Editing Hankel transform (section)

== Relation to the multidimensional Fourier transform ==
The Hankel transform appears when one writes the multidimensional Fourier transform in [[hyperspherical coordinates]], which is the reason why the Hankel transform often appears in physical problems with cylindrical or spherical symmetry.

Consider a function <math>f(\mathbf{r})</math> of a <math display="inline">d</math>-dimensional vector {{math|'''r'''}}. Its <math display="inline">d</math>-dimensional Fourier transform is defined as<math display="block">F(\mathbf{k}) = \int_{\R^d} f(\mathbf{r}) e^{-i\mathbf{k} \cdot \mathbf{r}} \,\mathrm{d}\mathbf{r}.</math>To rewrite it in hyperspherical coordinates, we can use the decomposition of a plane wave into <math display="inline">d</math>-dimensional hyperspherical harmonics <math>Y_{l,m}</math>:<ref>{{Cite book |last=Avery, James Emil |title=Hyperspherical harmonics and their physical applications |isbn=978-981-322-930-3 |oclc=1013827621}}</ref><math display="block">e^{-i\mathbf{k} \cdot \mathbf{r}} =  (2 \pi)^{d/2} (kr)^{1-d/2}\sum_{l = 0}^{+\infty}
(-i)^{l} J_{d/2-1+l}(kr)\sum_{m}
Y_{l,m}(\Omega_{\mathbf{k}}) Y^{*}_{l,m}(\Omega_{\mathbf{r}}),</math>where <math display="inline">\Omega_{\mathbf{r}}</math> and <math display="inline">\Omega_{\mathbf{k}}</math> are the sets of all hyperspherical angles in the <math>\mathbf{r}</math>-space and <math>\mathbf{k}</math>-space. This gives the following expression for the <math display="inline">d</math>-dimensional Fourier transform in hyperspherical coordinates:<math display="block">F(\mathbf{k}) =  (2 \pi)^{d/2} k^{1-d/2} \sum_{l = 0}^{+\infty} (-i)^{l} \sum_{m}Y_{l,m}(\Omega_{\mathbf{k}})
\int_{0}^{+\infty}J_{d/2-1+l}(kr)r^{d/2}\mathrm{d}r \int f(\mathbf{r})  Y_{l,m}^{*}(\Omega_{\mathbf{r}}) \mathrm{d}\Omega_{\mathbf{r}}. </math>If we expand <math>f(\mathbf{r})</math> and <math>F(\mathbf{k})</math> in hyperspherical harmonics:<math display="block">f(\mathbf{r}) = \sum_{l = 0}^{+\infty} \sum_{m}f_{l,m}(r)Y_{l,m}(\Omega_{\mathbf{r}}),\quad F(\mathbf{k}) = \sum_{l = 0}^{+\infty} \sum_{m} F_{l,m}(k) Y_{l,m}(\Omega_{\mathbf{k}}), </math>the Fourier transform in hyperspherical coordinates simplifies to<math display="block">k^{d/2-1}F_{l,m}(k) = (2 \pi)^{d/2}   (-i)^{l} 
\int_{0}^{+\infty}r^{d/2-1}f_{l,m}(r)J_{d/2-1+l}(kr)r\mathrm{d}r. </math>This means that functions with angular dependence in form of a hyperspherical harmonic retain it upon the multidimensional Fourier transform, while the radial part undergoes the Hankel transform (up to some extra factors like <math display="inline">r^{d/2-1}</math>).

=== Special cases ===

==== 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).

==== 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.

==== Fourier transform in {{math|''d''}} dimensions (radially symmetric case) ====
If a {{math|''d''}}-dimensional function {{math|''f''(''r'')}} does not depend on angular coordinates, then its {{math|''d''}}-dimensional Fourier transform {{math|''F''(''k'')}} also does not depend on angular coordinates and is given by<ref>{{cite web|url=http://math.arizona.edu/~faris/methodsweb/hankel.pdf|title=Radial functions and the Fourier transform: Notes for Math 583A, Fall 2008|last=Faris|first=William G.|date=2008-12-06|website=University of Arizona, Department of Mathematics|accessdate=2015-04-25}}</ref><math display="block">k^{d/2-1}F(k) = (2 \pi)^{d/2}
\int_{0}^{+\infty}r^{d/2-1}f(r)J_{d/2-1}(kr)r\mathrm{d}r.</math>which is the Hankel transform of <math>r^{d/2-1}f(r)</math> of order <math display="inline">(d/2-1)</math> up to a factor of <math>(2 \pi)^{d/2} </math>.

====2D functions inside a limited radius====

If a two-dimensional function {{math|''f''('''r''')}} is expanded in a [[multipole expansion|multipole series]] and the expansion coefficients {{math|''f<sub>m</sub>''}} are sufficiently smooth near the origin and zero outside a radius {{mvar|R}}, the radial part {{math|''f''(''r'')/''r<sup>m</sup>''}} may be expanded into a [[power series]] of {{math|1 − (''r''/''R'')^2}}:

:<math>f_m(r)= r^m \sum_{t \ge 0} f_{m,t} \left(1 - \left(\tfrac{r}{R}\right)^2 \right)^t, \quad 0 \le r \le R,</math>

such that the two-dimensional Fourier transform of {{math|''f''('''r''')}} becomes

:<math>\begin{align}
 F(\mathbf k)
  &= 2\pi\sum_m i^{-m} e^{i m\theta_k} \sum_t f_{m,t} \int_0^R r^m \left(1 - \left(\tfrac{r}{R}\right)^2 \right)^t J_m(kr) r\,\mathrm{d}r && \\
 &= 2\pi\sum_m i^{-m} e^{i m\theta_k} R^{m+2} \sum_t f_{m,t} \int_0^1 x^{m+1} (1-x^2)^t J_m(kxR) \,\mathrm{d}x && (x = \tfrac{r}{R})\\
 &= 2\pi\sum_m i^{-m} e^{i m\theta_k} R^{m+2} \sum_t f_{m,t} \frac{t!2^t}{(kR)^{1+t}} J_{m+t+1}(kR),
\end{align}</math>

where the last equality follows from §6.567.1 of.<ref>{{cite book
|last1=Gradshteyn|first1=I. S.
|last2=Ryzhik|first2=I. M.
|editor1-last=Zwillinger|editor1-first=Daniel
|title=Table of Integrals, Series, and Products
|date=2015
|publisher=Academic Press
|isbn=978-0-12-384933-5
|edition=Eighth
|page=687}}</ref> The expansion coefficients {{math|''f<sub>m,t</sub>''}} are accessible with [[discrete Fourier transform]] techniques:<ref>{{cite journal
|first1=José D.
|last1=Secada
|title=Numerical evaluation of the Hankel transform
|journal=Comput. Phys. Commun.
|volume=116
|issue=2–3
|pages=278–294
|bibcode=1999CoPhC.116..278S
|year=1999
|doi = 10.1016/S0010-4655(98)00108-8 }}</ref> if the radial distance is scaled with
:<math>r/R\equiv \sin\theta,\quad 1-(r/R)^2 = \cos^2\theta,</math>
the Fourier-Chebyshev series coefficients {{math|''g''}} emerge as
:<math>f(r)\equiv r^m \sum_j g_{m,j} \cos(j\theta)= r^m\sum_jg_{m,j} T_j(\cos\theta).</math>
Using the re-expansion
:<math>
\cos(j\theta) = 2^{j-1}\cos^j\theta-\frac{j}{1}2^{j-3}\cos^{j-2}\theta +\frac{j}{2}\binom{j-3}{1}2^{j-5}\cos^{j-4}\theta - \frac{j}{3}\binom{j-4}{2}2^{j-7}\cos^{j-6}\theta + \cdots
</math>
yields {{math|''f''<sub>''m,t''</sub>}} expressed as sums of {{math|''g''<sub>''m,j''</sub>}}.

This is one flavor of fast Hankel transform techniques.