The Euler-Maclaurin expansion and finite-part integrals

In this paper we compare G p the Mellin transform together with its analytic continuation and G p the related Hadamard nite part integral of a function g x which decays exponentially at in nity and has speci ed singular behavior at the origin Except when p is a nonpositive integer these coincide When p is a large negative integer G p is well de ned 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 nite 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 The One Dimensional Euler Maclaurin Expansion The prototype problem in numerical quadrature is that of approximating an integral If Z f x dx by a sum of function values of the form