I need to know how to calculate
$$ \int^\infty_{-\infty} \, d^2q \frac{e^{i\mathbf{q}\cdot\mathbf{x}}}{a +bq^2} $$
basically a 2-dimensional fourier transform of that function.
I think the answer is some sort of exponentially decaying function but I don't know how to proceed. Possibly using polar coordinates or complex integration or something is in order Please help
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Whenever $\displaystyle{\quad{a \over b} > 0}$:
\begin{align} &\color{#f00}{\int_{{\bf R}^{2}}{\rm d}^{2\,}\vec{q}\, {{\rm e}^{{\rm i}\,\vec{q}\cdot\vec{r}} \over a + bq^{2}}} = \int_{0}^{\infty}{\rm d}q\,{q \over a + bq^{2}} \int_{0}^{2\pi}{\rm d}\theta\,{\rm e}^{{\rm i}\,qr\cos\left(\theta\right)} \\[5mm]&= 2\pi\int_{0}^{\infty}{\rm d}q\,{q \over a + bq^{2}} \overbrace{\quad{1\over \pi}\int_{0}^{\pi}{\rm d}\theta\,{\rm e}^{{\rm i}\,qr\cos\left(\theta\right)}\quad}^{\displaystyle{{\rm J}_{0}\left(qr\right)}} = 2\pi\int_{0}^{\infty}{\rm d}q\,{q\,{\rm J}_{0}\left(qr\right) \over a + bq^{2}} \\[5mm]&= {2\pi \over b}\int_{0}^{\infty}{\rm d}q\,{q\,{\rm J}_{0}\left(qr\right) \over q^{2} + \left(\sqrt{a/b}\right)^{2}} =\color{#f00}{% {2\pi \over b}\ {\rm K}_{0}\left(\sqrt{a \over b\,}\ r\right)} \end{align}
where $\ds{\rm J_{0}}$ and $\ds{\rm K_{0}}$ are Bessel Functions.
\Largeis very seldom necessary, in the present case it is not (and the result rather horrendous). – Did Oct 06 '14 at 09:08