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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}](img1.gif) |
(1) |
| (2) |
To evaluate the integral, we use the fact that
where x0 is defined by f(x)=0
and the definition
.
| ![\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}](img4.gif) |
(3) |
| (4) |
Next: UTILITY OF THIS RESULT
Up: Claerbout: Random lines in
Previous: RESOLUTION OF THE PARADOX
Stanford Exploration Project
4/27/2000