Aequationes mathematicae

, Volume 81, Issue 1–2, pp 65–76 | Cite as

Uniqueness of holomorphic Abel functions at a complex fixed point pair

  • Henryk Trappmann
  • Dimitrii Kouznetsov


We give a simple uniqueness criterion (and some derived criteria) for holomorphic Abel functions and show that Kneser’s real analytic Abel function of the exponential is subject to this criterion.

Mathematics Subject Classification (2000)

Primary 30D05 


Abel function Abel equation exponential function fractional iterates holomorphic solution real analytic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bromer N.: Superexponentiation. Math. Mag. 60(3), 169–174 (1987)MathSciNetGoogle Scholar
  2. 2.
    Contreras M.D., Madrigal S.D., Pommerenke C.: Some remarks on the Abel equation in the unit disk. J. Lond. Math. Soc. II. Ser. 75(3), 623–634 (2007)MATHCrossRefGoogle Scholar
  3. 3.
    Hooshmand M.H.: Ultra power and ultra exponential functions. Integral Transforms Spec. Funct. 17(8), 549–558 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Kneser H.: Reelle analytische Lösungen der Gleichung \({\varphi(\varphi(x))=e^x}\) und verwandter Funktionalgleichungen. J. Reine Angew. Math. 187, 56–67 (1949)MathSciNetMATHGoogle Scholar
  5. 5.
    Kouznetsov D.: Solution of f(x + 1) = exp(f(x)) in complex z-plane. Math. Comput. 78, 1647–1670 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Kouznetsov D., Trappmann H.: Superfunctions and square root of factorial. Mosc. Univ. Phys. Bull. 65(1), 6–12 (2010)CrossRefGoogle Scholar
  7. 7.
    Kuczma M., Choczewski B., Ger R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  8. 8.
    Markushevich A.I.: Theory of Functions of a Complex Variable. Prentice-Hall, Englewood Cliffs (1965)Google Scholar
  9. 9.
    Szekeres G.: Regular iteration of real and complex functions. Acta Math. 100, 203–258 (1958)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Szekeres G.: Fractional iteration of exponentially growing functions. J. Aust. Math. Soc. 2, 301–320 (1961)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Walker P.L.: Infinitely differentiable generalized logarithmic and exponential functions. Math. Comput. 57(196), 723–733 (1991)MATHCrossRefGoogle Scholar
  12. 12.
    Walker P.L.: On the solutions of an Abelian functional equation. J. Math. Anal. Appl. 155(1), 93–110 (1991)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.BerlinGermany
  2. 2.Institute for Laser ScienceUniversity of Electro-CommunicationsChofushi, TokyoJapan

Personalised recommendations