Solving a 2D Fourier Transform

Questions
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1

HI Folks,
I don’t do much in the Fourier arena so please forgive the naivete of this question. :slight_smile:
I am trying to solve the following 2D Fourier transform (z is a constant):

FourierTransform[1/Sqrt[x^2 + y^2 + z^2], {x, y}, {u, t}]

After some time, Mathematica simply returns the original formula. Similar scenario, if I try to solve the corresponding integral directly:

Integrate[Exp[2 Pi I (m*x + n*y)]/Sqrt[x^2 + y^2 + z^2], {x, 0, 1}, {y, 0, 1}]

Any advice?

Cheers,
Nick

2

It may not be an integral that is doable in closed form. If you first do the x-Integral, the remaining y-integral is (assuming that z is Real)

FourierTransform[
 Sqrt[2/Pi] BesselK[0, u Sqrt[y^2 + z^2] Sign[u]], {y}, {t}, 
 Assumptions -> {z \[Element] Reals}]

Which returns unevaluated.

3

Hi David,
Thanks very much for your comment. Yes, z is Real and positive. I will focus on a numerical solution. I had thought that perhaps I was missing something obvious with respect to a closed form solution.

Cheers.

4

Also note, FourierTransform transform is defined over {- Infinity, + Infinity}, while your 2nd integrals have range {0,1}.

5

Thanks Sam.