CONSTRUCTION OF A RIGHT INVERSE OPERATOR TO THE DISCRETE CAUCHY–RIEMANN OPERATOR

We approximate the Cauchy-Riemann operators in the complex plane by finite differences, such that the factorization of the Laplacian into two adjoint Cauchy-Riemann operators is preserved in the discrete case. Integral representations of the fundamental solutions of these difference operators are given. Furthermore, we define a discrete version of the complex T-operator. Based on this operator a discrete Borel-Pompeiu formula is formulated.