Generalized Cauchy Theorem in Clifford Analysis and Boundary Value Problems for Regular Functions

Abstract

In this paper, we establish the generalized Cauchy theorems on the para-sphere and the generalized Cauchy integral formulae on the strong para-sphere in Clifford analysis. As applications, the generalized Cauchy theorems and the generalized Cauchy integral formulae on the closed smooth surface and the cylindroid with crooked tips are respectively obtained. And these directly result in the Painleve theorem and the generalization of the Sochocki–Plemelj formula for the difference of boundary values in Clifford analysis. Then, by using these results the Riemann jump boundary value problems and Dirichlet boundary value problems for regular functions in Clifford analysis are discussed. Some singular integral equations are also solved and the inversion formula for Cauchy principal value is obtained by the results based on these boundary value problems solved.