Electrostatic Fields in Some Special Inhomogeneous Media and New Generalizations of the Cauchy–Riemann System

This paper extends approach of our recent paper together with Kähler to building special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media by means of different generalizations of the Cauchy–Riemann system with variable coefficients. A new class of three-dimensional solutions of the static Maxwell system in some special cylindrically layered media is obtained using class of exact solutions of the elliptic Euler–Poisson–Darboux equation in cylindrical coordinates. The principal invariants of the electric field gradient tensor within a wide range of meridional fields are described using a family of Vekua type systems in cylindrical coordinates. Analytic models of meridional electrostatic fields in accordance with different generalizations of the Cauchy–Riemann system with variable coefficients allow us to introduce the concept of α -meridional mappings of the first and second kind depending on the values of a real parameter α . In particular, in case α = 0 , geometric properties of harmonic meridional mappings of the second kind are demonstrated explicitly within meridional fields in homogeneous media.