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.

2. Originally Posted by grahamlee
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 $f(z) = \cos \tfrac{z}{2009}$. This is an entire function. If $|f|\leq 2009$ on the complex plane then $f$ would be a constant function which it is not (by Louiville's theorem). Therefore, there has to exists at least one point $z_0\in \mathbb{C}$ so that $|f(z_0)| > 2009$.

3. thank you so much