Need help with finding residue of a simple function

[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 + ...)
But now I'm stuck ! please help :)

 

[Mute]

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!)

 

[joris_pixie]

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

Thank you ! :)