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Here are my problems:

  1. Find an entire function $f(z)$ such that $|f(z)| < e^{\operatorname{Im} f(z)}$ for all $z \in \mathbb{C}$ and $f(0)=2$. I am trying to guess $2\cos(z), 2e^z$, etc, but they are all failed in the end.

  2. Let $f(z)$ be analytic in the punctured disk $0<|z|<1$,and $f(1/n)=\sqrt{n}$ for every integer $n>1$. Prove that $f(z)$ has an essential singularity at $z=0$. I am trying to use the definition of essential singularity to prove. But I don't know how to express the $f(z)$

I am sorry that I wrote wrong condition for problem #1. The condition should be $|f(z)| > e^{\operatorname{Im} f(z)}$

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@user9706: it is more typical on this website to ask separate questions in separate questions, unless the several parts of your questions are very closely related. I would suggest you keep this one question about the first of your two, edit the title to be more descriptive, and ask the second question in a separate thread. Thanks. –  Willie Wong Apr 18 '11 at 20:02
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I don't think @Willie intended to say you should change this question after someone has answered, rather that you should ask separate questions separately, next time. With the new version, @user9325's answer didn't make much sense anymore, so I've rolled back to the last version. –  t.b. Apr 18 '11 at 21:13
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@user9706: right, sorry about that. When I made that comment there were no answers posted. Please follow Theo's suggestion now. –  Willie Wong Apr 18 '11 at 21:34
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2 Answers

  1. You ask a lot: $2=|f(0)|< e^{Im f(0)}=e^0=1$. (For the new version: Have you tried $e^z+e^{-z}$?)

  2. It would be a better idea to show that the singularity is neither removable nor a pole and then use the classification of singularities.

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2) if there were a pole at zero $$ f(z)=\sum_{k=-N}^{\infty}a_kz^k, N>0 $$ then $f(1/n)\sim a_{-N}n^N$ but the assumption is $f(1/n)\sim n^{1/2}$

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