Cauchy-Pompeiu Formula for Discrete Monogenic Functions

Abstract

For the integral theory of discrete monogenic functions, we establish a new version of Cauchy-Pompeiu formula via the notions ‘discrete boundary measure’ and ‘discrete normal vector’. It shares the same form with the continuous version of Cauchy-Pompeiu formula in contrast to the original Cauchy-Pompeiu formula in discrete Clifford analysis. It has applications in the boundary theory of discrete monogenic functions. We can thus set up the discrete Sokhotski-Plemelj formula and provide an equivalent characterization of the Dirichlet problem with the discrete Dirac operator in terms of the eigenvectors of certain operator.