Asymptotic Expansions for Two-Dimensional Hypersingular Integrals

Summary. 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.Mathematics Subject Classification (2000): 65D30, 65B15, 65R10This author was supported in part by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract W-31-109-Eng 38. Part of this work was performed while this author was visiting professor at the Politecnico di Torino, under the sponsorship of the Italian C.N.R.-Gruppo Nazionale per lInformatica Matematica.This author was supported by the Ministero dellUniversitá e della Ricerca Scientifica e Tecnologica of Italy, and by the C.N.R.-Comitato n.11.