Generalized cauchy—riemann systems with a singular point

The elliptic system of 1st order partial differential equation of the following type is called a generalized Cauchy–Riemann system. Here u and v are unknown and a,b,c,d are given of the real variables x and y in some domain G. This system extends the classical Cauchy-Riemann system which can be obtained from (1) for a=b=c=d=0. If we introduce the complex variable z=x+iy i 2=−1, the complex unknown function w=u+iv, and the notations then (1) and (2) can be written as The theory of w-functionsz∊G, satisfying (3), was constructed by L. Bers and I. Vekua assuming that A(z) and B(z) belong to L p (G)-class, p>2, [1]. In this case (3) is called the regular generalized Cauchy-Riemann system and its solutions are named the generalized analytic functions. Such system coefficients can allow small singularities. In particular, if A(z) and B(z) tend to infinity in some isolated point of G, then the order of the singularity in this point must be strictly less than 1.