Computational Methods and Function Theory

, Volume 16, Issue 4, pp 585–608 | Cite as

Conformal Equivalence of Analytic Functions on Compact Sets

  • Trevor RichardsEmail author


In this paper we present a geometric proof of the following fact: Let D be a Jordan domain in \(\mathbb {C}\), and let f be analytic on cl(D). Then there is an injective analytic map \(\phi :D\rightarrow \mathbb {C}\), and a polynomial p, such that \(f\equiv p\circ \phi \) on D (that is, f has a polynomial conformal model p).


Conformal equivalence Level curves Analytic functions Conformal invariant 

Mathematics Subject Classification

30C 30A 


  1. 1.
    Ebenfelt, P., Khavinson, D., Shapiro, H.S.: Two-dimensional shapes and lemniscates. Contemp. Math. 553, 45–59 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hilbert, D.: Über die Entwicklung einer beliebigen analytischen Funktion einer Variablen in eine unendliche anch ganzen rationalen Funktionen fortschreitende Reihe. Göttinger Nachrichten, pp. 63–70, (1897)Google Scholar
  3. 3.
    Lowther, G., Speyer D.: Conjecture: every analytic function on the closed disk is conformally a polynomial. (2013). Accessed: 22 June 2015
  4. 4.
    Pfluger, A.: Über die Konstruktion Riemannscher Flächen durch Verheftung. J. Indian Math. Soc. 24, 401–412 (1961)Google Scholar
  5. 5.
    Richards, T., Younsi, M.: Conformal models and fingerprints of pseudo-lemniscates. Submitted to Constructive Approximation (2015)Google Scholar
  6. 6.
    Richards, T.: Level curve configurations and conformal equivalence of meromorphic functions. Comput. Methods Funct. Theory 15(2), 323–371 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Younsi, M.: Shapes, fingerprints and rational lemniscates. Proc. Amer. Math. Soc. 144, 1087–1093 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematics DepartmentWashington and Lee UniversityLexingtonUSA

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