I have to show that:

$$ \int_{0}^{\infty} \frac{x^{-p}}{x^2+2x\cos\theta +1}\,dx = \frac{\pi \sin (p\theta)}{\sin (p\pi) \sin(\theta)}$$

for $0<p<1$ and $0 < \theta < \pi$.

We can write the denominator as $(x+e^{i\theta})(x+e^{-i\theta})$ and I now want to evaluate this integral using contour integration. However, I have no idea how to do this...

  • $\begingroup$ Have you tried $x=e^u$? $\endgroup$ – J.G. Aug 23 '19 at 13:57
  • $\begingroup$ As one of the two sides of the "keyhole" in the Hankel contour? $\endgroup$ – K B Dave Aug 23 '19 at 14:02
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    $\begingroup$ Duplicate of Symmetry of function defined by integral, or maybe it was better to mark it as a duplicate of this one. $\endgroup$ – Zacky Aug 23 '19 at 14:06