Finding Laurent Series of for the annulus $1<|z|<3$

Question

I have the following function and I want to find the Laurent Series. We already have the answer but I don't know/understand how I can get there :

Let

$$f(z) = \frac{1}{1-z^2}+\frac{1}{3-z}$$

For the annulus : $1<|z|<3$

Any help would be a lot appreciated.

Answer 1

Just pull out $-1/{z^{2}}$ from the first term and $\frac 1 3$ from the second.

$f(z)=-\frac 1{z^{2}} \sum\limits_{k=0}^{\infty}(\frac 1 {z^{2}})^{k}+\frac 1 3 \sum\limits_{k=0}^{\infty}(\frac z 3)^{k} $.