I've got an exam tomorrow so any quick responses would be appreciated. I'm following the Boas section on Laurent series... Anyway, here's my problem:
In an example Boas starts with f(z) = 12/(z(2-z)(1+z), and then using partial fractions arrives at f(z) = (4/z)(1/(1+z) + (1/2-z)). Fine. So there are three singular points, at z = 0, z = 2, and z = -1. So, we have two circles about z = 0 and should be able to obtain three Laurent series, one valid for 0 < |z| < 1, 1 < |z| < 2, and |z| > 2. I'll skip the details of the rest of this example but she expands the partial fraction in terms of z to obtain f(z) for 0 < |z| < 1, then proceeds to expand in terms of 1/z to obtain f(z) for |z| > 2 and then one of the partial fractions in terms of z and the other one in terms of 1/z to obtain f(z) for 1 < |z| < 2.
I'm a bit confused as to why z, 1/z, and then a combination and also how you know which will correspond to which solution for f(z).
As I said, the exam is tomorrow so any quick responses would be helpful.