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Fix nonzero ω1,ω2 ∈R. Suppose that f is an entire function which satisfies $$f(z + ω1) = f(z + iω2) = f(z)$$ for all $z ∈\Bbb C$. Prove that f must be constant.

My first and immediate thought upon seeing this problem is to use Liouville's theorem somehow, because if I can prove that the function's $n$th derivative is entire and bounded for some fixed positive integer $c ∈\Bbb C$, then the function is constant throughout the plane.

So can I say, by the Cauchy Integral formula, that since we know that f is entire, all of its derivatives are entire? Thus, by the conditions of Liouville, everything follows from there?

user340423
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  • If $f$ is entire then all its derivatives are entire as well. But you don't need that here, Liouville can be applied to $f$ directly. – Martin R May 18 '16 at 14:57
  • @MartinR I see. So the f(z + ω1) = f(z + iω2) would be extraneous information because the problem already tells us that f is entire? – user340423 May 18 '16 at 15:05
  • Sorry, I don't get what you mean. $f$ is given as an entire function. $f(z + ω1) = f(z + iω2) = f(z)$ for all $z$ tells us that $f$ is doubly-periodic. It follows that $f$ is bounded and therefore constant. – Martin R May 18 '16 at 18:04

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