@MISC{Bogatyrev_conformalmapping, author = {A B Bogatyrev}, title = {Conformal mapping of rectangular heptagons}, year = {} }

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Abstract

Abstract. A new effective approach to calculating the direct and inverse conformal mapping of rectangular polygons onto a half-plane is put forward; it is based on the use of Riemann theta functions. Bibliography: 14 titles. Keywords: Christoffel-Schwarz integral, Riemann surface, Jacobian, Siegel space, theta functions. § 1. Introduction There exists an impressive list of numerical methods for conformal mapping of (simply connected) polygons [1]. A simple observation allows one to extend this list. Once the angles of the polygon are rational multiples of π, the Christoffel-Schwarz (CS) integral which maps the upper half-plane H to the polygon is an Abelian integral on a compact Riemann surface. The full power of the function theory on Riemann surfaces may be applied now to attack the evaluation of the integral as well as its auxiliary parameters. In this paper we consider a simply connected polygon with six right angles and one zero angle. This is the simplest case beyond the elliptic CS-integral (4 right angles in the polygon), described in As usual, there are several auxiliary parameters of the mapping which are determined by the geometrical dimensions of the polygon. Many problems, for instance, in classical mechanics, become simpler when written in a suitable system of coordinates. We shall see that this also holds for problems relating to conformal mappings: a good portion of those determining equations are linear if entries of the period matrix are taken for independent coordinates. Say, if we map an L-shaped rectangular hexagon, we have to solve just one nonlinear equation to determine the mapping.