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We show a close relation between the Schrödinger equation and the conductivity equation to a Vekua equation of a special form. Under quite general conditions we propose an algorithm for explicit construction of pseudoanalytic positive formal powers for the Vekua equation that as a consequence gives us a complete system of solutions for the Schrödinger and the conductivity equations. Besides the construction of complete systems of exact solutions for the above mentioned second order equations and the Dirac equation, we discuss some other applications of pseudoanalytic function theory.

Electromagnetic wave propagation is currently present in the vast majority of situations which occur in veryday life, whether in mobile communications, DTV, satellite tracking, broadcasting, etc. Because of this the study of increasingly complex means of propagation of lectromagnetic waves has become necessary in order to optimize resources and increase the capabilities of the devices as required by the growing demand for such services.
Within the electromagnetic wave propagation different parameters are considered that characterize it under various circumstances and of particular importance are the reflectance and transmittance. There are several methods or the analysis of the reflectance and transmittance such as the method of approximation by boundary condition, the plane wave expansion method (PWE), etc., but this work focuses on the WKB and SPPS methods.
The implementation of the WKB method is relatively simple but is found to be relatively efficient only when working at high frequencies. The SPPS method (Spectral Parameter Powers Series) based on the theory of pseudoanalytic functions, is used to solve this problem through a new representation for solutions of Sturm Liouville equations and has recently proven to be a powerful tool to solve different boundary value and eigenvalue problems. Moreover, it has a very suitable structure for numerical implementation, which in this case took place in the Matlab software for the valuation of both conventional and turning points profiles.
The comparison between the two methods allows us to obtain valuable information about their perfor mance which is useful for determining the validity and propriety of their application for solving problems where these parameters are calculated in real life applications.

We show a transformation K which allows us to rewrite the Dirac equation in its covariant form in a purely real quaternionic equation. We discuss how this transformation allows us for obtaining a involutive symmetry of the Dirac equation, as well as one simplification of the traditional vector of currents of the Dirac equation in traditional form. We also show the corresponding quaternionic equation for the problem of charge conjugation in the hole theory, and the quaternionic equation of conservation of currents. Finally, we discuss one decomposition of the quaternionic Dirac operator in two Maxwell's operators corresponding to time-harmonic case in homogeneous media, without sources which surprisingly agrees with the well known relation in quantum mechanics between the frequency ù and the impulse p E=p²c²+m²c, where E denotes the energy.

With the aid of factorization of the Schrödinger operator by quaternionic differential operators of first order proposed in recent works by S. Bernstein and K. Gürlebeck we study the system describing forcefree magnetic fields with nonconstant proportionality factor, the static Maxwell system for inhomogeneous media, the Beltrami condition and the Dirac equation with different types of potentials depending on one variable. We obtain integral representations for solutions of these systems.