A numerical method of computing Hadamard finite-part integrals with an integral power singularity at the endpoint on a half infinite interval

TL;DR

This paper introduces a numerical technique for accurately computing Hadamard finite-part integrals with power singularities at the endpoint on a half-infinite interval, using complex integrals and the DE formula.

Contribution

The paper presents a novel numerical method that expresses finite-part integrals via complex integrals and evaluates them with the DE formula, including error estimates.

Findings

Method effectively computes divergent integrals with singularities.

Theoretical error estimates support the method's accuracy.

Numerical examples demonstrate the method's practicality.

Abstract

In this paper, we propose a numerical method of computing Hadamard finite-part integrals with an integral power singularity at the endpoint on a half infinite interval, that is, a finite value assigned to a divergent integral with an integral power singularity at the endpoint on a half infinite interval. In the proposed method, we express a desired finite-part integral using a complex integral, and we obtain the integral by evaluating the complex integral by the DE formula. Theoretical error estimate and some numerical examples show the effectiveness of the proposed method.