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On the Periodicity of Compositions of Entire Functions

Published online by Cambridge University Press:  20 November 2018

Fred Gross
Fred Gross*
Affiliation:
U.S. Naval Research Laboratory, Washington, B.C.
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For two entire functions f(z) and g(z) the composition f(g(z)) may or may not be periodic even though g(z) is not periodic. For example, when f(u) = cos √u and g(z) = z2, or f(u) = eu and g(z) = p(z) + z, where p(z) is a periodic function of period 2πi, f(g(z)) will be periodic. On the other hand, for any polynomial Q(u) and any non-periodic entire function f(z) the composition Q(f(z)) is never periodic (2).

The general problem of finding necessary and sufficient conditions for f(g(z)) to be periodic is a difficult one and we have not succeeded in solving it. However, we have found some interesting related results, which we present in this paper.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 724 - 730
Copyright
Copyright © Canadian Mathematical Society 1966

References

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3. Hayman, W. K., Symmetrization in the theory of functions, Stanford University Tech. Rep. No. 11 (1950), 21.Google Scholar
4. Polya, G., On an integral function of an integral function, J. London Math. Soc, 1 (1926), 12.Google Scholar
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