Evaluate $\int_{0}^{\infty} \frac{x^{-p}}{x^2+2x\cos\theta +1}\,dx$

Asked Aug 23 '19 at 13:44

Active Aug 23 '19 at 13:59

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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...

edited Aug 23 '19 at 13:59

StubbornAtom

17,052

asked Aug 23 '19 at 13:44

Mee98

1,133

Have you tried $x=e^u$? – J.G. Aug 23 '19 at 13:57
As one of the two sides of the "keyhole" in the Hankel contour? – K B Dave Aug 23 '19 at 14:02
4

Duplicate of Symmetry of function defined by integral, or maybe it was better to mark it as a duplicate of this one. – Zacky Aug 23 '19 at 14:06

Evaluate $\int_{0}^{\infty} \frac{x^{-p}}{x^2+2x\cos\theta +1}\,dx$

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