Editing Hankel transform (section)

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

:<math>f(r, \theta) = \sum_{m=-\infty}^\infty f_m(r) e^{im\theta_{\mathbf{r}}},</math>

then its two-dimensional Fourier transform is given by<math display="block">F(\mathbf k) = 2\pi \sum_m i^{-m} e^{im\theta_{\mathbf{k}}} F_m(k),</math>where<math display="block">F_m(k) = \int_0^\infty f_m(r) J_m(kr) \,r\,\mathrm{d}r</math>is the <math display="inline">m</math>-th order Hankel transform of <math>f_m(r)</math> (in this case <math display="inline">m</math> plays the role of the angular momentum, which was denoted by <math display="inline">l</math> in the previous section).