Euler-Maclaurin expansions and approximations of hypersingular integrals

Abstract

This article presents the Euler-Maclaurin expansions of thehypersingularintegrals $\int_{a}^{b}\frac{g(x)}{|x-t|^{m+1}}dx$ and $\int_{a}^{b}%\frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ andarbitrary non-negative integer $m$. These general expansions areapplicable to a large range of hypersingular integrals, includingboth popular hypersingular integrals discussed in the literature andother important ones which have not been addressed yet. Thecorresponding mid-rectangular formulas and extrapolations, which canbe calculated in fairly straightforward ways, are investigated.Numerical examples are provided to illustrate the features of thenumerical methods and verify the theoretical conclusions.