0
$\begingroup$

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

|cite|improve this question
$\endgroup$
  • $\begingroup$ For the first, how about $e^{\Gamma(z)}$? $\endgroup$ – ajotatxe Jul 12 '15 at 9:15
  • $\begingroup$ i think you need Mittag-Lefler theorem $\endgroup$ – Adelafif Jul 12 '15 at 9:26
1
$\begingroup$

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.)

|cite|improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.