On the solutions of the electrical impedance equation, applying quaternionic analysis and pseudoanalytic function theory

We review the state of the art of solutions for the electrical impedance equation div (γ gradu) = 0 where the function div γ = σ + iωε is the admitivity, σ denotes the conductivity, ε is the frequency, ε is the permittivity, i is the imaginary unit, and u denotes the electric potential. When γ is a function of three spatial variables, we show how to rewrite the equation into a stationary Schrödinger equation, and using elements of quaternionic analysis, we study one method for factorizing this Schrödinger equation’s operator into two first-order differential operators. For the two-dimensional case we show that the electrical impedance equation is equivalent to a particular Vekua equation, and using recent discoveries in Pseudoanalytic Function Theory, we analyze the structure of its solutions. We broach how to solve the inverse Calderon problem in the plane, and finally we mention the concepts that allows us to express the general solution of the electrical equation through Taylor series in formal powers.