# Math Help - laurent series and residues

1. ## laurent series and residues

Hi
Any help would be greatful!

The question is: Find and classify all singularities of the following functions and where appropriate, find the residues?

(c) exp(z)/(z-1)^2
the singularity is at z=1 how do i calculate the residue via laurent series.

(d) (z-i)^(-3/2)
the singularity is at z=i how do i calculate the residue via laurent series.

Thanks

2. Originally Posted by jules
(c) exp(z)/(z-1)^2
the singularity is at z=1 how do i calculate the residue via laurent series.
Why use Laurent here? Notice that $z=1$ is a double pole. To find the residue define $f(z) = (z-1)^2 \cdot \frac{e^z}{(z-1)^2} = e^z$ and compute $\frac{1}{(2-1)!}\cdot f'(1) = e$.

(d) (z-i)^(-3/2)
the singularity is at z=i how do i calculate the residue via laurent series.
Do it with derivatives like I did above. What what of pole is this? Meaning what is smallest $n$ so that $\lim_{z\to i} (z-i)^n (z-i)^{-3/2}$ exists?