Article
Estimating the Error in the Koebe Construction
Computational Methods and Function Theory
(2011)
Abstract
In 1912, Paul Koebe proposed an iterative method, the Koebe construction, to construct a conformal mapping of a non-degenerate, finitely connected domain D onto a circular domain C[9]. In 1959, Gaier provided a convergence proof of the construction which depends on prior knowledge of the circular domain [5]. We demonstrate that it is possible to compute the convergence rate solely from information about D. We do so by computing a suitable bound on the error in the Koebe construction (but, again, without knowing the circular domain in advance) by using a relatively recent result on the distortion of capacity by Thurman [12] and a generalization of Schwarz-Pick Lemma by He and Schramm [7].
Keywords
- multiply-connected domains,
- potential theory,
- capacity
Disciplines
Publication Date
2011
DOI
10.1007/BF03321883
Publisher Statement
The final publication is available at Springer via https://doi.org/10.1007/BF03321883.
Copyright 2011 Heldermann Verlag
Citation Information
Valentin V. Andreev and Timothy H. McNicholl. "Estimating the Error in the Koebe Construction" Computational Methods and Function Theory Vol. 11 Iss. 2 (2011) p. 707 - 724 Available at: http://works.bepress.com/timothy-mcnicholl/7/