2024/12/27 by Dmitry Bryukhov, Bryukhov, Dmitry
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2412.19536
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 R3 are compared. The original properties of the Jacobian matrix J( V) of potential meridional fields V in cylindrically layered media, where ϕ( ρ) = ρ-α (α∈ \mathbb R), lead to the concept of α-meridional mappings of the first and second kind. The concept of α-Meridional functions of the first and second kind naturally arises in this way. When α=1, the special concept of Radially holomorphic functions in \mathbb R3, introduced by Gürlebeck, Habetha and Sprössig 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\breveβ x (\breveβ ∈ \mathbb R) and function reversed with respect to e\breveβ x are demonstrated explicitly. The basic properties of the radially holomorphic potential represented by the radially holomorphic extension of the Joukowski transformation in \mathbb R3 are studied.