Integral in $n$−dimensional euclidean space I've asked this question in Mathematics Stack Exchange, but unfortunately there is no answer yet. I repost it because this integral comes from QFT and maybe someone here did it before or could help me. I merely copy this post. 

I want to calculate this integral in $n$-dimensional euclidean space.
$$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$
  where $k^2=(k\cdot k)$,
  $k=(k_1,\ldots,k_n)\in\mathbb{R}^n$,
  $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$,$a\in \mathbb{R}$.
I've done this integral for $n=3$ by spherical coordinates and residue
  theorem. I have 
$$I(r)=\frac{1}{4\pi r}e^{-ar},$$ where $r=|x|$
But in $n$-dimensions I failed in using spherical coordinates,
  because I have never done it before. Also I see that this integral is
  Fourier transform of $\frac{1}{k^2+a^2}$, but I failed here too,
  because I can't find Fourier pair in my reference books.
If someone could guide me in this integration it would be great.

 A: WARNING: The function is not absolutely integrable for $n>1$, so the integral strongly depends on how you decide to compute it if you break the integration into iterated integrals. 
Use instead cylindric coordinates. $k = (z, \vec{r})$, where $\vec{r} \in \mathbb R^{n-1}$ and $z\in \mathbb R$. You have this way, assuming that $x$ is directed along $z$:
$$I(x) = \frac{1}{(2\pi)^n}\int_{\mathbb R^{n-1}} d\vec{r} \int_{\mathbb R} dz \frac{e^{i|x|z}}{\vec{r}^2 + z^2 +a^2}=\frac{\omega_{n-1}}{(2\pi)^n}\int_{0}^{+\infty} dr \int_{\mathbb R} dz \frac{e^{i|x|z}r^{n-2} }{r^2 + z^2 +a^2}$$
So:
$$I(x) = \frac{2\omega_{n-1}}{(2\pi)^n}\int_{0}^{+\infty} dr \int_0^{+\infty} dz \frac{r^{n-2}\cos(|x|z) }{r^2 + z^2 +a^2}$$
where $\omega_{n-1} = \frac{2\pi^{(n-1)/2}}{\Gamma((n-1)/2)}$ is the measure of the surface of the unit sphere in $\mathbb R^{n-1}$.
The internal integral can be found in several books e.g. identity 3.723(2) in Gradshteyn - Ryzhik book (seventh edition). Performing it one has:
$$I(x) = \frac{\pi\omega_{n-1}}{(2\pi)^n}\int_{0}^{+\infty} dr  \frac{r^{n-2}e^{-|x|\sqrt{r^2  +a^2}} }{\sqrt{r^2  +a^2}} $$
The remaining integral, passing to integrate in $d(r^2/a^2)$, can be computed in terms of Bessel functions $K_\nu$ using identity 3.479(1) in Gradshteyn - Ryzhik book (seventh edition).
Please check everything since, as usual, I am not confident in my computations!
