Numerical evaluation of hypersingular integrals (Q1334749)

The paper is motivated by the occurrence of singular and hypersingular integrals in applied mathematics, for example Cauchy principal value and Hadamard finite-part integrals in boundary integral equations. Attention is concentrated on the less familiar two-dimensional Cauchy principal value integrals and one- and two-dimensional integrals with stronger singularities requiring interpretation as finite-part integrals. Basic definitions and properties are discussed. Methods for the numerical evaluation of these integrals are critically reviewed and a number of new results on convergence, expressions for weights, asymptotic error estimates and conditioning of rules (stability factors) are obtained. Numerical examples are included.