finding residue for complex analysis

Question

I am having a tough time finding the residue for a function, suppose my test function is

$$\frac{z^2}{{(z^2+a^2)}^2}$$

while I could determine the poles to be $+-ai$ and I know the formula to find the residue to be

$$a_{-1}= \frac{1}{2\pi i} \int f(z) dz$$

but I am confused, if I have to integrate the aforementioned function now and divide it by $2\pi i$ to find the residue or what. nay hint or suggestion would be appreciated .

You need to integrate $f(z)$ along a contour surrounding one of the poles. — Henricus V., Dec 06 '15 at 03:25
can you give me a little hint on how to do contour integrals? — Xfrglkv, Dec 06 '15 at 04:05

score 0 · Accepted Answer · answered Dec 06 '15 at 04:14

The contour for calculating residue is a closed curve surrounding the pole you want to calculate. The contour integral is $$ \oint_\gamma f(z)dz = \oint_a^b f(z(t)) z'(t) dt $$ where $z$ is a parametrisation of the curve $\gamma$.

In your case, the parametrisation could be $e^{i t} \pm ai, t \in [0, 2\pi]$

finding residue for complex analysis

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