# Math Help - Residue Theorem

1. ## Residue Theorem

Let $f(z) = \frac{e^{i z}}{(1+z^2)^2}$

Find the singularities and classify them. Find the residue when z = i.

Singularities found to be i and -i, both poles of order 2. So i figure ill need the taylor expansion at some point.
Is there a set way to calculate residues? The question i did before this involved differentiating the denominator but that wont work here since there will still be a zero on the bottom line. So how to i find residues if they are of order 2?

2. Actually i may have solved it, although its not at all like the working in the answer, though its does get the right one.

$Res(f, i) = lim_{z \rightarrow i} \frac{d}{dz} \frac{(z-i)^2(e^{iz})}{(z-i)^2(z+i)^2}$. Then simplifying and differentiating the bottom line gives $\lim_{z \rightarrow i} \frac{e^{iz}}{2(z + i)} = \frac{e^{-1}}{2i} = \frac{-i}{2e}$.

Which is the correct answer but not at all like the working given in the solutions...