SOKHOTSKI–PLEMELJ FORMULAS AND THEIR ROLE IN THE THEORY OF SINGULAR INTEGRAL EQUATIONS

Abstract

This paper discusses the Sokhotski–Plemelj formulas, which play a fundamental role in the regularization and solution of singular integral equations. The formulas describe the boundary behavior of Cauchy-type integrals and serve as the theoretical foundation for the Carleman–Mikhlin method and related analytical approaches. Through a synthesis of classical results and modern interpretations, this work presents the derivation, interpretation, and boundary properties of Cauchy-type integrals under Hölder continuity assumptions. The study highlights the analytical importance of these formulas in resolving discontinuities and establishing the existence and uniqueness of boundary values in complex analysis and mathematical physics.