I recently discovered this relationship but I cannot understand why it holds.
$$\int_0^\infty \frac{x^\alpha}{1+x^2} \, dx = \frac{\sin(\pi\alpha/2)}{\sin(\pi\alpha)}$$ when $0<\alpha<1$.
As we already know, the pdf of standard Cauchy is given by $f(x)=\frac{1}{1+x^2}$ so the left hand side of the equation above can be interpreted as the expectation of $X^\alpha I\{X>0\}$ with $X$ being standard Cauchy.
I tried with Mathematica and this equation seems to hold. I just really want to see why. Thanks.
