Thread: 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

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?

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?

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