Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-08T05:21:53.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Fractional iteration of exponentially growing functions

Published online by Cambridge University Press:  09 April 2009

G. Szekeres
G. Szekeres
Affiliation:
The University of Adelaide, South Australia.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The fractional iteration of ex and solutions of the functional equation have frequently been discussed in literature. G. H. Hardy has shown (in [3], and in greater detail in [4]) that the asymptotic behaviour of the solutions of (1) cannot be expressed in terms of the logarithmico-exponential scale, although they are comparable with each member of the scale.1 Hence solutions of (1) provide a remarkably simple instance of functions whose manner of growth does not fit into the scale of L-functions but requires non-elementary orders of infinity for an accurate representation. This raises quite naturally the question whether there exists a most regularly growing solution of equation (1) which might serve as a prototype for this kind of growth. In a slightly more general context we may ask whether there exists a ‘best’ family of fractional iterates fσ(x), satisfying.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 3 , February 1962 , pp. 301 - 320
Copyright
Copyright © Australian Mathematical Society 1962

References

[1]Baker, I. N., Zusammensetzungen ganzer Funktionen, Math. Z., 69 (1958), 121–163.CrossRefGoogle Scholar
[2]Bourbaki, N., Eléments de Mathématique, Book 4, Chapter 7 (Paris, 1951).Google Scholar
[3]Hardy, G. H., Orders of Infinity, Cambridge Tracts No. 12, 2-nd Edn. (Cambridge, 1924).Google Scholar
[4]Hardy, G. H., Properties of logarithmico-exponential functions, Proc. London Math. Soc. (2) 10 (1912), 54–90.CrossRefGoogle Scholar
[5]Kneser, H., Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex, J. f. reine angew. Math., 187 (1950), 66–67.Google Scholar
[6]Königs, G., Recherches sur les intégrales des certaines equations fonctionelles, Ann. Sci. Ecole Norm. Sup. (3) 1 (1884), Supplément, 3–41.CrossRefGoogle Scholar
[7]Kuczma, M., On convex solutions of the functional equation g[α(x)] –g(x) = φ(x), Publ. Math. Debrecen, 6 (1959) 40–47.CrossRefGoogle Scholar
[8]Lévy, P., Fonctions à croissance régulière et itération d'ordre fractionnaire, Ann. Mat. Pura Appl. (4) 5 (1928), 269–298.CrossRefGoogle Scholar
[9]Morris, K. W. and Szekeres, G., Tables of the logarithm of iteration of ex — 1.Google Scholar
[10]Szekeres, G., Regular iteration of real and complex functions, Acta Math. 100 (1958), 203–258.CrossRefGoogle Scholar
[11]Szekeres, G., On a Theorem of P. Lévy, Publ. Math. Inst., Hungarian Acad. Sci., Ser A, 5 (1960), 277–282.Google Scholar