Uniqueness of holomorphic Abel functions at a complex fixed point pair


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.

This is a preview of subscription content, access via your institution.


  1. 1

    Bromer N.: Superexponentiation. Math. Mag. 60(3), 169–174 (1987)

    MathSciNet  Google 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)

    MATH  Article  Google Scholar 

  3. 3

    Hooshmand M.H.: Ultra power and ultra exponential functions. Integral Transforms Spec. Funct. 17(8), 549–558 (2006)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  5. 5

    Kouznetsov D.: Solution of f(x + 1) = exp(f(x)) in complex z-plane. Math. Comput. 78, 1647–1670 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6

    Kouznetsov D., Trappmann H.: Superfunctions and square root of factorial. Mosc. Univ. Phys. Bull. 65(1), 6–12 (2010)

    Article  Google Scholar 

  7. 7

    Kuczma M., Choczewski B., Ger R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)

    MATH  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10

    Szekeres G.: Fractional iteration of exponentially growing functions. J. Aust. Math. Soc. 2, 301–320 (1961)

    MathSciNet  Article  Google Scholar 

  11. 11

    Walker P.L.: Infinitely differentiable generalized logarithmic and exponential functions. Math. Comput. 57(196), 723–733 (1991)

    MATH  Article  Google Scholar 

  12. 12

    Walker P.L.: On the solutions of an Abelian functional equation. J. Math. Anal. Appl. 155(1), 93–110 (1991)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Henryk Trappmann.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Trappmann, H., Kouznetsov, D. Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequat. Math. 81, 65–76 (2011). https://doi.org/10.1007/s00010-010-0021-6

Download citation

Mathematics Subject Classification (2000)

  • Primary 30D05


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