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

  • $\begingroup$ 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

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 $+4\pi i$.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.