Finding residue of a function(complex analysis)

[gametheory]

Homework Statement

I need to find the residue of $\frac{e^{iz}}{(1+9z^{2})^{2}}$ so I can use it as part of the residue theorem for a problem.

Homework Equations

Laurent Series
R(z$_{0}$) = $\frac{g(z_{0})}{h^{'}(z_{0})}$

The Attempt at a Solution

I tried using the laurent series but after expanding I got a 0 in the denominator.
For the second equation I used I also got a 0 in the denominator and I don't believe it converges. Any help would be appreciated. Thanks!

 

[Gregg]

Look at the inverse of the function (call it f(z)), you have a zero of order 2. So you have a pole of order 2 for your function. The formula for this goes like $\text{Res}(f,i/3) = \lim_{z \to a}\frac{d}{dz}\left( (z-i/3)^2 \frac{\exp (i z)}{(1+9z^2)^2} \right)$

you can do the others

$\text{Res}(f,a) = \lim_{z \to a} \frac{1}{(m-1)!}\frac{d^{m-1}}{dz^{m-1}} (z-a)^m f(z)$

If the pole is of order m. I think. Maybe look that up.