- #1
Hertz
- 180
- 8
Homework Statement
Specifically, I'm trying to find the laurent series for [itex]f(z)=\frac{z^2}{z+1}[/itex] around the point [itex]z=-1[/itex]. My real problem is my procedure in general though. I'm not sure what I'm doing wrong on a lot of these Laurent Series but for some reason I'm struggling with them.
(Even more specifically, I'm trying to find the type of singularity and the residue at z=-1.)
Homework Equations
[itex]\sum^{\infty}_{n=-\infty}{c_n (z-z_0)^n}[/itex] where
[itex]c_n=\frac{1}{2\pi i}\int_{C}{\frac{f(z)dz}{(z-z_0)^{n+1}}}[/itex]
The Attempt at a Solution
What I did is take the [itex]z^2[/itex] out and set [itex]z_0=-1[/itex] because that's the point I want to expand around. Then I set C such that [itex]z=-1+e^{i\theta}[/itex] where [itex]-\pi < \theta < \pi[/itex] and integrated.
I found the integral, which was [itex]\frac{sin(n+1)\pi}{(n+1)\pi}[/itex] which equals zero for all values of n besides [itex]n=-1[/itex].
I could probably find out more about the coefficient at -1 if I evaluated the integral for n=-1, but at this point I realized I still had the [itex]x^2[/itex] in there that would throw off the degree of my Laurent Series anyways...
So I basically feel like I have spent way more time on this problem than I should have and have almost no results to show for it. Clearly, my procedure is not spot on :\. Can anybody help me out? How should I start out on a problem like this?
Normally, when I try to find a Laurent Series, this is what I do:
1. Decide where it should be centered.
2. Think about other series representations that I've memorized to maybe do a quick easy substitution or break up the function into multiple parts.
3. If I have no success this far, I'll usually just result to the laurent series formula that I have posted above, but clearly, I'm running into problems with it.