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 October 16th, 2012, 10:19 PM #1 Newbie   Joined: Aug 2012 Posts: 7 Thanks: 0 Find all entire functions f Find all the entire functions f such that $f(0)= i$ and $\left |f(z)- \cos z \right |\geq{2}$ for all ${z\in{\mathbb{C}}}$. My attempt at this question... Let $g(z)= \frac{1}{f(z)-\cos z}$. Since $f(z)$ and $\cos z$ are entire and $f(z) - \cos z \neq 0, \forall {z\in{\mathbb{C}}}$ (since $\left |f(z)- cos z \right |\geq{2}$). $\Rightarrow g(z)$ is entire & $\left |g(z) \right |= \frac{1}{\left |f(z)-\cos z\right |} \leq \frac{1}{\sqrt{2}}, \forall {z\in{\mathbb{C}}}$. By Liouville's Theorem, $g(z) \equiv k$ for some ${k\in{\mathbb{C}}$. $\Rightarrow \frac{1}{f(z)-\cos z} \equiv k$ for some ${k\in{\mathbb{C}}, k \neq 0$, since $f(z) - \cos z \neq 0, \forall {z\in{\mathbb{C}}}$. Hence $f(z)= \frac{1}{k} + \cos z$ Now it is given that $f(0)= i$, and $cos(0)= 1$, hence $\frac{1}{k}= i-1$. $\Rightarrow f(z)= i - 1 + \cos z$ Is this correct? Are there any other solutions? Any improvement/suggestion is welcomed! Thanks in advance!.
 October 17th, 2012, 12:49 AM #2 Newbie   Joined: Aug 2012 Posts: 7 Thanks: 0 Re: Find all entire functions f Sorry, for the question it should be $\left |f(z) - \cos z\right | \geq \sqrt{2}$ instead of $\left |f(z) - \cos z\right | \geq 2$

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