Originally Posted by

**punkstart** Get the Laurent series expansion or f(z) = $\displaystyle \frac{1}{z+1} - \frac{4}{(z-2)^2}$ in the intervals;

(i) |z-1|<1 and

(ii) 1<|z-1|< 2

Now I don't have much trouble finding the Laurent series expansions but am not sure how to do it for specific intervals.

For example in (ii) of this problem i would get the Laurent expansions for $\displaystyle \frac{1}{z+1}$

for interval 1<|z-1|<2

and then get the expansions for $\displaystyle - \frac{4}{(z-2)^2}$

for 1<|z-1|<2,and add them all together. The textbook(correctly, just not sure why yet) just uses the expansion of $\displaystyle \frac{1}{z+1}$ for |z-1|<2 and adds to it the expansion of

$\displaystyle - \frac{4}{(z-2)^2}$ for 1<|z-1|. I'm not sure as to the procedure or the thinking behind getting Laurent expansions for these types of intervals

r1<|z-zo|< r2. Any help is much appreciated.