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Variational Aspects of the Abel and Schroder Functional Equations

Published online by Cambridge University Press:  20 November 2018

M.A. McKiernan  and
A. Rényi
M.A. McKiernan
Affiliation:
University of Waterloo
A. Rényi
Affiliation:
University of Waterloo
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Given an analytic function f, the successive iterates of f are defined by

f[0](z) = z, f[n+1](z) = f{f[n](z)} for every z.

In particular f[1](z) = {f[0](z)} f(z). Extensive study has been given [1] to the problem of generalizing the iterates f[n], for integer n, to f[t] for arbitrary t, where the iterative character of f[t] is to be preserved by the conditions,

f[0](z) = z and f[s]{f[t](z)} = f[s+t](z) for arbitrary s and t.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 2 , May 1963 , pp. 257 - 265
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. See for example, Szekeres, G., Regular Iteration of Real and Complex Functions, Acta Mathematica, 100 (1958), pp. 203-258. Also P. Erdős and E. Jabotinsky, On Analytic Iteration, Journal D'Analyse Mathematique, Vol. 8, part 2, 1960/61, Jerusalem.10.1007/BF02559539Google Scholar
2. Schrőder, E., Uber iterierte Funktionen, Math. Ann., 2(1870) pp. 317-365.Google Scholar
3. The equations are given in tensor form for example in Sokolnikoff, I. S., Tensor Analysis, John Wiley and Sons, 1951, p. 161.Google Scholar