An Analog of the Cauchy Formula for Certain Beltrami Equation

Abstract

The Beltrami differential equations are intrinsic generalizations of the Cauchy–Riemann system in complex analysis. Their solutions generalize holomorphic functions. As known, solutions to many problems of the complex analysis are based on application of the Cauchy formula, i.e., on the integral representation of analytical functions by curvilinear integrals over boundaries of the domains of analyticity. Particularly, this representation enables us to solve the Riemann boundary-value problem for holomorphic functions, to prove the Painleve theorem on erasing of singularities of analytical functions, and to obtain many other important results. A. Tungatarov established an analog of this representation of solutions to a certain simple case of the Beltrami equation (so-called beta-analytic functions). A. Tungatarov’s representation was used by R. Abreu-Blaya, J. Bory-Reyes, and D. Pe˜na-Pe˜na for solving the problems stated by B. Riemann, P. Painleve, and other researchers. In this paper, we constructed integral representations for the solutions of more extensive classes of the Beltrami equations, which are analogs of the integral Cauchy formula, and described their applications.