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Can anybody help me with this question:

If $f(z)$ is an entire periodic function and it has to periods $2$ and $2i$, how can I find all other periods?

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If the function is entire, and has periods $2$ and $2i$, then it is bounded, and by Liouville, it is then a constant, so that any non-zero complex number is a period.

The above is well-known result in the early stages of the theory of elliptic functions. See, for example, Apostol's book "Modular Functions and Dirichlet Series in Number Theory" on page 5.

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Umm ... not Halmos ... – GEdgar Dec 17 '12 at 14:46
Absolutely right - it is Apostol, sorry. – Old John Dec 17 '12 at 15:18

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