- #1
Dixanadu
- 254
- 2
Homework Statement
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
[itex]f(z)=\frac{30}{(1+z)(z-2)(3+z)}[/itex]
Homework Equations
None that I know of...just binomial expansion
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: [itex]-1<|z|<2[/itex]
Region 2: [itex]-3<|z|<-1[/itex]
Region 3: [itex]|z|>2[/itex]
Then I split f(z) into partial fractions:
[itex]f(z)=\frac{2}{z-2}+\frac{3}{3+z}-\frac{5}{1+z}[/itex]
Then I expanded for the region |z|>2, using the first term of the partial fractions (ignoring the other ones...right?) and I got
[itex]f(z)_{z>2}=\Sigma_{n=1}^{\infty}(\frac{2}{z})^{n}[/itex]
So now the problem is... first of all I don't 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 don't 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!