A numerical method of computing oscillatory integral related to hyperfunction theory

TL;DR

This paper introduces a numerical technique for computing oscillatory integrals using complex analysis and hyperfunction theory, enabling accurate evaluation of integrals with slowly decaying oscillatory behavior.

Contribution

The paper presents a novel numerical method that employs analytic continuation and continued fractions to evaluate oscillatory integrals related to hyperfunction theory.

Findings

Effective computation of oscillatory integrals demonstrated through numerical examples.

Method leverages complex analytic functions and hyperfunction theory for improved accuracy.

Applicable to integrals with slowly decaying oscillatory integrands.

Abstract

In this paper, we propose a numerical method of computing an integral whose integrand is a slowly decaying oscillatory function. In the proposed method, we consider a complex analytic function in the upper-half complex plane, which is defined by an integral of the Fourier-Laplace transform type, and we obtain the desired integral by the analytic continuation of this analytic function onto the real axis using a continued fraction. We also remark that the proposed method is related to hyperfunction theory, a theory of generalized functions based on complex function theory. Numerical examples show the effectiveness of the proposed method.