A numerical method of computing oscillatory integral related to hyperfunction theory

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.