complex integral around contour

I'm looking for some advice on talking this simple contour integration question. I am unsure of which theorem I should apply.

Let $C$ be the circle centred at the origin and with radius 2 and let

$$f(z)= \frac {z^2}{z^2+2z+2}$$

Find the integral of $f(z)$ around the contour C

• You may want to check out residual theorem – Yujie Zha May 3 '17 at 16:26
• Yes I am famaliar with the theorem however am unsure how to find the poles. – juper May 3 '17 at 16:56

Start by factoring the denominator and doing a partial fraction decomposition, to get $$f(z) = 1 - \frac1{z+1+i}-\frac1{z+1-i}$$ So there are two simple poles, each with strength $-1$, located at $-1-i$ and $-1+i$. These locations are within the contour $C$ (they are $\sqrt{2}<2$ from the origin.
So if the contour used goes counterclockwise around $C$, the integral will be $$(2\pi i) ([-1]+[-1]) = -4\pi i$$ Of course if the contour were meant to go in the clockwise direction the answer would be $+4\pi i$.
• @juper: I think there's a typo here. $\frac{1}{2+1-i}$ should be $\frac{1}{z+1-i}$, I think. – WB-man May 4 '17 at 5:44