Numerical method for computing Hadamard finite-part integrals with a non-integral power singularity at an endpoint

TL;DR

This paper introduces a numerical approach to accurately compute Hadamard finite-part integrals with non-integral power singularities at endpoints, using complex contour integration and trapezoidal rule, supported by error analysis and numerical validation.

Contribution

The paper presents a novel numerical method employing complex contour integrals and trapezoidal rule for Hadamard finite-part integrals with non-integral singularities, including error estimates.

Findings

Method effectively computes finite-part integrals with singularities.

Error estimates demonstrate the method's accuracy.

Numerical examples confirm practical applicability.

Abstract

In this paper, we propose a numerical method of computing a Hadamard finite-part integral with a non-integral power singularity at an endpoint, that is, a finite part of a divergent integral as a limiting procedure. In the proposed method, we express the desired finite-part integral using a complex loop integral, and obtain the finite-part integral by evaluating the complex integral by the trapezoidal formula. Theoretical error estimate and some numerical examples show the effectiveness of the proposed method.