The Study Of The Spin-Weighted Spheroidal Wave Function

It's well known that the general spheroidal wave functions play a premier role in mathematical physics. Recently, they have made strong contributions to extensively theoretical and technological applications in science and engineering. However, in comparison to simpler spherical special functions, their properties are still difficult for study than their counterpart. In this paper, we testly adopt several methods to study the spheroidal wave functions' eigenvalue problem.Specifically, this paper is divided into three parts. In the first part, which include the second chapter, the numerical method is used to solve the integral equation for the spin-weighted spheroidal wave functions when m=s=0.Since the good performance of the Ritz-Galerkin method, we can obtain the eigenvalues with different c. In the another part, the main efforts are devoted to the analytic property of the spheroidal wave equation in case of s=1/2.we adopt the perturbation method in supersymmetric quantum mechanics(SUSYQM) to calculate the ground eigenvalues and eigenfunctions of spheroidal functions by transforming them into the Schrodinger equation. Due to the good form of the first three term in the superpotential and its shape-invariant property in the third order, we furtherly obtain the eigenvalues and eigenfunctions of excited state in second term from that of ground state. At the end of the paper, we shortly analyse the relation between the spheroidal wave function in case of s=1/2 and one in case of s=-1/2.