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I am trying to show that

$$\int_{\gamma} \frac{z^{p-1}}{z^2+1} d{z} = 2\pi i\cos\left(\frac{\pi p}{2}\right)e^{i\pi(p-1)}\,\,,\,\gamma:=\{z\;\;;\;\;|z|=R\}$$

for $0 < p < 2$. This integral computes the path of the contour excluding the branch line along the positive real axis. So I'm really computing $\int_{\gamma}$ part in:

$$\int_{\gamma} = \int_{C_1} + \int_{C_2} + \int_{C_3} + \int_{C_4}$$

where $C_1, C_2, C_3$ and $C_4$ are all pieces of the contour.

I've computed the residues at $z=i$ and $z=-i$ but for some reason my calculations don't check out. I got $$\displaystyle\frac{e^{i\frac{\pi}{2}(p-1)}}{2i} - \frac{e^{-i\frac{\pi}{2}(p-1)}}{2i}.$$

  • 4
    What's your curve of integral?2012-04-22
  • 0
    $\gamma$ is a circle of radius $R$2012-04-22
  • 0
    Well I solved my problem. I thought that this question would make sense to all of you. Funny how the thoughts in my mind don't translate well in basic English.2012-04-22
  • 1
    That's why they sell books on communication :-). What is your solution?2012-04-22
  • 0
    Which radius $R$?2012-04-22
  • 0
    What are the $C_k$?2012-07-26

2 Answers 2