Editing Hankel transform

{{Short description|Mathematical operation}}
{{Distinguish|Hankel matrix transform}}

In [[mathematics]], the '''Hankel transform''' expresses any given function ''f''(''r'') as the weighted sum of an infinite number of [[Bessel functions|Bessel functions of the first kind]] {{math|''J<sub>ν</sub>''(''kr'')}}. The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor ''k'' along the ''r'' axis. The necessary coefficient {{math|''F<sub>ν</sub>''}} of each Bessel function in the sum, as a function of the scaling factor ''k'' constitutes the transformed function. The Hankel transform is an [[integral transform]] and was first developed by the mathematician [[Hermann Hankel]]. It is also known as the '''Fourier–Bessel transform'''. Just as the [[Fourier transform]] for an infinite interval is related to the [[Fourier series]] over a finite interval, so the Hankel transform over an infinite interval is related to the [[Fourier–Bessel series]] over a finite interval.

==Definition==
The '''Hankel transform''' of order  <math>\nu</math> of a function ''f''(''r'') is given by

: <math>F_\nu(k) = \int_0^\infty f(r) J_\nu(kr) \,r\,\mathrm{d}r,</math>

where <math>J_\nu</math> is the [[Bessel function]] of the first kind of order <math>\nu</math> with <math>\nu \geq -1/2</math>. The inverse Hankel transform of {{math|''F<sub>ν</sub>''(''k'')}} is defined as

: <math>f(r) = \int_0^\infty F_\nu(k) J_\nu(kr) \,k\,\mathrm{d}k,</math>

which can be readily verified using the orthogonality relationship described below.

===Domain of definition===
Inverting a Hankel transform of a function ''f''(''r'') is valid at every point at which ''f''(''r'') is continuous, provided that the function is defined in (0,&nbsp;∞), is piecewise continuous and of [[bounded variation]] in every finite subinterval in (0,&nbsp;∞), and

: <math>\int_0^\infty |f(r)|\,r^{\frac{1}{2}} \,\mathrm{d}r < \infty.</math>

However, like the Fourier transform, the domain can be extended by a density argument to include some functions whose above integral is not finite, for example <math>f(r) = (1 + r)^{-3/2}</math>.

===Alternative definition===
An alternative definition says that the Hankel transform of ''g''(''r'') is<ref>{{cite book |title=Hilbert spaces of entire functions |url=https://archive.org/details/hilbertspacesofe0000debr |url-access=registration |year=1968 |publisher=Prentice-Hall |location=London |isbn=978-0133889000 |author=Louis de Branges |authorlink=Louis de Branges de Bourcia |page=[https://archive.org/details/hilbertspacesofe0000debr/page/189 189]}}</ref>

: <math>h_\nu(k) = \int_0^\infty g(r) J_\nu(kr) \,\sqrt{kr}\,\mathrm{d}r.</math>

The two definitions are related:
: If <math>g(r) = f(r) \sqrt r</math>, then <math>h_\nu(k) = F_\nu(k) \sqrt k.</math>
This means that, as with the previous definition, the Hankel transform defined this way is also its own inverse:
: <math>g(r) = \int_0^\infty h_\nu(k) J_\nu(kr) \,\sqrt{kr}\,\mathrm{d}k.</math>
The obvious domain now has the condition
: <math>\int_0^\infty |g(r)| \,\mathrm{d}r < \infty,</math>
but this can be extended. According to the reference given above, we can take the integral as the limit as the upper limit goes to infinity (an [[improper integral]] rather than a [[Lebesgue integral]]), and in this way the Hankel transform and its inverse work for all functions in [[L2-space|L<sup>2</sup>]](0, ∞).

== Transforming Laplace's equation ==
The Hankel transform can be used to transform and solve [[Laplace's equation#Forms in different coordinate systems|Laplace's equation]] expressed in cylindrical coordinates. Under the Hankel transform, the Bessel operator becomes a multiplication by <math>-k^2</math>.<ref>{{Cite book |title=The transforms and applications handbook |date=1996 |publisher=CRC Press |author=Poularikas, Alexander D. |isbn=0-8493-8342-0 |location=Boca Raton Fla. |oclc=32237017}}</ref> In the axisymmetric case, the [[partial differential equation]] is transformed as

: <math>\mathcal{H}_0 \left\{ \frac{\partial^2 u}{\partial r^2} + \frac 1 r \frac{\partial u}{\partial r} 
+ \frac{\partial ^2 u}{\partial z^2} \right\} = -k^2 U + \frac{\partial^2}{\partial z^2} U,</math>

where <math>U = \mathcal{H}_0 u</math>. Therefore, the Laplacian in cylindrical coordinates becomes an ordinary differential equation in the transformed function <math>U</math>.

==Orthogonality==
The Bessel functions form an [[orthogonal basis]] with respect to the weighting factor ''r'':<ref>{{Cite journal|title=Revisiting the orthogonality of Bessel functions of the first kind on an infinite interval|last=Ponce de Leon|first=J.|journal= European Journal of Physics|volume=36|year=2015|issue=1|pages=015016|doi=10.1088/0143-0807/36/1/015016|bibcode=2015EJPh...36a5016P}}</ref>

: <math>\int_0^\infty J_\nu(kr) J_\nu(k'r) \,r\,\mathrm{d}r = \frac{\delta(k - k')}{k}, \quad k, k' > 0.</math>

==The Plancherel theorem and Parseval's theorem==
If ''f''(''r'') and ''g''(''r'') are such that their Hankel transforms {{math|''F<sub>ν</sub>''(''k'')}} and {{math|''G<sub>ν</sub>''(''k'')}} are well defined, then the [[Plancherel theorem]] states

: <math>\int_0^\infty f(r) g(r) \,r\,\mathrm{d}r = \int_0^\infty F_\nu(k) G_\nu(k) \,k\,\mathrm{d}k.</math>

[[Parseval's theorem]], which states

: <math>\int_0^\infty |f(r)|^2 \,r\,\mathrm{d}r = \int_0^\infty |F_\nu(k)|^2 \,k\,\mathrm{d}k,</math>

is a special case of the Plancherel theorem.  These theorems can be proven using the orthogonality property.

== 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.

==Relation to the Fourier and Abel transforms==
The Hankel transform is one member of the [[Projection-slice theorem|FHA cycle]] of integral operators. In two dimensions, if we define {{mvar|A}} as the [[Abel transform]] operator, {{mvar|F}} as the [[Fourier transform]] operator, and {{mvar|H}} as the zeroth-order Hankel transform operator, then the special case of the [[projection-slice theorem]] for circularly symmetric functions states that

: <math>FA = H.</math>

In other words, applying the Abel transform to a 1-dimensional function and then applying the Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions.

== Numerical evaluation ==
A simple and efficient approach to the numerical evaluation of the Hankel transform is based on the observation that it can be cast in the form of a [[convolution]] by a logarithmic change of variables<ref>{{cite journal |last=Siegman |first=A.E. |date=1977-07-01 |title=Quasi fast Hankel transform |journal=Optics Letters |volume=1 |issue=1 |pages=13 |doi=10.1364/ol.1.000013 |pmid=19680315 |bibcode=1977OptL....1...13S |issn=0146-9592}}</ref>
<math display="block">r = r_0 e^{-\rho}, \quad k = k_0 \, e^{\kappa}.</math>
In these new variables, the Hankel transform reads
<math display="block">\tilde F_\nu(\kappa) = \int_{-\infty}^\infty \tilde f(\rho) \tilde J_\nu(\kappa - \rho) \,\mathrm{d}\rho,</math>
where
<math display="block">\tilde f(\rho) = \left(r_0 \, e^{-\rho} \right)^{1-n} \, f(r_0 e^{-\rho}),</math>
<math display="block">\tilde F_\nu(\kappa) = \left(k_0 \, e^{\kappa} \right)^{1+n} \, F_\nu(k_0 e^\kappa),</math>
<math display="block">\tilde J_\nu(\kappa-\rho) = \left(k_0 \, r_0 \, e^{\kappa-\rho} \right)^{1+n} \, J_\nu(k_0 r_0 e^{\kappa-\rho}).</math>

Now the integral can be calculated numerically with <math display="inline">O(N \log N)</math> [[Computational complexity|complexity]] using [[fast Fourier transform]]. The algorithm can be further simplified by using a known analytical expression for the Fourier transform of <math>\tilde J_\nu</math>:<ref>{{cite journal |last=Talman |first=James D. |date=October 1978 |title=Numerical Fourier and Bessel transforms in logarithmic variables |journal=Journal of Computational Physics |volume=29 |issue=1 |pages=35–48 |doi=10.1016/0021-9991(78)90107-9 |bibcode=1978JCoPh..29...35T |issn=0021-9991}}</ref>
<math display="block">
 \int_{-\infty}^{+\infty} \tilde J_\nu(x) e^{-i q x} \,\mathrm{d}x =
 \frac{\Gamma\left(\frac{\nu + 1 + n - iq}{2} \right)}{\Gamma\left(\frac{\nu + 1 - n + iq}{2}\right)} \, 2^{n - iq}e^{iq \ln(k_0 r_0)}.</math>
The optimal choice of parameters <math>r_0, k_0, n</math> depends on the properties of <math>f(r),</math> in particular its asymptotic behavior at <math>r \to 0</math> and <math>r \to \infty.</math>

This algorithm is known as the "quasi-fast Hankel transform", or simply "fast Hankel transform".

Since it is based on [[fast Fourier transform]] in logarithmic variables, <math>f(r)</math> has to be defined on a logarithmic grid. For functions defined on a uniform grid, a number of other algorithms exist, including straightforward [[Numerical integration|quadrature]], methods based on the [[projection-slice theorem]], and methods using the [[Bessel function#Asymptotic forms|asymptotic expansion]] of Bessel functions.<ref>{{Cite journal |last1=Cree |first1=M. J. |last2=Bones |first2=P. J. |date=July 1993 |title=Algorithms to numerically evaluate the Hankel transform |journal=Computers & Mathematics with Applications |volume=26 |issue=1 |pages=1–12 |doi=10.1016/0898-1221(93)90081-6 |doi-access=free |issn=0898-1221}}</ref>

== Some Hankel transform pairs ==
<ref>{{cite book |last=Papoulis |first=Athanasios |title=Systems and Transforms with Applications to Optics |year=1981 |publisher=Krieger Publishing Company |location=Florida USA |isbn=978-0898743586 |pages=140–175}}</ref>

{|class="wikitable"
! <math>f(r)</math>
! <math>F_0(k)</math>
|-
| <math>1</math>
| <math>\frac{\delta(k)}{k}</math>
|-
| <math>\frac{1}{r}</math>
| <math>\frac{1}{k}</math>
|-
| <math>r</math>
| <math>-\frac{1}{k^3}</math>
|-
| <math>r^3</math>
| <math>\frac{9}{k^5}</math>
|-
| <math>r^m</math>
| <math>\frac{\, 2^{m+1} \, \Gamma \left( \tfrac{m}{2} + 1 \right) \,}{k^{m+2} \, \Gamma\left( -\tfrac{m}{2} \right)}, \quad -2 < \mathcal{R_e} \{ m \} < -\tfrac{1}{2}</math>
|-
| <math>\frac{1}{\sqrt{r^2 + z^2\,}}</math>
| <math>\frac{\, e^{-k|z|} \,}{k}</math><!--  = \sqrt{ \frac{2|z| \,}{\pi k}}K_{-\tfrac{1}{2}}(k|z|) \, ref Smythe 1968  -->
|-
| <math>\frac{1}{\, z^2 + r^2 \,}</math>
| <math>K_0(kz), \quad z \in \mathbb{C}</math>
|-
|rowspan="2"| <math>\frac{e^{iar}}{r}</math>
| <math>\frac{i}{\, \sqrt{a^2 - k^2 \,} \,}, \quad a > 0, \; k < a</math>
|-
| <math>\frac{1}{\,\sqrt{k^2 - a^2\,}\,}, \quad a > 0, \; k > a</math>
|-
| <math>e^{-\frac{1}{2} a^2r^2}</math>
| <math>\frac{1}{\,a^2\,} \, e^{-\tfrac{k^2}{2\,a^2}}</math>
|-
| <math>\frac{1}{r} J_0(lr) \, e^{-sr}</math>
| <math>\frac{2}{\, \pi \sqrt{ (k + l)^2 + s^2 \,} \,} K\left( \sqrt{\frac{4kl}{(k + l)^2 + s^2} \,} \right)</math>
|-
| <math>-r^2 f(r)</math>
| <math>\frac{\, \mathrm{d}^2 F_0 \,}{\mathrm{d}k^2} + \frac{1}{k} \frac{\, \mathrm{d} F_0 \,}{\mathrm{d}k}</math>
|}

{|class="wikitable"
! <math>f(r)</math>
! <math>F_\nu(k)</math>
|-
| <math>r^s</math>
| <math>\frac{2^{s+1}}{\, k^{s+2} \,} \, \frac{\Gamma\left(\tfrac{1}{2}(2 + \nu + s)\right)}{\Gamma(\tfrac{1}{2} (\nu - s))}</math>
|-
| <math>r^{\nu-2s} \Gamma(s, r^2 h)</math>
| <math>\tfrac{1}{2} \left(\tfrac k 2\right)^{2s-\nu-2} \gamma\left(1 - s + \nu, \tfrac{k^2}{4h} \right)</math>
|-
| <math>e^{-r^2} r^\nu \, U(a, b, r^2)</math>
| <math>\frac{\Gamma(2 + \nu - b)}{\, 2\, \Gamma(2 + \nu - b + a)} \left(\tfrac k 2\right)^\nu \, e^{-\frac{k^2}{4} \,} \, _1F_1\left( a, 2 + a - b + \nu, \tfrac{k^2}{4} \right)</math>
|-
| <math>r^n J_\mu(lr) \, e^{-sr}</math>
| Expressable in terms of [[elliptic integral]]s.<ref>{{cite journal | last1 = Kausel | first1 = E. | last2 = Irfan Baig | first2 = M.M. | year = 2012 | title = Laplace transform of products of Bessel functions: A visitation of earlier formulas | journal = Quarterly of Applied Mathematics | volume = 70 | pages = 77–97 | hdl = 1721.1/78923 | url = http://dspace.mit.edu/bitstream/1721.1/78923/1/Kausel_Baig.pdf | doi = 10.1090/s0033-569x-2011-01239-2 | doi-access = free}}</ref>
|-
| <math>-r^2 f(r)</math>
| <math>\frac{\mathrm{d}^2 F_\nu}{\mathrm{d}k^2} + \frac{1}{k} \frac{\, \mathrm{d} F_\nu \,}{\mathrm{d}k} - \frac{\nu^2}{k^2} \, F_\nu</math>
|}

{{math|''K<sub>n</sub>''(''z'')}} is a [[modified Bessel function of the second kind]].
{{math|''K''(''z'')}} is the [[complete elliptic integral of the first kind]].

The expression
: <math>\frac{\, \mathrm{d}^2 F_0 \,}{\mathrm{d}k^2} + \frac{1}{k} \frac{\, \mathrm{d} F_0 \,}{\mathrm{d}k}</math>
coincides with the expression for the [[Laplace operator]] in [[polar coordinates]] {{math|( ''k'', ''θ'' )}} applied to a spherically symmetric function {{math| ''F''<sub>0</sub>(''k'') .}}

The Hankel transform of [[Zernike polynomial]]s are essentially Bessel Functions (Noll 1976):

: <math>R_n^m(r) = (-1)^{\frac{n-m}{2}} \int_0^\infty J_{n+1}(k) J_m(kr) \,\mathrm{d}k</math>

for even {{math|''n'' − ''m'' ≥ 0}}.

==See also==

* [[Fourier transform]]
* [[Integral transform]]
* [[Abel transform]]
* [[Fourier–Bessel series]]
* [[Neumann polynomial]]
* [[Y and H transforms]]

== References ==
{{Reflist|30em}}
{{div col|colwidth=30em}}
* {{cite book |last=Gaskill |first=Jack D. |title=Linear Systems, Fourier Transforms, and Optics|publisher=John Wiley & Sons|location=New York|year=1978|isbn=978-0-471-29288-3}}
* {{cite book
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|title=Handbook of Integral Equations
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|location=Boca Raton
|year=1998
|isbn=978-0-8493-2876-3
}}
* {{cite book |last=Smythe|first=William R.|title=Static and Dynamic Electricity |edition=3rd|publisher=McGraw-Hill|location=New York|year=1968|pages=179–223}}
* {{cite journal
|last1=Offord
|first1=A. C.
|title=On Hankel transforms
|journal=Proceedings of the London Mathematical Society
|volume=39
|issue=2
|pages=49&ndash;67
|year=1935
|doi=10.1112/plms/s2-39.1.49
}}
* {{cite journal
|first1=G.
|last1=Eason
|first2=B.
|last2=Noble
|first3=I. N.
|last3=Sneddon
|title=On certain integrals of Lipschitz-Hankel type involving products of Bessel Functions
|journal=[[Philosophical Transactions of the Royal Society A]]
|year=1955
|volume=247
|issue=935
|pages=529&ndash;551
| jstor = 91565
|doi=10.1098/rsta.1955.0005|bibcode=1955RSPTA.247..529E
}}
*{{cite journal
|first1=J. E.
|last1=Kilpatrick
|first2=Shigetoshi
|last2=Katsura
|first3=Yuji
|last3=Inoue
|title=Calculation of integrals of products of Bessel functions
|journal=Mathematics of Computation
|volume=21
|issue=99
|pages=407&ndash;412
|year=1967
|doi=10.1090/S0025-5718-67-99149-1
|doi-access=free
}}
* {{cite journal
|first1=Robert F.
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{{div col end}}
{{Authority control}}

[[Category:Integral transforms]]