3D-Mappings using monogenic functions

Abstract

Conformal mappings of plane domains are realized by holomorphic functions with non vanishing derivative. Therefore complex differentiability plays an important role in all questions related to fundamental properties of such mapping. In contrast to the planar case, in higher dimensions the set of conformal mappings consists only of M¨obius transformations. But unfortunately M¨obius transformations are not monogenic functions and therefore also not hypercomplex differentiable. However the equivalence between both concepts - hypercomplex differentiability in the sense of [9], [11] and monogenicity - suggests the question whether monogenic functions can play or not a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. Our goal is to present a case study of an approach to 3D-mappings by using particularly easy to handle monogenic homogeneous polynomials as basis for approximating the mapping function. Thereby we extend significantly the results obtained in [3]. From the numerical point of view we apply ideas from complex numerical analysis realizing the approximation via polynomials of a small real parameter.