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 (gamma gradu) = 0 where the function div gamma = sigma + iomegaepsiv is the admitivity, sigma denotes the conductivity, epsiv is the frequency, epsiv is the permittivity, i is the imaginary unit, and u denotes the electric potential. When gamma is a function of three spatial variables, we show how to rewrite the equation into a stationary Schrodinger equation, and using elements of quaternionic analysis, we study one method for factorizing this Schrodinger equationpsilas 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.