How Can I Show That the Function is Bounded?

[MidnightR]

Hi,

Suppose f is an entire function such that f(z) = f(z+2pi) = f(z+2(pi)i) for all z E C.

Use Liouville's theorem to show that f is constant.

Obviously I need to show that the function is bounded but I'm unsure of how to approach it.

The hint is: Consider the restriction of f to the square S = {z = x + iy : 0 <= x <= 2Pi, 0<= y <= 2Pi}

Any help/hints appreciated to get me started, thanks

 

[micromass]

HINT: the set S is closed and bounded...

 

[MidnightR]

Yea I realize that but f doesn't have to be restricted does it? Unless that's what they mean by the hint, in which case there's nothing to say?

 

[micromass]

Well, it suffices to show that f is bounded on S. Since values not in S, can be brought back to a value in S.

It's like checking that sin(x) is bounded. It suffices to do that on [0,2pi], since any other value can be brought back to a value on [0,2pi].