fourier transform question (math based)

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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? My research tells me this is OK. Since my function is in polar coordinates, 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!