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A Class of functional equations which have entire solutions

Published online by Cambridge University Press:  17 April 2009

Peter L. Walker
Peter L. Walker
Affiliation:
Department of Mathematical Sciences, Sultan Qaboos University, P.O. Box 32486, Al-Khoudh, Muscat, Sultanate of Oman
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Abstract

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We consider the Abelian functional equation

where φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfy

We show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 3 , December 1988 , pp. 351 - 356
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Abel, N.H., ‘Determination d'une fonction au moyen d'une equation qui ne contient qu'une seule variable’, in Oeuvres Completes de Niels Henrik Abel II, p. 246 (Christiana Impr. de Grondahl, 1881).Google Scholar
[2]Fatou, P., ‘Sur les equations fonctionelles’, Bull. Soc. Math. France 47 (1919), 161–271. 48, (1920) pp. 33–94, 208–314.CrossRefGoogle Scholar
[3]Szekers, G., ‘Fractional iteration of expronentially growing functions’, J. Austral. Math. Soc. 2 (1961–1902), 301–320.CrossRefGoogle Scholar
[4]Walker, P.L., ‘On the solutions of an Abelian functional equation’, (Submitted).Google Scholar