Computation of a 2D Fourier transform

Question

Is there an easy to compute the Fourier transform of $\frac{1}{1+x_1^2+x_2^2}$ in two variables ? And more generally, the Fourier transform of $\frac{1}{1+x_1^2+...+x_N^2}$, where $N$ denotes the dimension ?

Answer 1

These functions are radial functions so their Fourier transforms are too. You might consider the Hankel transform an easy way to compute them.

Answer 2

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With $\ds{\vec{k} \equiv k_{x}\,\hat{x} + k_{y}\,\hat{y}}$ ( where $\ds{k_{x}, k_{y} \in \mathbb{R}}$ ) and $\ds{\phi\ \equiv\ \angle\pars{\vec{k},\vec{r}}}$:

\begin{align} &\bbox[10px,#ffd]{\left.\iint_{\mathbb{R}^{\large 2}}{\expo{\ic\vec{k}\cdot\vec{r}} \over 1 + r^{2}}\,\dd^{2}\vec{r} \,\right\vert_{\ \vec{k}\ \not=\ \vec{0}}} = \int_{0}^{\infty}{r \over 1 + r^{2}} \int_{0}^{2\pi}{\exp\pars{\ic kr\cos\pars{\phi}}} \,\dd\phi\,\dd r \label{1}\tag{1} \end{align} However, \begin{align} \int_{0}^{2\pi}{\exp\pars{\ic kr\cos\pars{\phi}}}\,\dd\phi & = \int_{-\pi}^{\pi}{\exp\pars{-\ic kr\cos\pars{\phi}}}\,\dd\phi \\[5mm] & = 2\int_{0}^{\pi}{\cos\pars{kr\cos\pars{\phi}}}\,\dd\phi = \bbx{2\pi\,\mrm{J}_{0}\pars{kr}}\label{2}\tag{2} \end{align}

where $\ds{\mrm{J}_{0}}$ is a Bessel Function of the First Kind. The above link already reported the integration \eqref{2}.

\eqref{1} and \eqref{2} lead to \begin{align} &\bbox[10px,#ffd]{\left.\iint_{\mathbb{R}^{\large 2}}{\expo{\ic\vec{k}\cdot\vec{r}} \over 1 + r^{2}}\,\dd^{2}\vec{r} \,\right\vert_{\ \vec{k}\ \not=\ \vec{0}}} = 2\pi\int_{0}^{\infty}{\mrm{J}_{0}\pars{kr} \over 1 + r^{2}} \,r\,\dd r = \bbx{\large 2\pi\,\mrm{K}_{0}\pars{k}} \end{align}

$\ds{\mrm{K}_{0}}$ is a Modified Bessel Function. The last integration is reported in this link.