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# How to show that an entire function is a constant function using Liouville's Theorem watch

1. we have:
f is entire such that f(z) = f(z + 2Pi) and f(z) = f(z+2*i*Pi) for all z in the complex plane.
need to show that f is constant.

By Liouville's them, we only need to show f is bounded.

there's a hint which says to restrict f to the square S = {z=x+iy: 0=<x=<2Pi, 0=<y=<2pi}

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my working so far:

to show f is bounded maybe look at x=0, y=0, x=2Pi, y=2Pi? i'm not sure what this would yield though
2. yyug
3. what do you know about continuous functions on a compact set.
4. (Original post by around)
what do you know about continuous functions on a compact set.
they're bounded? thats it? then liouville's follows?
5. (Original post by anzerftum)
they're bounded? thats it? then liouville's follows?
Pretty much.

This is why all elliptic functions are meromorphic (the question is not utterly pointless).

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Updated: November 29, 2011
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