• Mar 14th 2009, 09:48 AM
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.
• Mar 14th 2009, 09:53 AM
ThePerfectHacker
Quote:

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 $|f(z_0)| > 2009$.
• Mar 14th 2009, 10:10 AM
grahamlee
thank you so much