# difficult question on complex analysis

• Nov 13th 2006, 02:58 PM
edgar davids
difficult question on complex analysis
Can some one help me on this difficult q. Spent ages trying and cant get it, thanks in advance

Suppose that f is an entire function such that f(z) = f(z + 2pie) and
f(z) = f(z +2ipie) for all z ∈ C. Use Liouville’s theorem to show that f is
constant.
Hint: Consider the restriction of f to suitable squares.
• Nov 13th 2006, 04:38 PM
ThePerfectHacker
Quote:

Originally Posted by edgar davids
Can some one help me on this difficult q. Spent ages trying and cant get it, thanks in advance

Suppose that f is an entire function such that f(z) = f(z + 2pie) and
f(z) = f(z +2ipie) for all z ∈ C. Use Liouville’s theorem to show that f is
constant.
Hint: Consider the restriction of f to suitable squares.

Here is a try (if that is not what you need tell me an I will delete it).

On the disk centered at the origin having radius of $\displaystyle 2 \pi$ the function is bounded for it is homolorphic, and every homolorphic function is continous, and every continous function is bounded. Then everywhere else this function is still bounded because of the property $\displaystyle f(z+2i\pi)=f(z)$. Therefore it is constant