Potential Vector Fields in $\mathbb R^3$ and $\alpha$-Meridional Mappings of the Second Kind $(\alpha \in \mathbb R)$

Abstract

This paper extends approach developed in a recent author's paper on analytic models of potential fields in inhomogeneous media. New three-dimensional analytic models of potential vector fields in some layered media are constructed. Properties of various analytic models in Cartesian and cylindrical coordinates in $\mathbb R^3$ are compared. The original properties of the Jacobian matrix $\mathbf{J}(\vec V)$ of potential meridional fields $\vec V$ in cylindrically layered media, where $\phi( \rho) = \rho^{-\alpha}$ $(\alpha \in \mathbb R)$, lead to the concept of \emph{$\alpha$-meridional mappings of the first and second kind}. The concept of \emph{$\alpha$-Meridional functions of the first and second kind} naturally arises in this way. When $\alpha =1$, the special concept of \emph{Radially holomorphic functions in $\mathbb R^3$}, introduced by G\"{u}rlebeck, Habetha and Spr\"{o}ssig in 2008, is developed in more detail. Certain key properties of the radially holomorphic functions $G$ and functions reversed with respect to $G$ are first characterized. Surprising properties of the radially holomorphic potentials represented by superposition of the radially holomorphic exponential function $e^{\breve{\beta} x}$ $(\breve{\beta} \in \mathbb R)$ and function reversed with respect to $e^{\breve{\beta} x}$ are demonstrated explicitly. The basic properties of the radially holomorphic potential represented by the radially holomorphic extension of the Joukowski transformation in $\mathbb R^3$ are studied.