Korean J. Math. Vol. 24 No. 1 (2016) pp.81-106
DOI: https://doi.org/10.11568/kjm.2016.24.1.81

Finding the natural solution to $f(f(x)) = \exp(x)$

Article Sidebar

PDF
Published: Mar 30, 2016
Keywords:
Tetration, Abel's functional equation, iteration
Mathematics Subject Classification:
26A18, 30D05, 39B12.

Main Article Content

William Paulsen
Department of Mathematics and Statistics Arkansas State University Jonesboro, AR 72401, USA

Abstract

In this paper, we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are an infinite number of solutions, and there is no agreement as to which solution is "best.'' We will approach the problem by first solving Abel's functional equation $\alpha(e^x) = \alpha(x) + 1$ by perturbing the exponential function so as to produce a real fixed point, allowing a unique holomorphic solution. We then use this solution to find a solution to the unperturbed problem. However, this solution will depend on the way we first perturbed the exponential function. Thus, we then strive to remove the dependence of the perturbed function. Finally, we produce a solution that is in a sense more natural than other solutions.


Article Details

References

[1] N. H. Abel, Untersuchung der Functionen zweier unabh ̈angig ver ̈anderlichen Gr ̈oßen x und y, wie f(x, y), welche die Eigenschaft haben, ..., Journal fu ̈r die reine und angewandte Mathematik, 1 (1826), 11–15. Google Scholar

[2] C. C. Cowen, Analytic solutions of B ̈ottcher’s functional equation in the unit disk, Aequationes Mathematicae 24 (1982), 187–194. doi:10.1007/BF02193043 Google Scholar

[3] P. Fatou, Sur les equations fonctionelles, Bull. Soc. Math. France 47 (1919), 161–271. Google Scholar

[4] E. Jabotinsky, Analytic iteration. Trans. Amer. Math. Soc., 108 (1963), 457– 477. Google Scholar

[5] H. Kneser, Reelle analytishe L ̈osungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen, J. reine angew. Math. 187 (1950), 56–67. Google Scholar

[6] G. Koenigs, Recherches sur les int egrales de certaines equations fonctionelles, Annales Scientifiques de l'E cole Normale Sup erieure, 1 (3, Suppl ement) (1884), 3-41. Google Scholar

[7] D. Kouznetsov, Solution of F (z + 1) = exp(F (z)) in the complex z-plane, Mathematics of Computation 78: 267 (2009), 1647–1670. Google Scholar

[8] M. Kuczma, B. Choczewski, and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990. Google Scholar

[9] H. O. Peitgen and D. Saupe, editors, The Science of Fractal Images, Springer-Verlag, New York, 1988. Google Scholar

[10] J. Ritt, On the iteration of rational functions, Trans. Amer. Math. Soc. 21:3 (1920), 348–356. doi:10.1090/S0002-9947-1920-1501149-6 Google Scholar

[11] E. Schr ̈oder, U ̈ber iterierte Funktionen, Math. Ann. 2 (1871), 296–322. Google Scholar

[12] H. Trappmann and D. Kouznetsov, Uniqueness of holomorphic Abel functions at a complex fixed point pair, Aequationes Mathematicae, 81:1 (2011), 65–76. Google Scholar

[13] P. Walker, The exponential of iteration of ex − 1, Proc. Am. Math. Soc. 110:3 (1990), 611–620. Google Scholar

[14] P. Walker, On the Solutions of an Abelian Functional Equation, Journal of Mathematical Analysis and Applications 155 (1991), 93–110. Google Scholar