Integral Representation of $\frac{\mathrm{sign}(x)}{|x|^{s}}$ In this paper, on page 433 (4.12), the authors used an integral formula of the function $\frac{\mathrm{sign}(x)}{|x|^{s}}$, which is 
$$|x|^{-s}\mathrm{sign}(x)=\frac{2}{\Gamma(\frac{s+1}{2})}\int_{0}^{\infty}dyy^{s}xe^{-x^{2}y^{2}}$$
for any real non-zero $x$.
This integral seems very unexptected. Can anybody tell me how one knows that the function can be expressed in the above integral? Is there any motivation behind it?

It's a Laplace transform. Roughly, powers of $y$ transform to powers of $x^{-1}$ because $x,y$ have opposite dimension (as seen from the $x^2y^2$ term in the exponential). Changing variable $u=y^2$ would tidy it up, and then it would be essentially the definition of the Gamma function, which is the Laplace transform of $x^a$.
