On finite difference Cauchy-Riemann operators
by
Angela Hommel
Bauhaus-University of Weimar, Institute of Mathematics and Physics, D-99421 Weimar, Germany

Main problems of discrete function theories are related to the requirements, that discrete analytic functions should be discrete harmonic functions, too, that one needs a discrete version of the Cauchy-integral and the fact that discrete analytic functions do not form an algebra. We describe finite difference approximations of the Cauchy-Riemann operator in the complex as well as in the hypercomplex case, such that the factorization of the real Laplacian into two adjoint Cauchy-Riemann operators is preserved in the discrete case, too. Fundamental solutions of these finite difference operators are calculated and their properties are studied in detail. Main goal is to prepare the basic tools for the definition of difference analogues of the Cauchy integral.

Date received: August 7, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-48.