Calculating residues of multiple poles

Question

I have $f(z)=\dfrac{z^2e^{iz}}{(z^2+1)^2}$, and I need its residue at $z=i$, which is a double pole. I tried expanding $f(w+i)$, where I got to $f(w+i)=w^{-2}\dfrac{(w^2+2iw-1)e^{iw-1}}{(w+2i)^2}$ and going further was a pain, and each time somehow gave me 0 as the residue, which isn't right. I don't really know of any other ways of finding residues at a multiple pole!

Answer 1

You have to calculate first the first derivative of $\;(z-i)^2f(z)\;$:

$$(z-i)^2f(z)=\frac{z^2e^{iz}}{(z+i)^2}\implies \frac d{dz}\left((z-i)^2f(z)\right)=\frac{(2z+iz^2)e^{iz}(z+i)-2z^2e^{iz}}{(z+i)^3}\implies$$

$$\lim_{z\to i}\frac d{dz}\left((z-i)^2f(z)\right)=\frac{-2e^{-1}+2e^{-1}}{2i}=0$$