As you helped me so well last time, I might as well ask a final question! Today I'm trying to prove this:
$$ \int_0^\infty \frac{x^{p}}{ 1+x^{2}}dx = \frac{\pi}{2}\cos\left(p\frac{\pi}{2}\right) $$
For $-1 < p < 1$.
I have no idea how to handle the varying $p$. I've been able to prove the relationship in the case $p = 0$. So now I could try showing this for $-1 < p < 0$ and $0 < p < 1$ but both of those seem to be tricky.
Any tips on dealing with the non-constant $p$? There are poles at $x = i$ and $x = -i$, and if $p < 0$ also at $x = 0$.
I think the best approach would be a semicircle in the top half, maybe with a small inner radius as well in the case $p<0$? Or should I not be trying to prove the relationship for these separate parts but is there a general way to do it?
is required for the convergence of the integral.
– 2012-08-10