# Laurent expansion for a complex function with 3 singularites

1. Nov 28, 2013

1. The problem statement, all variables and given/known data
Hey guys,

So I need a bit of help with this question:

Find three Laurent expansions around the origin, valid in three regions you should specify, for the function
$f(z)=\frac{30}{(1+z)(z-2)(3+z)}$

2. Relevant equations
None that I know of...just binomial expansion

3. The attempt at a solution

Okay so what I did was first specify the regions. Not sure if they are right though, although I think they are:

Region 1: $-1<|z|<2$
Region 2: $-3<|z|<-1$
Region 3: $|z|>2$

Then I split f(z) into partial fractions:
$f(z)=\frac{2}{z-2}+\frac{3}{3+z}-\frac{5}{1+z}$

Then I expanded for the region |z|>2, using the first term of the partial fractions (ignoring the other ones...right?) and I got

$f(z)_{z>2}=\Sigma_{n=1}^{\infty}(\frac{2}{z})^{n}$

So now the problem is... first of all I dont know if that's right. Even if it is, I have no idea how to expand for the other regions...for example, say I wanted to do region 2...I dont even know where to start, do I first expand for -3<|z|, then |z|<-1 and add them...or what?

Really need some help here guys! the fate of the universe hinges on this unfortunate question sheet!

2. Nov 29, 2013

### clamtrox

Let's start with the most obvious one: the regular Taylor series. Can you calculate it, and for what values of z does it converge?