Proving Analytic Functions are Constant: Liouville's Theorem

[Pyroadept]

Homework Statement

Q. (a) State Liouville's Theorem

(b) Suppose that f is analytic in C and satisfies f(z + m + in) = f(z) for all integers m,n . Prove f is constant.

Homework Equations

The Attempt at a Solution

(a) Liouville's Theorem - If f is bounded and analytic in C, then f is constant.

(b) I'm guessing I need to use Liouville's Theorem here i.e. need to show the f is bounded. But I'm so confused! The question states that f(z + m + in) = f(z), so f gives the same value regardless of the z. Doesn't this mean that f is constant?!

Could someone please point me in the right direction? Thanks for any help :)

 

[vela]

It doesn't quite say f(z) is constant. It tells you, for example, that

f(0) = f(1) = f(2) = f(3) = ...

and

f(1/2) = f(3/2) = f(5/2) = ...

but it doesn't directly tell you anything about how f(0) and f(1/2) are related.