1
$\begingroup$

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 $\displaystyle\frac{e^{i\frac{\pi}{2}(p-1)}}{2i} - \frac{e^{-i\frac{\pi}{2}(p-1)}}{2i}.$

  • 0
    What are the $C_k$?2012-07-26

2 Answers 2

1

Assuming that the contour of integration is as follows

$\hspace{4.5cm}$enter image description here

The only residues contained in $\gamma$ would be at $i$ and $-i$. $ \begin{align} \mathrm{Res}_{z=i}\left(\frac{z^{p-1}}{z^2+1}\right) &=\frac{e^{i\pi(p-1)/2}}{2i}\\ &=-e^{i\pi p}\frac{e^{-i\pi p/2}}{2}\tag{1} \end{align} $ and $ \begin{align} \mathrm{Res}_{z=-i}\left(\frac{z^{p-1}}{z^2+1}\right) &=\frac{e^{i3\pi(p-1)/2}}{-2i}\\ &=-e^{i\pi p}\frac{e^{i\pi p/2}}{2}\tag{2} \end{align} $ Thus, adding $(1)$ and $(2)$ we get $ \int_\gamma\frac{z^{p-1}}{z^2+1}\,\mathrm{d}z=-2\pi i\,e^{i\pi p}\cos\left(\frac{\pi p}{2}\right)\tag{3} $ and $(3)$ should be valid for all $p\in\mathbb{R}$.

0

Shouldn't there be a residue corresponding to the z=0 when p is different from 1? Hope this helps.