Analysis Of Singular And Near-singularIntegrals By The Continuation Approach

Abstract

We present a summary of the Continuation Approach for a broad class of singular and near-singular integrals. Insight is gained into the behaviour of these integrals, and general formulae for integration are obtained using conventional concepts in calculus. Moreover, conditions for boundedness of the singular integrals arise naturally, showing when a strongly singular integral coincides with the classical Cauchy Principal Value (and jump) or Hadamard Finite Part definitions. The continuation approach exploits the functional homogeneity shared by many Green's Functions, a property that has been mostly overlooked. Originally developed for homogeneous integrands on flat domains, the method has since been extended to more general integrands, as well as to curved surfaces and domains with corners.