Fourier transform of $1/ sqrt{m^2+p_1^2+p_2^2+p_3^2}$

Question

Let $m>0$ and consider the function $f:mathbb R^3tomathbb C$
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?

Felix Marin · Answer

$newcommand{bbx}[1]{,bbox[15px,border:1px groove navy]{displaystyle{#1}},}
 newcommand{braces}[1]{leftlbrace,{#1},rightrbrace}
 newcommand{bracks}[1]{leftlbrack,{#1},rightrbrack}
 newcommand{dd}{mathrm{d}}
 newcommand{ds}[1]{displaystyle{#1}}
 newcommand{expo}[1]{,mathrm{e}^{#1},}
 newcommand{ic}{mathrm{i}}
 newcommand{mc}[1]{mathcal{#1}}
 newcommand{mrm}[1]{mathrm{#1}}
 newcommand{pars}[1]{left(,{#1},right)}
 newcommand{partiald}[3][]{frac{partial^{#1} #2}{partial #3^{#1}}}
 newcommand{root}[2][]{,sqrt[#1]{,{#2},},}
 newcommand{totald}[3][]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
 newcommand{verts}[1]{leftvert,{#1},rightvert}$
$ds{underline{underline{mbox{With} vec{R} equiv verts{m}vec{r}}}}$:
begin{align}
&bbox[10px,#ffe]{iiint_{largemathbb{R}^{3}}{expo{-icvec{p}cdotvec{r}} over root{m^{2} + p^{2}}},dd^{3}vec{p}}  =
m^{2}iiint_{largemathbb{R}^{3}}{expo{-icvec{p}cdotvec{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_{Largevec{p}}}expo{-icvec{p}cdotvec{R}},{ddOmega_{vec{p}} over 4pi}}}
^{ds{sinpars{pR} over pR}} 4pi p^{2},dd p
\[5mm] = &
{4pi m^{2} over R}int_{0}^{infty}{psinpars{pR} over
root{p^{2} + 1}},dd p
\[5mm] = &
-,{4pi m^{2} over R},
partiald{}{R}int_{0}^{infty}{cospars{pR} over
root{p^{2} + 1}},dd p
\[5mm] = &
-,{4pi 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 $ds{bfcolor{black}{9.6.21}}$.
begin{align}
&bbox[10px,#ffe]{iiint_{largemathbb{R}^{3}}{expo{-icvec{p}cdotvec{r}} over root{m^{2} + p^{2}}},dd^{3}vec{p}}  =
-4pi m^{2},{mrm{K}_{1}pars{R} over R}
\[5mm] = &
bbox[10px,#ffd,border:1px solid navy]{-4pi verts{m},{mrm{K}_{1}pars{verts{m}r} over r}}
\ &
end{align}
See A & S $ds{bfcolor{black}{9.6.28}}$.

Fourier transform of $1/ sqrt{m^2+p_1^2+p_2^2+p_3^2}$

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