# Solving a 2D Fourier Transform

GROUPS:
 HI Folks,I don't do much in the Fourier arena so please forgive the naivete of this question. 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
Answer
3 months ago
4 Replies
 David Reiss 2 Votes 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.
Answer
3 months ago
 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.
Answer
3 months ago
 Also note, FourierTransform transform is defined over {- Infinity, + Infinity}, while your 2nd integrals have range {0,1}.
Answer
3 months ago
 Thanks Sam.
Answer
3 months ago