I have the question: "Find the Laurent series which represents the function
$$ f(z) = (z^2 - 1)/(z + 2)(z + 3)\ $$
in the regions
(i) $\mid z\mid < 2\ $
(ii) $ 2 < \mid z\mid < 3\ $
(iii) $\mid z\mid > 3\ $"
I can rewrite this function into partial fractions as:
$$ f(z) = 5/(z+2) - 10/(z+3) $$
I understand that for part (ii) this is the region within the annulus.
I also know that I can write:
$$ 5/(z+2) = 5/2 \sum (-1)^n (z/2)^n $$ for $ \mid z\mid < 2\ $
$$ 10/(z+3) = 10/3 \sum (-1)^n (z/3)^n $$ for $ \mid z\mid < 3\ $
I also know how to find the respective sums for $ \mid z\mid > 2\ $ and $ \mid z\mid > 3\ $ But I'm not at all sure how to combine these to get the answers for the whole function $f$, for parts (i), (ii) and (iii).
Any help would be much appreciated