Principle part of Laurent series

[wam_mi]

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

Find the Principle part of the Laurent Expansion of f(z) about z=0 in the region
0 < mod z < 1, where f(z) = exp(z) / [(z^2)*(z+1)]

2. Relevant equations

1/(1-z) = Summation (n = 0 to n = infinity) { z^n}


3. The attempt at a solution

First, by using partial fraction,
I got f(z) = exp(z) {-1/z + 1/(z^2) + 1/(z+1)}

Then f(z) = exp (z) {1/ (-1+1+z) + 1/ (-1+1+(z^2)) + (1/(z+1) }

Since the question were only after the principle parts, so I ignore 1/(z+1) term

Basically I need to evalute
exp (z) { 1/ (-1+1+z) + 1/ (-1+1+(z^2)) }

Is this step right?


Then I tried to do the following,

and I got something like

exp (z) { - summation (1/(1+z))^(n+1) + summation (1/(1+z^2))^(n+1)}


But is this right?

Thanks a lot!

[Dick]

You are making this harder than it is. Your function is 1/z^2 times e^z/(z+1). e^z/(z+1) is nonsingular at z=0. All you need is the first two terms in the series expansion of e^z/(z+1) around z=0. (Why only the first two?). Then multiply that series by 1/z^2.