Chapter 14.4, Problem 7P

For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin. (Warning: To find the residue, you must use the Laurent series which converges near the origin.) Hints: See Problem 2. Use partial fractions as in equations ( 4.5 ) and ( 4.7 ). Expand a term $1/(z-a)$ in powers of $z$ to get a series convergent for $\left|z\right|<a$, and in powers of $1/z$ to get a series convergent for $\left|z\right|>a$.

$\frac{2-z}{1-z^2}$