On conformal mapping of polygonal regions
type: Item
instance of: scholarly article
title: On conformal mapping of polygonal regions
zbMATH Open document ID: 0857.30006
DOI: 10.1007/BF01060857
author: Vladimir Gutlyanskiĭ
author: A. O. Zaidan
published in: Ukrainian Mathematical Journal
publication date: 1997-03-03T00:00:00Z
review text: Let \(\overline C\) denote a closed plane and \(H^+\) its upper half-plane. Let \(D_n\) stand for the interior of an \(n\)-gon whose internal angles at vertices \(A_k\) are equal to \(\alpha_k\pi\), \(k=1, \dots, n\). It is known that, for any polygon, the relation \(\sum^n_{k=1} \alpha_k=n-2\) holds. The following theorem is a central result in the theory of conformal mappings of polygonal regions [\textit{R. Schinzinger} and \textit{P. Laura}, Conformal mapping: methods and applications, Elsevier, Amsterdam (1991), p. 65]. Theorem, Suppose that \(D_n\) is a simply connected domain in the complex plane \(\mathbb{C}\), bounded by a polygon with vertices at points \(A_1, \dots, A_n\) and internal angles \(\pi \alpha_k\) where \(0\leq \alpha_k\leq 2\) for finite \(A_k\) and \(-2\leq \alpha_k\leq 0\) for \(A_k=\infty\). Then there exists a conformal mapping of the upper half-plane \(H^+\) onto \(D_n\) and, moreover, each mapping of this sort can be represented in the form \[ f(z)=c\int^z_0 \prod^n_{k=1} (z-a_k)^{\alpha_k-1} dz+c_1 \] where \(a_1, \dots, a_n\) are the preimages of the vertices \(A_1, \dots, A_n\). The complex constants \(a_1, \dots, a_n\), \(c\) and \(c_1\) are called the assessory parameters of the Schwarz-Christoffel integral. The principal problem is to determine these parameteter. The authors, developing \textit{P. P. Kufarev}'s method [Dokl. Akad. Nauk SSSR, N. Ser. 57, 535-537 (1947; Zbl 0041.24302)], solve this problem.
Mathematics Subject Classification ID: 30C20
zbMATH DE Number: 923078
zbMATH Keywords: polygonal regions
zbMATH Keywords: conformal mapping
zbMATH Keywords: assessory parameters
zbMATH Keywords: Schwarz-Christoffel integral
reviewed by: Leon Mikołajczyk
describes a project that uses: SCPACK
MaRDI profile type: Publication
cites work: Q3126755
cites work: Numerical Computation of the Schwarz–Christoffel Transformation
cites work: Q5801578
cites work: Q3913028
cites work: Kufarev's method for determining the Schwarz-Christoffel parameters
cites work: Q3749277
cites work: Q5582453
full work available at URL: https://doi.org/10.1007/bf01060857
OpenAlex ID: W1989403792
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description: scientific article; zbMATH DE number 923078