Fourier transform (math based question)

Questions
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1

Hello all,

I am trying to take Fourier Transform of a function and I would love some pointers on how to best approach it.

My function is a function of (r,phi) and it is a piecewise function where:

f(r,phi) = 0 , r < r_inner
f(r,phi) = cos(phi).^2 + (-0.5)*sin(phi).^2 ,  r_inner <= r <= r_outter
f(r,phi) = 0 , r > r_outter

I’ve attached a figure here. Can I take the FT of the pieces individually and then sum? I realize that I should take the FT in polar coordinates; is there an efficient (clever) way to go about this given the the nature of the function, perhaps that it is symmetric?

Any suggestions/insight would be really appreciated. Thanks in advance!

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2

“Can I take the FT of the pieces individually and then sum?”

Search in this

http://mathworld.wolfram.com/FourierTransform.html

for the word linear

3

The Fourier transform of 0 is 0 so you just need to do the ring.
It would probably be easier to advise if you tell us the goal of your calculation.