Asymptotic Expansions for Two-Dimensional Hypersingular Integrals

Abstract

We define and examine two-dimensional hypersingular integrals on [0,1)2 and on [0,∞)2 and relate their Hadamard finite-part (HFP) values to Mellin transforms. These integrands have algebraic singularities of a possibly unintegrable nature on the axes and at the origin. Extending our work on one-dimensional integrals reported in 1998, we obtain variants of the classical Euler-Maclaurin expansion for various two-dimensional integrals. In many cases, the constant term in the expansion (which is not necessarily the leading term) provides the value of the HFP integral. These expansions may be used as the basis for the numerical evaluation of a class of HFP integrals by extrapolation.