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].
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Andreev, V.V., McNicholl, T.H. Estimating the Error in the Koebe Construction. Comput. Methods Funct. Theory 11, 707–724 (2012). https://doi.org/10.1007/BF03321883
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DOI: https://doi.org/10.1007/BF03321883