I am trying to Fourier transform the following:
$\int \frac{d^2r}{(2\pi)^2}\sqrt{\frac{1}{r}} \; \delta (\rho-r) e^{i\alpha \phi_{r}} \; e^{i\vec{k} \cdot \vec{r}}$
where $\rho$ is a constant; $\vec{r}=\{\hat{x},\hat{y}\}$ and $\vec{k}=\{\hat{k_x},\hat{k_y}\}$. Analytically one would use polar coordinates and obtain something like
$\frac{\sqrt{\rho}}{2\pi} e^{i\alpha \phi_k} J_{\alpha}(k \rho)$
but numerically, I couldn't find out the way to calculate a Fourier transform over the finite integration range. So I went to cartesian coordinate system to take it at least in MMA:
\[Alpha] = 1.; k = 5.;
f[x_, y_] := Sqrt[(1)/(Sqrt[x^2 + y^2])] KroneckerDelta[
Sqrt[8] - Sqrt[x^2 + y^2]] Exp[I \[Alpha] ArcTan[x, y]] ;
NN = 100000;
signal = Table[0.0, {i, 1, NN}];
For[i = 0, i < NN, i++, x = i/NN * \[Pi]; signal[[i + 1]] = f[x, 1.]]
Show[ListLinePlot[Re[InverseFourier[signal]]]]
This produces zero. I am not that great at Fourier transforming, but shouldn't the result be independent of the choice of the coordinate system? Shouldn't DFT be analogous to continuous FT when the number of steps is big enough?
What am I doing wrong? Does anybody know if this is even possible to calculate this transform numerically in any coordinate system?
Thanks a lot for any help, prospective, guidance!
P.S.: final goal is to calculate it using FFTW library. MMA is just a draft to check if such thing is even possible.