The Static Maxwell System in Three Dimensional Axially Symmetric Inhomogeneous Media and Axially Symmetric Generalization of the Cauchy–Riemann System

Abstract

This paper extends our previous joint paper with Kähler, published in 2017, on problems of the static Maxwell system in three dimensional axially symmetric inhomogeneous media. Applied pseudoanalytic function theory, developed by Kravchenko, allows us to characterize new subclasses of meridional and transverse electrostatic fields in axially symmetric inhomogeneous media in two dimensional setting. Quaternionic analysis in R 3 , using analytic solutions of the system (R), allows us to characterize new subclasses of harmonic solutions of the static Maxwell system in homogeneous media in three dimensional setting. Leutwiler in 1992 initiated new approach of modified quaternionic analysis in R 3 , using analytic solutions of a hyperbolic non-Euclidean modification (H) of the system (R). Applications of modified quaternionic analysis and contemporary hyperbolic function theory in problems of geometrical optics and geo-electrostatics of inhomogeneous media are presented now. Some new classes of electrostatic potentials in three dimensional setting are implemented using Bessel functions. The singular sets of the electric field gradient tensor allow us to characterize new geometric properties of meridional fields in axially symmetric inhomogeneous media in the context of generalized axially symmetric potential theory (GASPT). Fueter construction allows us to present a wide range of meridional electrostatic models.Mathematics Subject Classification (2010). Primary 35Q61, 78A30; Secondary 35Q05, 30G20, 30C65, 30G35.Keywords. electrostatic fields; inhomogeneous isotropic media; modified quaternionic analysis; GASPT; the Fueter holomorphic potential; the Euler's Gamma function of the reduced quaternionic argument. Generalized Hyperbolic Non-Euclidean Modification of the System (R) and a Class of α-Hyperbolic Harmonic Electrostatic Potentials in Three Dimensional Setting in Cartesian CoordinatesAs seen from the system (2.3), new extensions of the system (H) require to study electrostatic models in special plane-layered inhomogeneous isotropic

Potential Vector Fields in $\mathbb R^4$ and New Generalizations of the Cauchy-Riemann System

“…Meanwhile, static potential vector fields in R 4 may be investigated in the context of Non-Euclidean geometry using the Laplace-Beltrami equation (see, e.g., [1,9,10,34,41])…”

“…(see, e.g., [9]). The second equation of the system (2.11) may be solved using linear independent solutions…”

“…When α = 2, the system (1.5) in cylindrical coordinates within Fueter's construction in R 4 (1.4) is reduced to the Cauchy-Riemann type system in the meridian half-plane (see, e.g., [2,9,10,13,16,32])…”