Here are my problems:
- 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.
- Let $f(z)$ be analytic in the punctured disk $0<|z|<1$, and $f(\frac 1n)=\sqrt{n}$ 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 $f(z).$