Discrete Quaternionic Function Theory

Abstract

A decisive disadvantage of the numerical implementation of continuous models seems to be that essential analytical and algebraic properties which are characteristic of the continuous theory have to be dropped out if we change over to a corresponding discrete problem. As far as the theory of quaternionic analysis is concerned, such properties are, for instance, algebraical relations of the continuous operators, orthogonality, statements of invertibility and the behaviour of H-regular functions in the neighbourhood of singular points. In order to change this state, in several papers [Z], [DD], [Hay], [DL], [Du], [Fe] attempts were made to develop a discrete analogue to the complex function theory. For functions over lattices which satisfy a well-defined difference equation similar to the CAUCHY-RIEMANN system it is possible to transfer some essential elements of the classical function theory, for instance, CAUCHY’s theorem, idea of the complex curve integral and power series expansions. Up to the present time a discrete BOREL-POMPEIU formula and the restriction to bounded domains have been wanted. Statements in this direction also contain the theory of finite difference methods.