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

• November 29th 2011, 12:43 PM
cassius
How to show that an entire function is a constant function using Liouville's Theorem?
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 :(
• November 29th 2011, 12:54 PM
TheEmptySet
Re: How to show that an entire function is a constant function using Liouville's Theo
Quote:

Originally Posted by cassius
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}

------------------------
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 :(

Hint: The unit square is a compact subset of the complex plane. What can we say about analytic functions on compact sets?
• November 29th 2011, 02:00 PM
cassius
Re: How to show that an entire function is a constant function using Liouville's Theo
Quote:

Originally Posted by TheEmptySet
Hint: The unit square is a compact subset of the complex plane. What can we say about analytic functions on compact sets?

...they are uniformly bounded? so liouville's follows?
• November 29th 2011, 02:08 PM
TheEmptySet
Re: How to show that an entire function is a constant function using Liouville's Theo
Quote:

Originally Posted by cassius
...they are uniformly bounded? so liouville's follows?

Yes and since the function is peroidic is must be bounded on all of $\mathbb{C}$