Cauchy principal values and finite parts of boundary integrals-revisited

The relationship between Finite Parts (FPs) and Cauchy Principal Values (CPVs) (when they exist) of certain integrals has been previously studied by Toh and Mukherjee [Toh K-C, Mukherjee S. Hypersingular and finite part integrals in the boundary element method. Int J Solids Struct 1994;31:2299-2312] and Mukherjee [Mukherjee S. CPV and HFP integrals and their applications in the boundary element method. Int J Solids Struct 2000;37:6623-6634, Mukherjee S. Finite parts of singular and hypersingular integrals with irregular boundary source points. Engrg Anal Bound Elem 2000;24:767-776]. This paper continues this study and presents and proves an interesting new relationship between the CPV and FP of certain boundary integrals (on closed boundaries) that occur in Boundary Integral Equation (BIE) formulations of some common Boundary Value Problems (BVPs) in science and engineering.