Laurent Expansions about $z_0=0$

Question

I'm really struggling to understand the concept of Laurent Expanaions.. I have gone throu lectures notes and a couple of examples, but don't seem to be able to compute fully understand..

Any help with the following example would be greatly appreciated...

$f(z)= \dfrac{1}{z(z-1)(z-2)} $

On

(i) $0< |z| < 1$

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

(iii) $ 2<|z|$

Workings

I have worked out the partial fractions for this equation as follows....

$\dfrac{1}{z(z-1)(z-2)} = \dfrac{A}{z} + \dfrac{B}{z-1} + \dfrac{C}{z-2}$

$\dfrac{1}{z(z-1)(z-2)}= \dfrac{1}{z} -\dfrac{1}{z-1}+ \dfrac{1}{2(z-2)}$

But now I'm lost as to what to do..

Answer 1

Hint:
Expand separately $\dfrac{1}{z-1}$ and $\dfrac{1}{z-2}$ using the geometric series and taking into account the domain of convergence.