# what are the poles of $\cot z$?

I have this in my textbook.

The function $$f_{2}(z)=\cot (z)=\frac{\cos z}{\sin z}$$ has a simple pole at $$z=0 .$$ The residue is $$\operatorname{Res}\left(f_{2}(z), z=0\right)=\lim _{z \rightarrow 0} \frac{z \cos z}{\sin z}=1$$

But why $$\cot z$$ has a simple pole at $$0$$? and no other poles elsewhere? thanks

• It does have other poles elsewhere. At $\ z=n\pi\$, to be precise, for any $\ n\in\mathbb{Z}\$. – lonza leggiera Jan 17 '20 at 16:53

It has a simple pole at $$0$$ because $$\lim_{z\to0}z\cot z$$ exists (in $$\mathbb C$$) and it is different from $$0$$. But it also has a simple pole at every multiple integer of $$\pi$$.