I would like to find the residue of $f(z)=\frac{e^{iz}}{z\,(z^2+1)^2}$ at $z=i$. One way to do it is simply to take the derivative of $\frac{e^{iz}}{z\,(z^2+1)^2}$. Another is to find the Laurent expansion of the function.
I managed to do it using the first way, and the answer is $-3/(4e)$. However, I'm out of ideas as to how to find the expansion.
Any help is greatly appreciated.