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 :(

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?

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?

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 $\displaystyle \mathbb{C}$