This is the equation im trying to prove
$$P_y(\textbf{x})=\frac{1}{(2\pi)^n}\int{e}^{i\textbf{k}\cdot\bf{x}}{e}^{-y|\textbf{k}|}\ d\textbf{k}=\frac{\Gamma(\frac{(n+1)}{2})}{\pi\frac{^{(n+1)}}{2}}\frac{y}{(|\textbf{x}|^2+y^2)^\frac{(n+1)}{2}}$$
And the help I have
$${e}^{-b}=\int_{0}^{\infty}\frac{{e}^{-s}}{\sqrt{\pi s}}{e}^\frac{{-b^2}}{4s}ds,\qquad b\gt0.$$
I am supposed to use the help to substitute y|k| with b and swap the integrals, then use the Fourier Transformation of the Gaussian. k is an element of R^n in this context.
I have no idea how to properly use the Fourier Transformation of a Gaussian for an element of $R^n$. I would appreciate any help with that.