Need help with finding residue of a simple function

1. Oct 20, 2009

joris_pixie

Need help with finding residue of a "simple" function

Hello,
I'm trying to find the residue z=0 of f(z) = (1 + z)e^(3/z)

I understand this is a essential singularity. I know the answer is 15/2 but I can't seem to find the solution.

I've tried this so far:

f(z) = (1 + z) ( (3/z) + 9/2z² + ...)

2. Oct 20, 2009

Mute

Re: Need help with finding residue of a "simple" function

Do you know what the residue is? You've expanded the exponential in a series - why? It's the correct thing to do, but do you know why?

It's because the residue of a function at $z = 0$ is given by the coefficient of $1/z$ in its Laurent series expansion. So, what's the coefficient of 1/z in your series expansion? (You have to multiply your two series together!)

3. Oct 20, 2009

joris_pixie

Re: Need help with finding residue of a "simple" function

Aha !
I'm going to study the Laurent series expansion part again... ;)

Thank you ! :)