Fourier transform of $1/ \sqrt{m^2+p_1^2+p_2^2+p_3^2}$ Let $m>0$ and consider the function $f:\mathbb R^3\to\mathbb C$
defined through
$$ f(p_1,p_2,p_3) = \frac{1}{\sqrt{m^2+p_1^2+p_2^2+p_3^2}}.$$
I would like to compute the Fourier transform of $f$.
This particular function is of interest as one which naturally appears in some problems of special relativity.

What I already know:

*

*Although $f$ is neither integrable nor square integrable, the Fourier transform of $f$ is well defined as the Fourier transform of a tempered distribution.

*Using symbolic calculus software, I expect that
$\int_{-\infty}^{+\infty}f(p_1,p_2,p_3) e^{-i x_1p_1} dp_1 = 2K_0(x_1 \sqrt{m^2+p_2^2+p_3^2})$, where $K_0$ is a modified Bessel function of the second kind.


Questions:

*

*Is the Fourier transform of $f$ explicitly computable?

*If it is, how could I compute it?
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\ds{\underline{\underline{\mbox{With}\ \vec{R} \equiv \verts{m}\vec{r}}}}$:
\begin{align}
&\bbox[10px,#ffe]{\iiint_{\large\mathbb{R}^{3}}{\expo{-\ic\vec{p}\cdot\vec{r}} \over \root{m^{2} + p^{2}}}\,\dd^{3}\vec{p}}  =
m^{2}\iiint_{\large\mathbb{R}^{3}}{\expo{-\ic\vec{p}\cdot\vec{R}} \over \sqrt{p^{2} + 1}}\,\dd^{3}\vec{p}
\\ = &\
m^{2}\int_{0}^{\infty}{1 \over \root{p^{2} + 1}}\
\overbrace{\pars{\int_{\Omega_{\Large\vec{p}}}\expo{-\ic\vec{p}\cdot\vec{R}}\,{\dd\Omega_{\vec{p}} \over 4\pi}}}
^{\ds{\sin\pars{pR} \over pR}}\ 4\pi p^{2}\,\dd p
\\[5mm] = &\
{4\pi m^{2} \over R}\int_{0}^{\infty}{p\sin\pars{pR} \over
\root{p^{2} + 1}}\,\dd p
\\[5mm] = &\
-\,{4\pi m^{2} \over R}\,
\partiald{}{R}\int_{0}^{\infty}{\cos\pars{pR} \over
\root{p^{2} + 1}}\,\dd p
\\[5mm] = &\
-\,{4\pi m^{2} \over R}\,
\partiald{\mrm{K}_{0}\pars{R}}{R}
\end{align}
$\ds{\mrm{K}_{0}}$ is a Modified Bessel Function.
See A & S $\ds{\bf\color{black}{9.6.21}}$.
\begin{align}
&\bbox[10px,#ffe]{\iiint_{\large\mathbb{R}^{3}}{\expo{-\ic\vec{p}\cdot\vec{r}} \over \root{m^{2} + p^{2}}}\,\dd^{3}\vec{p}}  =
-4\pi m^{2}\,{\mrm{K}_{1}\pars{R} \over R}
\\[5mm] = &\
\bbox[10px,#ffd,border:1px solid navy]{-4\pi \verts{m}\,{\mrm{K}_{1}\pars{\verts{m}r} \over r}}
\\ &
\end{align}
See A & S $\ds{\bf\color{black}{9.6.28}}$.
