Laurent Series for $\csc(z)$

Question

I have to find the Laurent series for $$\csc(z), \qquad |z|>0 $$ but I really don't know how to start.

Please, guys.

Answer 1

Hint. One may recall the Bernoulli Numbers, defined by the series, $$ \frac{z}{e^z-1}:=\sum\limits_{n=0}^{\infty}B_n\frac{z^n}{n!},\tag1$$ with $B_0 = 1$, $B_1 = -1/2$, $B_2 = 1/6$, $B_3 = -1/30$, and so on.

Then one may observe that

$$ \csc(z)=\frac1{\sin z}=\frac{e^{iz}}{z}\frac{2iz}{e^{2iz}-1}, \qquad 0<|z|<\pi,\tag2 $$

and conclude using $(1)$.