# holomorphy domain

Let $$f_n=\frac{\log\left(z-\frac{1}{n}\right)}{\log^\frac{3}{2}(n)\left(z^2+n\right)}$$ Where $$\log(z)$$ is the natural branch of the function logarithm.
Find the holomorphy domain of $$f(z)=\sum\limits_{n=2}^{\infty} f_n$$.
I tried to use the fact that a series of holomorphic functions witch converge nomally to a function $$f$$ implies the holomorphy of $$f$$ but i can't find a majoration.
I also would be happy to recive other approaches to this kind of exercises.

Hint: For each $$n$$ determine where $$f_n$$ is well defined. This will be a set of the form $$\mathbb C\setminus E_n.$$ The set where $$\sum f_n$$ makes sense can be no larger than $$\mathbb C\setminus (\cup \,E_n).$$
• Thank you, $E_n$ should be {$z\in \mathbb{C}:z\neq i\sqrt{n},$ $z\neq-i\sqrt{n}$ } $\cup$ {$z\in \mathbb{C}: Re(z)\leq \frac{1}{n}$ $and$ $Img(z)=0$}. Jan 7 '19 at 22:10
• I would write $E_n = \{i\sqrt n,-i\sqrt n\}\cup (-\infty,1/n].$