Riemann–Schwarz Reflection Principle and Asymptotics of Modules of Rectangular Frames

Abstract

We investigate asymptotical behavior of the conformal module of a doubly-connected domain which is the difference of two homothetic rectangles under stretching it along the abscissa axis. Thereby, we give the answer to a question put by Prof. M. Vuorinen. 2010 AMS Subject Classification. Primary 30C20.

Conformal mappings of stretched polyominoes onto half-plane

“…But they are equivalent to (9), Remark 1. Using the method of [18] we can obtain the following asymptotics:…”

“…which has the same form as (18) but correspond to the parameter (−c). Let us find the point α 0 ∈ (0, c) which is mapped on the point λ = a 2 , the endpoint of the slit of R. We have…”

“…Now we find the points α 1 ∈ (−1, 0) and α 3 ∈ (0, 1) which are mapped on λ and 1/λ under the map (18). The points can be found from the relations…”

“…One of the main model domains is so-called L-polygon, i. e., hexagon which is the difference of two rectangles with common vertex, in addition, one of them contains another. Such polygons were studied by many specialists, see , e. g., [2], [14], [15], [18]. We consider L-polygons of the form Fixation of four vertices turns it to a quadrilateral.…”

“…A conformal mapping of the trimino S, stretched by the map f H , onto the upper half-plane and its module are found in [8], [18]; we should note that a formula for the module for non-stretched trimino can be obtained from results of in [7], see also [3].…”