On the regularization of bilinear forms with hypersingular kernel

In this paper, we first recall the classical definitions of Cauchy Principal Value (P.V.) and Hadamard Finite Part (F.P.) of some singular and hypersingular integrals, and we point out the relationships that they have with the first and second derivatives of certain converging improper integrals, both in the classical and the distributional framework. Then, by using this theoretical analysis, conducted on both one-dimensional and two-dimensional examples, we define and analyze rigorously what is meant by regularization of bilinear forms with hypersingular kernels, that typically arise from the resolution in weak form of boundary integral equations related to elliptic and hyperbolic problems. Related numerical results are given.