integral equations for the spin-weighted spheroidal wave function

Integral equations for the spin-weighted spheroidal wave functions is given . For the prolate spheroidal wave function with m = 0, there exists the integral equation whose kernel is sin x x , and the sinc function kernel sin x x is of great mathematical significance. In the paper, we also extend the similar sinc function kernel sin x x to the case m 6= 0 and s 6= 0, which interestingly turn out as some kind of Hankel transformation. Perturbation to black-hole is important not only in principle but also astrophysically in reality. Teukolsky obtain separable equations of scalar, electromagnetic and gravitational fields’ perturbation to the kerr black-hole[1]-[2]. The separated angular equation are called the spin-weighted spheroidal wave equations, and are extension of the ordinary spherical harmonics equation. The spin-weighted spheroidal wave functions are not only connected to stability of kerr black hole in general relativity, but also to many other physical problems[15]. They are studied in many papers [1]-[11], though not by the method of the integral equation. It is well-known that the method of the integral equation play an important role in mathematical physics. Many important theoretical and mathematical problems are solved mainly by the method of the integral equation. They are also indispensable for numerical computation. Therefore, it is useful for one to find the integral equation of a given differential equation, or equivalently, to find the kernel of the integral equation. In this paper, we mainly give an integral equation of these spin-weighted spheroidal wave functions. When reducing to the prolate spheroidal wave function, there exists the integral equation whose kernel is sinx x when m = 0, the sinx x kernel is of great mathematical significance[13], [14]. In the paper, we also extend the similar sinx x kernel to the case m 6= 0 and s 6= 0, which interestingly turn out as some kind of Hankel transformation. The spin-weighted spheroidal wave functions Zslm connect with Sslm by the relation Zslm = [ 2l + 1 4π (l −m)! (l +m)! ] Sslm(−aω, cos θ)e imφ (1) Where Sslm satisfy the angular part of the perturbation wave equation