3D deformations by means of monogenic functions

Abstract

In this paper, the authors compute the coefficient of quasiconformality for monogenic functions in an arbitrary ball of the Euclidean space R 3 . This quantification may be needed in applications but also appears to be of intrinsic interest. The main tool used is a 3D Fourier series development of monogenic functions in terms of a special set of solid spherical monogenics. Ultimately, we present some examples showing the applicability of our approach. @OEf 2 @x 1 D 0.We may point out that the previous system is called the Riesz system [31], and it generalizes the classical Cauchy-Riemann system in the plane. Following [32], the solutions of the system (R) are called (R)-solutions. The subspace of polynomial (R)-solutions of degree

Local distortion of M-conformal mappings

“…Besides this, Morais et al in [37] computed the coefficient of quasiconformality of those mappings. From the point of view taken here, this is particularly rewarding since the computation of this coefficient gives us the information of the ratio of the major to minor axes of the aforementioned ellipsoids.…”

“…Together with the geometric interpretation of the (hypercomplex) derivative, dilatations and distortions of these mappings could be estimated, see [15]. To progress in this direction, in [37] the coefficient of quasiconformality of those mappings was calculated explicitly. This is particularly rewarding since the computation of this coefficient gives us the information of the ratio of the major to minor axes of the aforementioned ellipsoids.…”

Hadamard three-hyperballs type theorem and overconvergence of special monogenic simple series

“…In a series of papers [13,15,21,22], the authors have investigated higher dimensional counterparts of the well-known Bohr theorem and Hadamard real part theorems on the majorant of a Taylor's series, as well as Bloch's theorem, in the context of quaternionic analysis. These results provide powerful additional motivation to study the asymptotic growth behavior of monogenic functions from a given space, and to explore classical problems of the theory of monogenic quasi-conformal mappings [14,23] (see also [20,Ch. 3]).…”

Local distortion of monogenic functions

“…As demonstrated by Gürlebeck et al in a sequence of papers [11,12,13,14] (see also [25,Chap. IV] and [26]) the class of monogenic functions can be defined as a special subclass of quasiconformal mappings. More precisely, monogenic functions with nonvanishing Jacobian determinant and taking values in the reduced quaternions (identified with R 3 ) asymptotically map small balls onto explicitly characterized ellipsoids and vice versa.…”

“…Together with the geometric interpretation of the derivative, dilatations and distortions of these mappings could be estimated. Further progress has been made by Morais et al in [26]. The coefficient of quasiconformality of those mappings has been calculated.…”