Local distortion of M-conformal mappings

Abstract

A conformal mapping in a plane domain locally maps circles to circles. More generally, quasiconformal mappings locally map circles to ellipses of bounded distortion. In this work, we study the corresponding situation for solutions to Stein-Weiss systems in the (n + 1)D Euclidean space. This class of solutions coincides with the subset of monogenic quasiconformal mappings with nonvanishing hypercomplex derivatives (named M-conformal mappings). In the theoretical part of this work, we prove that an M-conformal mapping locally maps the unit sphere onto explicitly characterized ellipsoids and vice versa. Together with the geometric interpretation of the hypercomplex derivative, dilatations and distortions of these mappings are estimated. This includes a description of the interplay between the Jacobian determinant and the (hypercomplex) derivative of a monogenic function. Also, we look at this in the context of functions valued in non-Euclidean Clifford algebras, in particular the split complex numbers. Then we discuss quasiconformal radial mappings and their relations with the Cauchy kernel and p-monogenic mappings. This is followed by the consideration of quadratic M-conformal mappings. In the applications part of this work, we provide the reader with some numerical examples that demonstrate the effectiveness of our approach.

On 3D orthogonal prolate spheroidal monogenics

“…, Â/ as (20), and we omit vanishing-integrals. Thus, the remaining terms are, respectively, Using again (20) for B n1,m . , Â/, we are thus lead to integrals of the form as stated in Proposition 3.2.…”

“…The motivation for writing the present paper is to develop further general numerical methods to solve both basic initial‐boundary value and quasi‐conformal mapping problems. Most relevant to our study are the intimate connections between monogenic functions and spheroidal structures and the potential flexibility afforded by a spheroid's non‐spherical canonical geometry. The topic here is the prolate functions, but the principles can be extended to oblate spheroids as well (see Remark below).…”

Constructing prolate spheroidal quaternion wave functions on the sphere

“…Most relevant to our approach are the intimate connections between generalized holomorphic functions and spheroidal structures, and the potential flexibility afforded by a spheroid's nonspherical canonical geometry. Developments are described in a sequence of papers; see . The topic here is the prolate wave functions, but the principles can be extended to oblate spheroids as is shown in one of our preceding papers .…”

“…The corresponding solutions, or eigenfunctions, 0 .y; c/, 1 .y; c/, : : : can be chosen to be real and orthogonal on OE T, T. The variational problem that led to (1) only requires that equation to hold for jyj Ä T. With n .x; c/ on the left-hand side of (1) given for jxj Ä T, nevertheless, the left is well defined for all y. We use this to extend the range of definition of the n 's and so define papers; see [23][24][25][26]. The topic here is the prolate wave functions, but the principles can be extended to oblate spheroids as is shown in one of our preceding papers [27].…”