Generalized Analytic Functions

Abstract

On a complex plane, we consider the system of differential equations

$$ \begin{array}{*{20}{c}} {{\xi _x} - {\eta _y} = a\xi + b\eta }\\ {{\xi _x} - {\eta _y} = {a_1}\xi + {b_1}\eta .} \end{array} $$

(19.1)

This system is written in the complex form

$$ \begin{array}{*{20}{c}} {\bar \partial u = Au + B\bar u,}&{u = \xi + i\eta ,}\\ {\bar \partial = \frac{1}{2}\left( {\frac{\partial }{{\partial x}} + i\frac{\partial }{{\partial y}}} \right),}&{x = x + iy.} \end{array} $$

(19.2)

One calls the system (19.1), (19.2) the Carleman system or the Bers-Vekua system. We prefer to call this system the system of Carleman-Bers-Vekua (CBV). Solutions to this system are called generalized analytic functions or pseudo-analytic functions. The generalized analytic function theory was founded in the 1950s by Bers [1], [a] and Vekua [35], [36].