Infinitely (countably) many essential singularities

Question

Let $f$ be a holomorphic function on $\mathbb{C}\setminus A$, where $A$ is a set of isolated singularities. I know it is possible, that $A$ contains infinite (countable) number of poles, but:

1) Is it possible that $A$ contains infinite (countable) number of essential singularities?

2) Is it possible to construct a function that has exactly one pole of every order $n\in\mathbb{N}$?

3) If both $f$ and $g$ have essential singularities at some $z\in \mathbb{C}$, is it possible that $f+g$ does not have essential singularity there? (I think it's easy, just define $g\equiv-f$)

4) Does identity theorem hold for meromorphic functions? What is counterexample if not?

Thanks for help

Answer 1

1) Consider

$$\sum_{n=1}^{\infty}\frac{e^{1/(z-n)}}{2^n}.$$

This sum converges uniformly on each $U_\delta = \{z: d(z,\mathbb {N}>\delta\},$ hence defines a holomorphic function $f$ on $\mathbb {C}\setminus \mathbb {N}.$ This $f$ has an essential singularity at each point of $\mathbb {N}.$

2) Take a look at

$$\sum_{n=1}^{\infty} \frac{1}{n!(z-n)^n}.$$

3) You answered your own question.

4) How could it not hold? (Perhaps I don't understand the question.)