Advances in Computational Mathematics

, Volume 43, Issue 6, pp 1261–1282 | Cite as

Solving F(z + 1) = b F(z) in the complex plane

  • William Paulsen
  • Samuel Cowgill


The generalized tetration, defined by the equation F(z+1) = b F(z) in the complex plane with F(0) = 1, is considered for any b > e 1/e . By comparing other solutions to Kneser’s solution, natural conditions are found which force Kneser’s solution to be the unique solution to the equation. This answers a conjecture posed by Trappmann and Kouznetsov. Also, a new iteration method is developed which numerically approximates the function F(z) with an error of less than 10−50 for most bases b, using only 180 nodes, with each iteration gaining one or two places of accuracy. This method can be applied to other problems involving the Abel equation.


Tetration Abel’s functional equation Iteration Cross-track method 

Mathematics Subject Classification (2010)

26A18 30D05 39B12 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abel, N. H.: Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben. J. Reine Angew. Math. 1, 11–15 (1826)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ackermann, W.: Zum Hilbertschen Aufbau der reellen Zahlen. Math. Ann. 99, 118–133 (1928). doi: 10.1007/BF01459088 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Atkinson, K., Han, W.: Elementary numerical analysis, 3rd edn. Wiley, New York (2004)Google Scholar
  4. 4.
    Jabotinsky, E.: Analytic iteration. Trans. Amer. Math. Soc. 108, 457–477 (1963)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kneser, H.: Reelle analytishe Lösungen der Gleichung φ(φ(x)) = e x und verwandter Funktionalgleichungen. J. Reine Angew. Math. 187, 56–67 (1950)Google Scholar
  6. 6.
    Koenigs, G.: Recherches sur les intégrales de certaines équations fonctionelles. Ann. Scientifiques l’École Norm. Supérieure 1(3, Supplément), 3–41 (1884)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kouznetsov, D.: Solution of \(F(z+1) = \exp (F(z))\) in the complex z-plane. Math. Comput. 78(267), 1647–1670 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kouznetsov, D.: Tetration as special function. Vladikavkaz Math. J. 12(2), 31–45 (2010)MathSciNetMATHGoogle Scholar
  9. 9.
    Kouznetsov, D.: Evaluation of holomorphic ackermanns. Appl. Comput. Math. 3(6), 307–314 (2014)CrossRefGoogle Scholar
  10. 10.
    Kouznetsov, D., Trappmann, H.: Portrait of the four regular super-exponentials to base sqrt(2). Math. Comput. 79(271), 1727–1756 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kuczma, M., Choczewski, B., Ger, R.: Iterative functional equations. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  12. 12.
    Paulsen, W.: Finding the natural solution to \(f(f(x)) = \exp (x)\). Korean J. Math. 24(1), 81–106 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Schröder, E.: Über iterierte funktionen. Math. Ann. 2, 296–322 (1871)Google Scholar
  14. 14.
    Trappmann, H., Kouznetsov, D.: Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Math. 81(1), 65–76 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Walker, P.: The exponential of iteration of e x−1. Proc. Am. Math. Soc. 110(3), 611–620 (1990)MATHGoogle Scholar
  16. 16.
    Walker, P.: Infinitely differentiable generalized logarithmic and exponential functions. Math Comput. 57, 723–733 (1990)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Walker, P.: On the Solutions of an Abelian Functional Equation. J. Math. Anal. Appl. 155, 93–110 (1991)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Arkansas State UniversityArkansasUSA

Personalised recommendations