Bergman-Vekua Operators and "Generalized Axially Symmetric Potential Theory"

Abstract

The generalized complex integral operators of Bergman-Vekua type give a uniform approach to construct solutions to various linear partial differential equations with three independent variables. The operators associate holomorphic functions of two complex variables and the solutions of the mentioned equations. Here we specify the complex, integral operators for the three-dimensional case for application to a special equation for axisym metric, problems (i.e. instationary axisymmetric problems in the plane or stationary axisymmetric problems in the space). Furthermore, the relations between the "Generalized Axially Symmetric Potential Theory", due to A. Weinstein et al., and the Bergman-Vekua operators are given and used in the explicit representation of the solutions.