A New Concept in Near-Singular Integral Evaluation: The Continuation Approach

The functional homogeneity of Green functions is exploited in the derivation of continuation formulae for the accurate evaluation of near singular integrals. These formulae provide a means of systematically continuing such unconventional integrals as the principal value and finite part integrals of singular functions to conventional integrals of nonsingular functions. A new concept, the notion of continuum integral, which is applicable to singular as well as to nonsingular integrals, is introduced as a generalization of principal value and finite part integrals. Numerical examples are provided that illustrate the computational advantages of the continuation method and give a new perspective on the subtleties of singular and near-singular integrals.