Numerical method of computing Hadamard finite-part integrals with a non-integral power singularity at the endpoint over a half infinite interval

TL;DR

This paper introduces a numerical approach to accurately compute Hadamard finite-part integrals with non-integral power singularities on half-infinite intervals, using complex integral representations and the DE formula, demonstrating exponential convergence.

Contribution

The paper presents a novel numerical method combining complex integral representation and the DE formula for Hadamard finite-part integrals with singularities, with proven exponential convergence.

Findings

Method achieves exponential convergence.

Numerical examples confirm accuracy and efficiency.

Theoretical error estimates support the method's reliability.

Abstract

In this paper, we propose a numerical method of computing an Hadamard finite-part integral, a finite value assigned to a divergent integral, with a non-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 finite part integral by evaluating the complex integral by the DE formula. Theoretical error estimate and some numerical examples show the exponential convergence of the proposed method.