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Entire Solutions Of The Functional Equation f(f(z)) = g(z)

  • W. J. Thron (a1)

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In this note it is proved that: the functional equation

(1) f(f(z)) = g(z),

where g{z) is an entire function of finite order, which is not a polynomial, and which takes on a certain value p only a finite number of times, does not have a solution f(z) which is an entire function.

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References

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1. Hadamard, J., Two works on iteration and related questions, Bull. Amer. Math. Soc, 50 (1944), 6775.
2. Isaacs, R., Iterates of fractional order, Can. J. Math., 2 (1950), 409416.
3. Kneser, H., Reelle analytische Lösungen der Gleichung und verwandter Funktionalgleichungen, J. reine angew. Math., 187 (1950), 5667.
4. Pólya, G., On an integral function of an integral function, J. London Math. Soc, 1 (1926), 1215.
5. Titchmarsh, E. C., The Theory of Functions (2nd ed., Oxford, 1939).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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