Could someone advise me how to use Liouville's Theorem to show that there is at least one value z0 in complex plain for which:
|cos (zo/2009)| > 2009
Thank you.
Consider the function $\displaystyle f(z) = \cos \tfrac{z}{2009}$. This is an entire function. If $\displaystyle |f|\leq 2009$ on the complex plane then $\displaystyle f$ would be a constant function which it is not (by Louiville's theorem). Therefore, there has to exists at least one point $\displaystyle z_0\in \mathbb{C}$ so that $\displaystyle |f(z_0)| > 2009$.