Recent developments in applied pseudoanalytic function theory

Abstract

We present recently obtained results in the theory of pseudoanalytic functions and its applications to elliptic second-order equations. The operator (divpgrad+q) with p and q being real valued functions is factorized with the aid of Vekua type operators of a special form and as a consequence the elliptic equation (divpgrad+q)u=0, (1) reduces to a homogeneous Vekua equation describing generalized analytic (or pseudoanalytic) functions. As a tool for solving the Vekua equation we use the theory of Taylor and Laurent series in formal powers for pseudoanalytic functions developed by L. Bers. The series possess many important properties of the usual analytic power series. Their applications until recently were limited mainly because of the impossibility of their explicit construction in a general situation. We obtain an algorithm which in a really broad range of practical applications allows us to construct the formal powers and hence the pseudoanalytic Taylor series in explicit form precisely for the Vekua equation related to equation (1). In other words, in a bounded domain this gives us a complete (in C-norm) system of exact solutions of (1).