A conformal map formula for difficult cases
type: Item
instance of: scholarly article
title: A conformal map formula for difficult cases
zbMATH Open document ID: 0589.30007
DOI: 10.1016/0377-0427(86)90139-1
author: Charles Zemach
published in: Journal of Computational and Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The standard problem for a difficult case is the conformal mapping of the set of all points in \({\mathbb{R}}^ 2\) (called z-plane) above a given boundary curve onto the upper half-plane of \({\mathbb{R}}^ 2\) (called w- plane). By applying exp(iz), exp(iw), respectively this problem can be transferred to a mapping between a simply connected region and the unit disk. The difficulties arise either when the given boundary curve varies considerably or when the given simply connected region is ''long''. The author mentions as examples a region bounded below by \(y(x)=-c \cos x\) with c large, say \(c=100\), or ellipses with large ratio c of large and small axis. In all these cases ''crowding'' is an apparent feature. That means that portions of the boundary are enormously compressed by the mapping. This compression factor in the mentioned examples is exponential in c. In an earlier paper [\textit{R. Menikoff} and the author, J. Comput. Phys. 36, 366-410 (1986; Zbl 0434.30007)] the integral equation \[ (*)\quad y(x)=const+\frac{1}{\pi}\int^{\pi}_{-\pi}\log | \sin \frac{1}{2}(u(x)-u(\quad x'))| dx' \] is derived, where y represents the given boundary and u the wanted boundary correspondence. The author calls a boundary point (x,y(x)) crowded, if the (unknown) compression factor \(du/dx<1\), he calls it severely crowded if \(du/dx\ll 1\). Several considerations lead the author to the simplified, approximate compression \[ (**)\quad \frac{\sin \{0.5(u(x)-u_ 0)\}}{\sin \{0,5(u(x_ 1)-u_ 0)\}}=\frac{\sinh Q(x)\quad}{\sinh Q(x_ 1)} \] for the unknown boundary correspondence u, where Q can be expressed in terms of the known y, \(u_ 0=u(x_ 0)\), \(x_ 0\) a minimum point of y (with high compression) and x, \(x_ 1\) severely crowded points near \(x_ 0\). Thus (**) is a local (asymptotic) formula. There are several numerical results including comparisons with the exact mapping which show that calculations of u based on (**) yield results such that the ''crowding'' du/dx has the righ…
Mathematics Subject Classification ID: 30C30
zbMATH DE Number: 3946525
zbMATH Keywords: crowding
reviewed by: Gerhard Opfer
MaRDI profile type: Publication
full work available at URL: 0377-0427(86)90139-1
OpenAlex ID: W2071481758
cites work: Methods for numerical conformal mapping
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description: scientific article; zbMATH DE number 3946525