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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.
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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

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