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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

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  • $\begingroup$ You may want to check out residual theorem $\endgroup$ – Yujie Zha May 3 '17 at 16:26
  • $\begingroup$ Yes I am famaliar with the theorem however am unsure how to find the poles. $\endgroup$ – juper May 3 '17 at 16:56
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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 $+4\pi i$.

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  • $\begingroup$ How do you arrive at the partial fractions and then see that the poles are as you've stated? I understand when a pole occurs I'm just unsure how you've arrived at it $\endgroup$ – juper May 3 '17 at 16:57
  • $\begingroup$ @juper: I think there's a typo here. $\frac{1}{2+1-i}$ should be $\frac{1}{z+1-i}$, I think. $\endgroup$ – WB-man May 4 '17 at 5:44
  • $\begingroup$ yes that makes more sense. thank to you both! $\endgroup$ – juper May 4 '17 at 15:48

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