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On Factorization of Elliptic Functions

Published online by Cambridge University Press:  20 November 2018

Fred Gross
Fred Gross*
Affiliation:
Bellcomm, Inc., Washington, D.C.
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In this paper we shall be concerned with the following problem: If h is an elliptic function and h(z) = ƒ(g(z)), what can be said about the functions ƒ and g? In order to simplify the discussion we introduce some basic definitions.

Definition 1. A meromorphic function h(z) = ƒ(g(z)) is said to have ƒ(z) and g(z) as left and right factors, respectively, provided that either ƒ(z) is non-linear and meromorphic and g(z) is non-linear and entire or f(z) is rational and g(z) is meromorphic.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 486 - 494
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

The author is indebted to the referee for useful remarks including the statement and proof of Lemma 2.

References

1. Edrei, A. and Fuchs, W. H., On the zeros off(g(z)) where f and g are entire functions, J. Analyse Math., 12 (1964).Google Scholar
2. Gross, F., On the periodicity of compositions of entire functions, Can. J. Math., 18 (1966), 724–730.Google Scholar
3. Gross, F., On factorization of meromorphic functions, Trans. Amer. Math. Soc. (to appear).Google Scholar
4. Gross, F., Handbook of mathematical functions, National Bureau of Standards, (Applied Math. Series 1964), p. 645.Google Scholar
5. Hayman, W. K., Meromorphic functions, Oxford Mathematical Monographs (Oxford, 1964), Chap. 2.Google Scholar
6. Hille, E., Analytic function theory (Ginn and Company, 1962), Vol. II, Chaps. 13, 14.Google Scholar
7. Neville, E. H., Jacobian elliptic functions (Oxford, 1964), p. 76.Google Scholar