Avtandil Tsitskishvili

Solution of the Schwarz Differential Equation

abstract:
A circular polygon of a general form with a finite number of vertices and arbitrary angles at these vertices is given. A single-valued analytic function mapping conformally a half-plane onto the given circular polygon is constructed in a general form. The function is proved to be a general solution of the Schwarz equation. First we construct functional series uniformly and rapidly convergent near all singular points and then fundamental local matrices which are connected by analytic continuation. The constructed analytic function satisfies nonlinear boundary conditions. In a general form, we compose and investigate all higher transcendental equations connecting geometric characteristics of circular polygons with unknown parameters of the Schwarz equation. Possible intervals of variation of unknown accessory parameters are established.

Mathematics Subject Classification: 34A20, 34B15.

Key words and phrases: Analytic function, differential equations, conformal mapping, circular polygons, fundamental matrices.