0
$\begingroup$

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

$\endgroup$
1
  • 2
    $\begingroup$ It does have other poles elsewhere. At $\ z=n\pi\ $, to be precise, for any $\ n\in\mathbb{Z}\ $. $\endgroup$ – lonza leggiera Jan 17 '20 at 16:53
0
$\begingroup$

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 $\pi$.

$\endgroup$

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.