# 2D Fourier transform of annulus

I have an annulus and I'd like to the take the 2D Fourier transform of it, my code:

a[x_, y_] := 1 < (x^2 + y^2) <= 2
RegionPlot[a[x, y], {x, -3, 3}, {y, -3, 3}]


I have tried FourierTransform and Fourier and they do nothing.

Am I doing something visibly stupid and wrong?

• It's a little unclear what you mean by the Fourier transform of a region. Are you trying to do the 2D Fourier transform of the piecewise defined function a[x_, y_] = Piecewise[{{1, 1 < (x^2 + y^2) <= 2}, {0, True}}]? Or are you trying to do the Fourier transform of a constant over that region (like, if you're computing the far-field diffraction pattern created by light passing through an annular aperture)? Or something else? Nov 21 '17 at 16:56
• Just change from cartesian to polar coordinates and do HankelTransfrom Nov 21 '17 at 17:54
• Just an FYI, RegionPlot returns a graphics object that is a polygonal representation of the input, and is meant for visualization only. Nov 21 '17 at 18:56
• My Mathematica (11.0) does not recognise HankelTransform as a command. It's not even in the help documentation! What could be the reason? Nov 22 '17 at 0:31
• Solved!I went to 11.2. Nov 22 '17 at 0:55

Define your region as a Piecewise function (as suggested by march) and then apply FourierTransform:

a[x_, y_] := Piecewise[{{1, 1 < (x^2 + y^2) <= 2}, {0, True}}];
FourierTransform[a[x, y], {x, y}, {u, v}] You may want to check to make sure that the FourierParameters are using the definition you are used to.

• Thanks, I'm trying it but its takes ages to do the FT. Is there any way of speeding it up? Nov 21 '17 at 19:30
• Its a bit faster if you use set instead of set delayed ("=" instead of ":="). If it is taking longer than a few seconds, then restart the kernel because something is wrong. Nov 21 '17 at 20:11
• I'm doing:<code>ring[x_, y_] = Piecewise[{{1,1<(x^2+y^2)<=2}, {0, True}] </code>, then <code>ft[u,v] = FourierTransform[ring[x,y],{x,y},{u,v}]</code>, it takes ages and it does not return anything! Anything wrong wiht me? Nov 21 '17 at 23:30
• Start with a fresh session (quit the kernel), copy paste the code above as it is, and you will get the answer I just pasted after a few seconds. Nov 21 '17 at 23:47
• Yes that's what I did... I'll try on a different machine tomorrow... Which version of Mathematica are you on by the way? Nov 21 '17 at 23:54