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# Showing a function is bounded. Watch

1. 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}

Cheers
2. (Original post by 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}

Cheers
Not too sure on this, but my initial idea would be to expand f in terms of a taylor series or something along these lines
3. (Original post by 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}

Cheers
Seriously, use the hint. Note that S is compact. What do you know about the image of a compact set?

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