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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:

$${\rm FT}\left({1\over r}\right) = \int \int \exp[i k_x r \cos\theta + i k_y r \sin\theta]\ {1\over r} r dr d\theta$$ (1)

$${\rm FT}\left({1\over r}\right) = \int \delta[k_x \cos\theta + k_y \sin\theta] d\theta$$ (2)

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

$${\rm FT}\left({1\over r}\right) = {1\over \vert-k_x \sin\theta_0 + k_y \cos\theta_0\vert}$$ (3)

$${\rm FT}\left({1\over r}\right) = {1\over\sqrt{k_x^2 + k_y^2}} = {1\over k_r}$$ (4)

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