Editing Hankel transform (section)

==== Fourier transform in three dimensions ====
If a three-dimensional function {{math|''f''('''r''')}} is expanded in a [[multipole expansion|multipole series]] over [[spherical harmonics]],

:<math>f(r,\theta_{\mathbf{r}},\varphi_{\mathbf{r}}) = \sum_{l = 0}^{+\infty} \sum_{m=-l}^{+l}f_{l,m}(r)Y_{l,m}(\theta_{\mathbf{r}},\varphi_{\mathbf{r}}),</math>

then its three-dimensional Fourier transform is given by<math display="block">F(k,\theta_{\mathbf{k}},\varphi_{\mathbf{k}}) = (2 \pi)^{3/2} \sum_{l = 0}^{+\infty} (-i)^{l} \sum_{m=-l}^{+l} F_{l,m}(k) Y_{l,m}(\theta_{\mathbf{k}},\varphi_{\mathbf{k}}),</math>where<math display="block">\sqrt{k} F_{l,m}(k) = 
\int_{0}^{+\infty}\sqrt{r} f_{l,m}(r)J_{l+1/2}(kr)r\mathrm{d}r.</math>is the Hankel transform of <math>\sqrt{r} f_{l,m}(r)</math> of order <math display="inline">(l+1/2)</math>.

This kind of Hankel transform of half-integer order is also known as the spherical Bessel transform.