Editing Hankel transform (section)

====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.