Computational Methods and Function Theory

, Volume 11, Issue 2, pp 707–724 | Cite as

Estimating the Error in the Koebe Construction

Article

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].

En]Keywords

multiply-connected domains potential theory capacity 

2000 MSC

30C20 30C30 30C85 

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Copyright information

© Heldermann  Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsLamar UniversityBeaumontUSA

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