3D-mappings and their approximations by series of powers of a small parameter

Abstract

Conformal mappings in the plane are closely linked with holomorphic functions and their property of complex differentiability. In contrast to the planar case, in higher dimensions the set of conformal mappings consists only of M¨obius transformations which are not monogenic and therefore also not hypercomplex differentiable. But due to the equivalence between being hypercomplex differentiable and being monogenic the question arises if from this point of view monogenic functions can still play 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 3Dmappings which is an extension of ideas of L. V. Kantorovich to the 3-dimensional case by using para-vectors and a suitable series of powers of a small parameter. In the case of the application of Bergman’s reproducing kernel approach (BKM) to 3D- mapping problems the recovering of the mapping function itself and its relation to the kernel function is still an open problem. The approach that we present here avoids such difficulties and leads directly to an approximation by monogenic polynomials depending on that small parameter.