Using Liouville's Theorem to Show f is Constant

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An entire function f that satisfies f(z) = f(z + 2π) and f(z) = f(z + 2πi) for all z in C is periodic with respect to both the real and imaginary axes. This periodicity implies that f is completely determined by its values within a square of side length 2π. To apply Liouville's theorem, it must be shown that f is bounded within this square. Since f is periodic and entire, it maps the square onto its range of values, indicating that f must be constant. Therefore, by Liouville's theorem, f is indeed constant.
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Suppose f is an entire function such that f(z) = f(z+2\pi)
and f(z)=f(z+2\pi i) for all z \epsilon C. How can you use Liouville's theorem to show f is constant..

any help on that please to get me started off.. thnx a lot :)
 
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The two given relations tell you that f is completely determined by its values in a square of side length 2\pi... what do you need to show about f to use Liouville? Can you get it from this info now?
 
not just completely determined, but actually that it maps that square onto its range of values.
 

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