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Mathematica Calculus Wolfram Language Wavelets and Fourier Series

Solving a 2D Fourier Transform

Nick Pizzi, IMT

Posted 11 years ago

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 (mx + ny)]/Sqrt[x^2 + y^2 + z^2], {x, 0, 1}, {y, 0, 1}] Any advice? Cheers, Nick

POSTED BY: Nick Pizzi

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Posted 11 years ago

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.

POSTED BY: David Reiss

Nick Pizzi, IMT

Posted 11 years 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.