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.

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- Mar 14th 2009, 08:48 AMgrahamleeLiouville's Theorem - advice needed please.
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, 08:53 AMThePerfectHacker
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 $\displaystyle |f(z_0)| > 2009$.

- Mar 14th 2009, 09:10 AMgrahamlee
thank you so much