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Estimating the Error in the Koebe Construction
Computational Methods and Function Theory (2011)
  • Valentin V. Andreev, Lamar University
  • Timothy H. McNicholl, Lamar University
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].
  • multiply-connected domains,
  • potential theory,
  • capacity
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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
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