Find the Laurent series about $z_0=0$ for the following function, valid in the indicated regions.

Question

$$\frac{1}{z-1} - \frac{1}{z+1}, \qquad 2<|z|<\infty$$

I am not sure how to approach this question. Can anyone help me with this question? Thank you.

Answer 1

Assuming you meant for $|z| > 2$ (and so $\frac{1}{|z|} < \frac{1}{2} < 1$):

$\frac{1}{z+1} = \frac{1}{z} \frac{1}{1+\frac{1}{z}} = \frac{1}{z} (1-\frac{1}{z} + \frac{1}{z} - \cdots)$.

Similarly for the other term.