# The Static Maxwell System in Three Dimensional Inhomogeneous Isotropic Media, Generalized Non-Euclidean Modification of the System $(R)$ and Fueter's Construction

## TL;DR

This paper advances the mathematical modeling of the static Maxwell system in inhomogeneous media using pseudoanalytic function theory, introducing new meridional models and integral representations relevant to generalized potential theory.

## Contribution

It introduces the concept of α-meridional mappings, develops new meridional models for α=1, and establishes integral representations of Bessel functions with quaternionic arguments.

## Key findings

- New meridional models for α=1 in GASPT
- Integral representations of Bessel functions with quaternionic arguments
- Geometric properties of harmonic meridional mappings for α=0

## Abstract

This paper extends approach of our joint paper with K\"{a}hler and recent paper of the author, published in 2021, on problems of the static Maxwell system in three dimensional inhomogeneous media. Applied pseudoanalytic function theory developed by Kravchenko et al. allows to characterize, in particular, meridional and transverse fields in cylindrically layered media. Geometric properties of the electric field gradient ($EFG$) tensor within a wide range of meridional fields allows us to introduce the concept of $\alpha$-meridional mappings of the first and second kind depending on the values of a real parameter $\alpha$. In case $\alpha =1$ tools of the radially holomorphic potential provide essentially new meridional models in the context of generalized axially symmetric potential theory (GASPT). Integral representations of Bessel functions of the first kind of integer order and the reduced quaternionic argument are first established. In case $\alpha =0$ geometric properties of harmonic meridional mappings of the second kind are described. Some open problems in three dimensional inhomogeneous anisotropic media are discussed using a generalized Riemannian modification of the system $(R)$.