Computational Methods and Function Theory

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

Estimating the Error in the Koebe Construction

  • Valentin V. Andreev
  • Timothy H. McNicholl


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 

2000 MSC

30C20 30C30 30C85 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. I. Akhiezer, Aerodynamical Investigations, vol. 7, Ukrain. Akad. Nauk Trudi Fiz.-Mat. Viddilu, 1928.Google Scholar
  2. 2.
    J. B. Conway, Functions of one Complex Variable II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, 1995.Google Scholar
  3. 3.
    D. Crowdy, Geometric function theory: a modern view of a classical subject, Nonlinearity 21 no.10 (2008),, 205–219.MathSciNetCrossRefGoogle Scholar
  4. 4.
    T. K. DeLillo, A. R. Elcrat and J. A. Pfaltzgraff, Schwarz-Christoffel mapping of multiply connected domains, J. Anal. Math. 94 (2004), 17–47.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    D. Gaier, Untersuchung zur Durchführung der konformen Abbildung mehrfach zusammenhängender Gebiete, Arch. Rat. Mech. Anal. 3 (1959), 149–178.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    J. B. Garnett and D. E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005.zbMATHCrossRefGoogle Scholar
  7. 7.
    Z. He and O. Schramm, Fixed points, Koebe uniformization, and circle packings, Ann. of Math. 137 (1993), 369–406.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    P. Henrici, Applied and Computational Complex Analysis. Vol. 3, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1986.Google Scholar
  9. 9.
    P. Koebe, Uber eine neue Methode der Konformen Abbildung und Uniformisierung, Nachr. Kgl. Ges. Wiss. Göttingen, Math.-Phys. K1. 1912 (1912), 844–848.Google Scholar
  10. 10.
    M. Schiffer, Some recent developments in the theory of conformal mapping, Appendix to: R. Courant, Dirichlet Principle, Conformal Mapping and Minimal Surfaces, Interscience Publishers, 1950.Google Scholar
  11. 11.
    K. Strebel, Uber quadratische Differentiale mit geschlossenen Trajektorien und extremale quasikonforme Abbildungen, H. P. Künzi and A. Pfluger (eds.), Festband zum 70. Geburtstag von Rolf Nevanlinna (Berlin), Vorträge, gehalten anlässlich des Zweiten Rolf Nevanlinna-Kolloquiums in Zürich vom 4–6. November 1965, Springer-Verlag, 1966, pp. 105–127.Google Scholar
  12. 12.
    R. E. Thurman, Upper bound for distortion of capacity under conformal mapping, Trans. Amer. Math. Soc. 346 no.2 (1994), 605–616.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    M. Tsuji, Potential Theory, Chelsea, New York, 1975.zbMATHGoogle Scholar

Copyright information

© Heldermann  Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsLamar UniversityBeaumontUSA

Personalised recommendations