FOURIER TRANSFORM OF 1/r

We would like to know the 2-D Fourier transform of 1/r. Everywhere I found tables of 1-D Fourier transforms but only one place did I find a table that included this 2-D Fourier transform. It was at http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip-Statisti.html

Sergey Fomel showed me how to work it out: Express the FT in radial coordinates:

$$\begin{eqnarray} {\rm FT}\left({1\over r}\right)&=& \int \int \exp[i k_x r \cos\... ...r}\right)&=& \int \delta[k_x \cos\theta + k_y \sin\theta]\ d\theta\end{eqnarray}$$ (1) (2)

To evaluate the integral, we use the fact that $\int\delta(f(x))dx = 1/\vert f'(x_0)\vert$ where x₀ is defined by f(x)=0 and the definition $\theta_0 =\arctan (-k_x/k_y)$.

$$\begin{eqnarray} {\rm FT}\left({1\over r}\right)&=& {1\over \vert-k_x \sin\theta... ...\over r}\right)&=& {1\over\sqrt{k_x^2 + k_y^2}} \ =\ {1\over k_r} \end{eqnarray}$$ (3) (4)