Help with integral $\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx$

Question

How should I proceed to work out following convolution integral:

$\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx$

for real $\alpha$ > 0.

It is the convolution of a powerlaw decaying impulse response with itself. My goal is to find the decay exponent of the autocorrelation function.

Answer 1

Rescale with $x=yt$, then

$$\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx=\int_0^1 (yt)^{-\alpha} (y-yt)^{-\alpha} y\,dt=y^{1-2\alpha}\int_0^1 t^{-\alpha} (1-t)^{-\alpha}\,dt\\ =y^{1-2\alpha}B(1-\alpha,1-\alpha).$$

Answer 2

Hint write it as $(xy-x^2)^{-\alpha}dx$ now y is a constant as variable is x which is easy to integrate.