I am given the following function: $R(z) = (z^2-9)/(z^2+9)^2 $
I need to let $R = P/Q$ be a rational function with $deg P < deg Q$. I will let $ξ$ be a pole of $R$ and the coefficient of $1/(z-ξ)$ in the partial fraction expansion of $R$ be the residue of $R(z)$ at $ξ$ (denoted $Res(ξ)$).
I need to find the decomposition of the above function $R(z)$ and then find the corresponding residues.
I don't know where to start. The reference books I have do not cover this section well at all.
you can write denominator as follows:$$\frac{z^2-9}{(z+3i)^2(z-3i)^2}$$
z=$-3i,+3i$ come to be two poles of order 2
$Res(3i)=\lim_{z\to 3i}\;\frac{1}{2}\frac{d}{dz}\frac{z^2-9}{(z+3i)^2}$
similarly residue at -3i can be found also