Integrating Singular Functions on the Sphere

TL;DR

This paper develops a rigorous complex analysis approach for evaluating integrals of both regular and singular functions on the sphere, reducing the problem to residue calculations via a limiting process.

Contribution

It introduces a novel method for computing integrals of singular functions on the sphere using complex analysis and residue theory.

Findings

Integral evaluation reduces to residue calculation.

Applicable to functions with poles on the sphere.

Provides a rigorous framework for singular integrals.

Abstract

We obtain rigorous results concerning the evaluation of integrals on the two sphere using complex methods. It is shown that for regular as well as singular functions which admit poles, the integral can be reduced to the calculation of residues through a limiting procedure.