I'm sorry because I'm not a mathematician so that my question may look a little bit messy.
I have tabulated values [1] of a 3 dimensional radial function $f(r)$: $f(x_1,x_2,x_3)=f(\sqrt{x_1^2+x_2^2+x_3^2})=f(r)$
I'd like to Fourier transform it in order to get $f(k)$ (or when having $f(k)$ transforming it to $f(r)$. I have looked around and understood this is a Hankel transform of order? (2 or 3, did not get it). And I did not find a practical way of doing this transformation. I have access to web, fortran and mathematica 7.
Could you please, please, please help me?
[1] To be more practical: I have a picture where is plotted the static structure factor $S(k)$ of a inhomogeneous fluid as obtained by neutron diffraction. I can extract points $(k:S(k))$ from this picture. $S(k)$ is related to the direct correlation function $C(k)$ as defined by Ornstein-Zernike through $S(k)=(1-nC(k))^{-1}$ where $n$ is a constant. I'd like to plot $C(r)$.
Radial symmetric Fourier transforms in 2 D are Hankel transforms. Spherically symmetric ones in 3D I think give you something like
$$ f(k) = 4 \pi \int f(r) {\rm sinc}(kr) ~ r^2 dr $$
To show this you need to do the integral of $$f(k) = 2\pi \int \int f(r) e^{i kr\cos \theta} \sin \theta d \theta r^2 dr $$
Which follows from $$ f(k,\theta', \phi') = \int \int \int e^{i {\bf k \cdot r}} f(r,\theta,\phi) r^2 dr \sin \theta d\theta d\phi $$ and ${\bf k \cdot r} = k r (\cos \theta \cos \theta' + \sin \theta \sin \theta' \cos(\phi -\phi')) $.