Numerical integration based on hyperfunction theory

摘要

In this paper, we propose an application of hyperfunction theory to numerical integration. Hyperfunction theory is a generalized version of function theory where functions with singularities such as poles, discontinuities and delta impulses are expressed in terms of complex analytic functions. This feature of hyperfunction theory allows us to construct a numerical integration method for analytic functions. Theoretical error estimates show that our method converges geometrically, and numerical examples show that our method is very efficient especially for integrals with strong end-point singularities. In addition, we present an automatic integration method based on our hyperfunction method.