Conformal Mapping onto a Doubly Connected Circular Arc Polygonal Domain
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Conformal Mapping onto a Doubly Connected Circular Arc Polygonal Domain Ulrich Bauer1 · Wolfgang Lauf1 Received: 27 October 2017 / Revised: 31 May 2018 / Accepted: 1 June 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper presents a construction principle for the Schwarzian derivative of conformal mappings from an annulus onto doubly connected domains bounded by polygons of circular arcs. Keywords Conformal mapping · Circular arc polygon · Schwarzian derivative Mathematics Subject Classification 30C20 · 30C30
1 Introduction The Schwarz–Christoffel mapping is a well-known and well-investigated tool for conformal mappings. It maps a canonical domain like the upper half plane or the unit disk onto a domain bounded by a polygon. This idea was modified by introducing changes to the boundary [14, Ch. 4]. One of these modifications is the mapping onto domains bounded by so-called circular arc polygons, where the straight sides of a polygon can be replaced by circular arcs. This concept was investigated by Bjørstad and Grosse [1] and by Howell [18]. Schwarz–Christoffel mappings were again in focus when DeLillo, Elcrat and Pfaltzgraff [2] obtained a formula for the mapping onto multiply connected polygonal domains [12]. From this point onwards, two lines of research have evolved: improve-
Communicated by Darren Crowdy.
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Wolfgang Lauf [email protected] Ulrich Bauer [email protected]
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Department of Computer Science and Mathematics, University of Applied Sciences OTH Regensburg, 93053 Regensburg, Germany
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U. Bauer, W. Lauf
Fig. 1 Annulus mapped onto a doubly connected circular arc polygonal domain
ments in the method developed by DeLillo, Elcrat and Pfaltzgraff with its focus on numerics and that by Crowdy [4] using the Schottky–Klein prime function. The method of DeLillo, Elcrat and Pfaltzgraff, which uses reflections for the construction of the mapping function, was also used to find a formula for bounded domains [8]. At present, there are some well-developed concepts for the numerical evaluation of the mapping based on this strategy [9,10,13]. The second method using the Schottky–Klein prime function was used in the construction of mappings for several different, also non-polygonal multiply connected domains [5–7] including especially the mapping onto doubly connected circular arc polygonal domains [5]. Lately, a third method has been presented in [19] which uses the method of boundary value problems to prove the Schwarz–Christoffel formula. This approach has the advantage of having no restrictions regarding the geometry of the preimage domain (Fig. 1). The main result presented in this paper is the construction of the Schwarzian derivative for a conformal mapping from an annulus onto a doubly connected circular arc polygonal domain using a construction similar to the one in the initial paper of DeLillo, Elcrat and Pfaltzgraff [11] instead of the Schottky–Klein prime function [5]. Theorem 1 Let P be a doubly connected circular arc polygonal domain with modul
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