ON FUNCTIONS SEMI-ANALYTICAL IN THE POLYDISK

Abstract

In the present paper the class of semi-analytical functions in the polydisk $U^n\subset\mathbb{C}^n$ is introduced. This class is an extension of the set of holomorphic functions. For $n=1$ the concept of semi-analyticity coincides with analyticity. The Dirichlet problem with values given on the distinguished boundary of the polydisk always has a solution in the set of real parts of semi-analytical functions. Therefore, to investigate semi-analytical functions one can apply the potential theory methods, like one does it for the one-dimensional case. In the present paper the Schwarz type integral representation for the above-mentioned functions is obtained.