Fourier Transform in spherical coordinates

Question

I have a problem finding the Inverse Fourier transform of the

$$e^{i\bf{k}\cdot\bf{r}}\dfrac{1}{k^2+a^2}$$

Any suggestion?

Answer 1

Converting to spherical coordinates in $k$-space and rotating the system so that $\vec r$ aligns with the $k_z$ axis, we have

$$\begin{align} \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty \frac{e^{i\vec k\cdot \vec r }}{k^2+a^2}\,dk_x\,dk_y\,dk_z&=\int_0^{2\pi}\int_0^\pi \int_0^\infty \frac{e^{ikr\cos(\theta)}}{k^2+a^2}\,k^2\,dk\,\sin(\theta)\,d\theta\,d\phi\\\\ &=2\pi \int_0^\pi \int_0^\infty \frac{e^{ikr\cos(\theta)}}{k^2+a^2}\,k^2\,dk\,\sin(\theta)\,d\theta\\\\ &=2\pi \int_0^\infty \frac{k^2}{k^2+a^2} \int_0^\pi e^{ikr\cos(\theta)}\sin(\theta)\,d\theta\,dk\,\\\\ &=4\pi \int_0^\infty \frac{k}{k^2+a^2}\left(\frac{\sin(kr)}{r}\right)\,dk \end{align}$$

The integral over $k$ can be evaluated using, for example, contour integration. The result is $\displaystyle2\pi ^2\frac{e^{-|a|r}}r$

Can you finish now?