only take the values “

Under the condition β = 0, they reduce to the Spherical equations and the solutions to the Sturm-Liouville eigenvalue problem are the associated Legendre-functions P^{m}_{l}(x) (the spherical functions) with the eigenvalues^{2}x^{2} than the spherical ones (the associated Lendgre’s equations). However, the extra term β^{2}x^{2} in the equation presents many mathematical difficulties [

Usually, one studied the spheroidal equations by the perturbation method in the basis of the spherical functions and resulting in the continued fraction to determine the eigenvalues and eigenfunctions [

In this paper, we mainly concern ourselves with the integral equations for them. For example, the integral equation for the prolate spheroidal wave equation is already existed [

where the kernel

with

In order to obtain a new integral equation for the spheroidal equation with

to the Equation (1) and obtain the following

The above equation becomes very simple when

where

It is easy to find the Green function for the Equation (6), that is

The Green function

and the boundary conditions

Hence the the Sturm-Liouville eigenvalue problem turns into the integral equation form:

The great advantage of the new integral Equation (12) lies in that the relation between the integral eigenvalues

Though the Green function

as desired by our requirement. It is well-known that one could easily to study the integral equations if their kernels are symmetry. Hence, the usual method to solve the integral equations could be used to treat the problem here too. We will stop here.

The Green function

where

If one supposes the parameter

The Green function in the Equation (16) will give much information about the eigenvalues and eigenfunctions in this special case. Now it could be regarded as the functions of the parameter

The eigenvalues are determined by the poles of the Green functions, that is

Hence,

and

the nth eigenfunction is

Except for the normalization constants, these results are the same as those in Ref. [

are one kind of the eigenfunctions for the fixed parameter

for the fixed parameter

The above example just provides some clues on the connection between the Green function and the solutions to the corresponding the Sturm-Liouville eigenvalue problem. If the Green function is the one corresponding with the parameter

This Green function

Our Green function

Of course,

The work was partly supported by the National Natural Science of China (No. 10875018) and the Major State Basic Research Development Program of China (973 Program: No.2010CB923202).