Three dimensional Fourier Transform of a rational function Given $s\in(0,2)$, can we find an explicit representation for the (three-dimensional) Fourier transform of the function
$$f_s(x):=\frac{|x|^s}{1+|x|^2},$$
or at least some pointwise bound?
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\begin{align}
\hat{\mrm{f}}_{s}\pars{\vec{k}} & \equiv
\iiint_{\large\mathbb{R}^{3}}{r^{s} \over 1 + r^{2}}\,
\expo{-\ic\vec{k}\cdot\vec{r}}\dd^{3}\vec{r} =
\int_{0}^{\infty}{r^{s} \over 1 + r^{2}}\,
4\pi r^{2}\
\overbrace{\int_{\Omega_{\large\vec{r}}}
\expo{-\ic\vec{k}\cdot\vec{r}}{\dd\Omega_{\vec{r}} \over 4\pi}}^{\ds{\sin\pars{kr} \over kr}}
\\[5mm] &=
{4\pi \over k}\int_{0}^{\infty}{r^{s + 1} \over 1+ r^{2}}\,\sin\pars{kr}\,\dd r =
{4\pi \over k^{s + 1}}\int_{0}^{\infty}{r^{s + 1} \over  r^{2} + k^{2}}\,\sin\pars{r}\,\dd r
\end{align}

It's still a cumbersome task but some CAS yields

\begin{align}
\hat{\mrm{f}}_{s}\pars{\hat{k}} & =
{1 \over 2}\,\sec\pars{\pi s \over 2}\bracks{%
\sin\pars{\pi s}\Gamma\pars{s}\,{}_{1}\mrm{F}_{2}\pars{1;{1 \over 2} - {s \over 2}, 1 - {s \over 2};{k^{2} \over 4}} - \pi\verts{k}^{s}\sinh\pars{\verts{k}}}
\\[5mm] & \mbox{with}\quad-3 < s < 1
\end{align}

${}_{1}\mrm{F}_{2}$ is the Generalized Hypergeometric PFQ.

