Integral equations for the spin-weighted spheroidal wave function

TL;DR

This paper derives integral equations for spin-weighted spheroidal wave functions, extending known kernels to cases where parameters are non-zero, revealing connections to Hankel transformations.

Contribution

It introduces new integral equations with sinc function kernels for general spin-weighted spheroidal wave functions, expanding previous special cases.

Findings

Integral equations with sinc kernel for m=0 case.

Extension of sinc kernel to cases where m and s are non-zero.

Connection to Hankel transformations for generalized kernels.

Abstract

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 and s both are not zero, which interestingly turn out as some kind of Hankel transformation.