Estimating the Error in the Koebe ConstructionComputational Methods and Function Theory (2011)
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. In 1959, Gaier provided a convergence proof of the construction which depends on prior knowledge of the circular domain . 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  and a generalization of Schwarz-Pick Lemma by He and Schramm .
- multiply-connected domains,
- potential theory,
Citation InformationValentin 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/