In the paper "Stochastic Processes in Several Dimensions" by Peter Whittle (1963), the following definitions and integration appear as equations (3.14) and (3.15). $n$ is the dimensionality of the space. $J$ is Bessel function and $K$ is modified Bessel funciton of the second kind. I checked the Digital Library of Mathematical Functions but there's so much stuff there that I'm just lost.
If anyone can shed some light on the following derivation, I'd be thankful.
$$ f_{\xi}(\vec{\nu}) := \frac{1}{(|\vec{\nu}|^2 + a^2)^{2p}} $$
\begin{align} \Gamma_{\xi} (|\vec{s}|) &= \frac{1}{(2\pi)^n} \int e^{i \langle \vec\nu, \vec s \rangle } f_{\xi}(\vec \nu) d\vec\nu \\ &= \frac{s^{(-n/2) + 1}}{ (2\pi)^{n/2} } \int_{0}^{\infty} \frac{ J_{(n/2)-1}(\nu s) \nu^{n/2} d\nu}{ (\nu^2 + \alpha^2)^{2p}} \\ &=\frac{ (s/\alpha)^{2p-n/2}K_{2p-n/2}(\alpha s)}{2^{2p-1} \Gamma(2p)} \end{align}
If you want the paper
The paper took some effort to find by Carol Hutchins, the amazing (former) librarian at the Courant Institute, so if you want to look at the original you can email me ([email protected]). Also, if I emailed you this paper, feel free to leave a comment with your contact, in case someone else wants this paper and cannot reach me.
If I understand correctly, $f_\xi(\nu)$ is a function of the modulus of $\nu$, only, then $\Gamma_\xi(s)$ is the n-dimensional radial Fourier transform (see here or here for example), it is then a Hankel transform. Finally, the modified Bessel function is tabulated in DLMF.