The Euler-Maclaurin expansion and finite-part integrals

Abstract

In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and $\overline{\overline{G}}(p)$ , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, $\overline{\overline{G}}(p)$ is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the Euler-Maclaurin expansion for the quadrature error functional.