Complex Hypersingular and Finite-Part Integrals

We want to justify the operations used when deriving equations, first of all hypersingular equations. What do we need to prove? In our discussion and derivations of hypersingular integrals, we employed the following formal elements: a)we assumed that there is a definition of divergent integrals that endows them with the usual properties of integrals, specifically with linearity and additivity;b)we used differentiation under the integral sign;c)we used integration by parts;d)we used formulae of Sokhotski-Plemelj type that connect direct values of divergent integrals with their limiting values.