Schwarz-Christoffel formula for multiply connected domains
type: Item
instance of: scholarly article
title: Schwarz-Christoffel formula for multiply connected domains
zbMATH Open document ID: 1293.30021
DOI: 10.1007/BF03321837
published in: Computational Methods and Function Theory
publication date: 2013-01-11T00:00:00Z
review text: The author derives a Schwarz-Christoffel formula for the conformal map of an arbitrary \(n\)-connected domain \(\mathbb{D}\) bounded by \(n\) mutually disjoint circles \(|z-a_k|=r_k\), \(k=1,..,n\), onto the exterior of \(n\) mutually disjoint polygons. To construct the conformal map the author solves the Riemann-Hilbert problem \[ \Re[(t-a_k)\psi (t)]=-1,\,\,\,\,\,\,\,\,|t-a_k|=r_k,\,\,\,\,k=1,..,n, \] where the function \(\psi (z)\) is analytic in \(\mathbb{D}\) and continuous on \(\mathbb{D}\cup \partial\mathbb{D}\) except for certain singularities on the boundary of \(\partial \mathbb{D}\). The derivation utilizes the uniform convergence of the Poincaré series for the classical Schottky groups that was established in [\textit{V. Mityushev}, Proc. Am. Math. Soc. 126, No. 8, 2399--2406 (1998; Zbl 0899.30029)]. As a result the author is able to obtain the Schwarz-Christoffel formula for arbitrary \(n\)-connected domains \(\mathbb{D}\) without any geometric restriction on the location of the non-overlapping disks \(|z-a_k|\leq r_k\).
reviewed by: Valentin V. Andreev
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30E25
zbMATH DE Number: 6124796
zbMATH Keywords: multiply connected domain
zbMATH Keywords: Schwarz-Christoffel formula
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: conformal mapping
zbMATH Keywords: functional equation
zbMATH Keywords: Poincaré series
author: Vladimir Mityushev
describes a project that uses: SC Toolbox
describes a project that uses: Schwarz-Christoffel
MaRDI profile type: Publication
cites work: The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains
cites work: Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions
cites work: Conformal mappings between canonical multiply connected domains
cites work: Conformal mappings to a doubly connected polycircular arc domain
cites work: Conformal mappings to multiply connected polycircular arc domains
cites work: Schwarz-Christoffel mapping of bounded, multiply connected domains
cites work: Computation of multiply connected Schwarz-Christoffel maps for exterior domains
cites work: Schwarz-Christoffel mapping of multiply connected domains
cites work: Q5515593
cites work: Q4322901
cites work: Convergence of the Poincaré series for some classical Schottky groups
cites work: Hilbert boundary value problem for multiply connected domains
cites work: Scalar Riemann–Hilbert Problem for Multiply Connected Domains
cites work: Riemann-Hilbert problems for multiply connected domains and circular slit maps
cites work: Q4935954
full work available at URL: https://doi.org/10.1007/bf03321837
OpenAlex ID: W2026947034
description: scientific article; zbMATH DE number 6124796