On the advances of two and three-dimensional electrical impedance equation

Abstract

We review the state of the art of solutions for the electrical impedance equation div( grad ') = 0;(1)where the function = + i!" is the admitivity, denotes the conductivity, ! is the frequency, " is the permittivity, i is the imaginary unit, and ' denotes the electric potential. Considering the three-dimensional case, we show that this equation is equivalent to a Schrödinger equation, and applying the algebra of quaternions, we study one method for factorizing (1) into two rst-order differential operators. We also discuss a method for rewriting equation (1) directly into a quaternionic rst-order linear differential equation and we cite some cases when it is possible to solve it analytically. For the two-dimensional case, we show that (1) is equivalent to a Vekua equation, and applying recently discovered properties of pseudoanalytic functions, we explore how to obtain solutions of (1) starting with solutions of a two-dimensional Schrödinger equation. Once we have discussed some elements of Pseudoanalytic Function Theory, we expose how to express the general solution of the Electrical Impedance Equation through Taylor series in formal powers, remarking the impact of this tools in such important questions as the Calderon problem in the plane.