Hankel transform

 ( Integral Transforms ) 

In mathematics, the Hankel transform expresses any given function f( r) as the weighted sum of an infinite number of Bessel functions . 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 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.

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Definition The Hankel transform of order $\backslash nu$ of a function f( r) is given by

$F\_\backslash nu(k)\; =\; \backslash int\_0^\backslash infty\; f(r)\; J\_\backslash nu(kr)\; \backslash ,r\backslash ,\backslash mathrm\{d\}r,$

where $J\_\backslash nu$ is the Bessel function of the first kind of order $\backslash nu$ with $\backslash nu\; \backslash geq\; -1/2$. The inverse Hankel transform of is defined as

$f(r)\; =\; \backslash int\_0^\backslash infty\; F\_\backslash nu(k)\; J\_\backslash nu(kr)\; \backslash ,k\backslash ,\backslash mathrm\{d\}k,$

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

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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, ∞), is piecewise continuous and of bounded variation in every finite subinterval in (0, ∞), and

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 $f(r)\; =\; (1\; +\; r)^\{-3/2\}$.

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Alternative definition An alternative definition says that the Hankel transform of g( r) is

Louis de Branges (1968). 9780133889000, Prentice-Hall. . ISBN 9780133889000

$h\_\backslash nu(k)\; =\; \backslash int\_0^\backslash infty\; g(r)\; J\_\backslash nu(kr)\; \backslash ,\backslash sqrt\{kr\}\backslash ,\backslash mathrm\{d\}r.$

The two definitions are related:

If $g(r)\; =\; f(r)\; \backslash sqrt\; r$, then $h\_\backslash nu(k)\; =\; F\_\backslash nu(k)\; \backslash sqrt\; k.$

This means that, as with the previous definition, the Hankel transform defined this way is also its own inverse:

$g(r)\; =\; \backslash int\_0^\backslash infty\; h\_\backslash nu(k)\; J\_\backslash nu(kr)\; \backslash ,\backslash sqrt\{kr\}\backslash ,\backslash mathrm\{d\}k.$

The obvious domain now has the condition

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(0, ∞).

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Transforming Laplace's equation The Hankel transform can be used to transform and solve Laplace's equation expressed in cylindrical coordinates. Under the Hankel transform, the Bessel operator becomes a multiplication by $-k^2$.

Poularikas, Alexander D. (1996). 9780849383427, CRC Press. ISBN 9780849383427

In the axisymmetric case, the partial differential equation is transformed as

$\backslash mathcal\{H\}\_0\; \backslash left\backslash \{\; \backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; r^2\}\; +\; \backslash frac\; 1\; r\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}$

+ \frac{\partial ^2 u}{\partial z^2} \right\} = -k^2 U + \frac{\partial^2}{\partial z^2} U,

where $U\; =\; \backslash mathcal\{H\}\_0\; u$. Therefore, the Laplacian in cylindrical coordinates becomes an ordinary differential equation in the transformed function $U$.

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Orthogonality The Bessel functions form an orthogonal basis with respect to the weighting factor r:

$\backslash int\_0^\backslash infty\; J\_\backslash nu(kr)\; J\_\backslash nu(k\text{'}r)\; \backslash ,r\backslash ,\backslash mathrm\{d\}r\; =\; \backslash frac\{\backslash delta(k\; -\; k\text{'})\}\{k\},\; \backslash quad\; k,\; k\text{'}\; >\; 0.$

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The Plancherel theorem and Parseval's theorem If f( r) and g( r) are such that their Hankel transforms and are well defined, then the Plancherel theorem states

$\backslash int\_0^\backslash infty\; f(r)\; g(r)\; \backslash ,r\backslash ,\backslash mathrm\{d\}r\; =\; \backslash int\_0^\backslash infty\; F\_\backslash nu(k)\; G\_\backslash nu(k)\; \backslash ,k\backslash ,\backslash mathrm\{d\}k.$

Parseval's theorem, which states

$\backslash int\_0^\backslash infty\; |f(r)|^2\; \backslash ,r\backslash ,\backslash mathrm\{d\}r\; =\; \backslash int\_0^\backslash infty\; |F\_\backslash nu(k)|^2\; \backslash ,k\backslash ,\backslash mathrm\{d\}k,$

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

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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 $f(\backslash mathbf\{r\})$ of a $d$-dimensional vector . Its $d$-dimensional Fourier transform is defined as$$F(\backslash mathbf\{k\})\; =\; \backslash int\_\{\backslash R^d\}\; f(\backslash mathbf\{r\})\; e^\{-i\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\}\; \backslash ,\backslash mathrm\{d\}\backslash mathbf\{r\}.$$To rewrite it in hyperspherical coordinates, we can use the decomposition of a plane wave into $d$-dimensional hyperspherical harmonics $Y\_\{l,m\}$:

$$e^\{-i\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\}\; =\; (2\; \backslash pi)^\{d/2\}\; (kr)^\{1-d/2\}\backslash sum\_\{l\; =\; 0\}^\{+\backslash infty\}\; (-i)^\{l\}\; J\_\{d/2-1+l\}(kr)\backslash sum\_\{m\}\; Y\_\{l,m\}(\backslash Omega\_\{\backslash mathbf\{k\}\})\; Y^\{*\}\_\{l,m\}(\backslash Omega\_\{\backslash mathbf\{r\}\}),$$where $\backslash Omega\_\{\backslash mathbf\{r\}\}$ and $\backslash Omega\_\{\backslash mathbf\{k\}\}$ are the sets of all hyperspherical angles in the $\backslash mathbf\{r\}$-space and $\backslash mathbf\{k\}$-space. This gives the following expression for the $d$-dimensional Fourier transform in hyperspherical coordinates:$$F(\backslash mathbf\{k\})\; =\; (2\; \backslash pi)^\{d/2\}\; k^\{1-d/2\}\; \backslash sum\_\{l\; =\; 0\}^\{+\backslash infty\}\; (-i)^\{l\}\; \backslash sum\_\{m\}Y\_\{l,m\}(\backslash Omega\_\{\backslash mathbf\{k\}\})\; \backslash int\_\{0\}^\{+\backslash infty\}J\_\{d/2-1+l\}(kr)r^\{d/2\}\backslash mathrm\{d\}r\; \backslash int\; f(\backslash mathbf\{r\})\; Y\_\{l,m\}^\{*\}(\backslash Omega\_\{\backslash mathbf\{r\}\})\; \backslash mathrm\{d\}\backslash Omega\_\{\backslash mathbf\{r\}\}.$$If we expand $f(\backslash mathbf\{r\})$ and $F(\backslash mathbf\{k\})$ in hyperspherical harmonics:$$f(\backslash mathbf\{r\})\; =\; \backslash sum\_\{l\; =\; 0\}^\{+\backslash infty\}\; \backslash sum\_\{m\}f\_\{l,m\}(r)Y\_\{l,m\}(\backslash Omega\_\{\backslash mathbf\{r\}\}),\backslash quad\; F(\backslash mathbf\{k\})\; =\; \backslash sum\_\{l\; =\; 0\}^\{+\backslash infty\}\; \backslash sum\_\{m\}\; F\_\{l,m\}(k)\; Y\_\{l,m\}(\backslash Omega\_\{\backslash mathbf\{k\}\}),$$the Fourier transform in hyperspherical coordinates simplifies to$$k^\{d/2-1\}F\_\{l,m\}(k)\; =\; (2\; \backslash pi)^\{d/2\}\; (-i)^\{l\}\; \backslash int\_\{0\}^\{+\backslash infty\}r^\{d/2-1\}f\_\{l,m\}(r)J\_\{d/2-1+l\}(kr)r\backslash mathrm\{d\}r.$$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 $r^\{d/2-1\}$).

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Special cases

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Fourier transform in two dimensions If a two-dimensional function is expanded in a multipole series,

$f(r,\; \backslash theta)\; =\; \backslash sum\_\{m=-\backslash infty\}^\backslash infty\; f\_m(r)\; e^\{im\backslash theta\_\{\backslash mathbf\{r\}\}\},$

then its two-dimensional Fourier transform is given by$$F(\backslash mathbf\; k)\; =\; 2\backslash pi\; \backslash sum\_m\; i^\{-m\}\; e^\{im\backslash theta\_\{\backslash mathbf\{k\}\}\}\; F\_m(k),$$where$$F\_m(k)\; =\; \backslash int\_0^\backslash infty\; f\_m(r)\; J\_m(kr)\; \backslash ,r\backslash ,\backslash mathrm\{d\}r$$is the $m$-th order Hankel transform of $f\_m(r)$ (in this case $m$ plays the role of the angular momentum, which was denoted by $l$ in the previous section).

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Fourier transform in three dimensions If a three-dimensional function is expanded in a multipole series over spherical harmonics,

$f(r,\backslash theta\_\{\backslash mathbf\{r\}\},\backslash varphi\_\{\backslash mathbf\{r\}\})\; =\; \backslash sum\_\{l\; =\; 0\}^\{+\backslash infty\}\; \backslash sum\_\{m=-l\}^\{+l\}f\_\{l,m\}(r)Y\_\{l,m\}(\backslash theta\_\{\backslash mathbf\{r\}\},\backslash varphi\_\{\backslash mathbf\{r\}\}),$

then its three-dimensional Fourier transform is given by$$F(k,\backslash theta\_\{\backslash mathbf\{k\}\},\backslash varphi\_\{\backslash mathbf\{k\}\})\; =\; (2\; \backslash pi)^\{3/2\}\; \backslash sum\_\{l\; =\; 0\}^\{+\backslash infty\}\; (-i)^\{l\}\; \backslash sum\_\{m=-l\}^\{+l\}\; F\_\{l,m\}(k)\; Y\_\{l,m\}(\backslash theta\_\{\backslash mathbf\{k\}\},\backslash varphi\_\{\backslash mathbf\{k\}\}),$$where$$\backslash sqrt\{k\}\; F\_\{l,m\}(k)\; =\; \backslash int\_\{0\}^\{+\backslash infty\}\backslash sqrt\{r\}\; f\_\{l,m\}(r)J\_\{l+1/2\}(kr)r\backslash mathrm\{d\}r.$$is the Hankel transform of $\backslash sqrt\{r\}\; f\_\{l,m\}(r)$ of order $(l+1/2)$.

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

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Fourier transform in dimensions (radially symmetric case) If a -dimensional function does not depend on angular coordinates, then its -dimensional Fourier transform also does not depend on angular coordinates and is given by$$k^\{d/2-1\}F(k)\; =\; (2\; \backslash pi)^\{d/2\}\; \backslash int\_\{0\}^\{+\backslash infty\}r^\{d/2-1\}f(r)J\_\{d/2-1\}(kr)r\backslash mathrm\{d\}r.$$which is the Hankel transform of $r^\{d/2-1\}f(r)$ of order $(d/2-1)$ up to a factor of $(2\; \backslash pi)^\{d/2\}$.

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2D functions inside a limited radius If a two-dimensional function is expanded in a multipole series and the expansion coefficients are sufficiently smooth near the origin and zero outside a radius , the radial part may be expanded into a power series of :

$f\_m(r)=\; r^m\; \backslash sum\_\{t\; \backslash ge\; 0\}\; f\_\{m,t\}\; \backslash left(1\; -\; \backslash left(\backslash tfrac\{r\}\{R\}\backslash right)^2\; \backslash right)^t,\; \backslash quad\; 0\; \backslash le\; r\; \backslash le\; R,$

such that the two-dimensional Fourier transform of becomes

$\backslash begin\{align\}$

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}

where the last equality follows from §6.567.1 of.

The expansion coefficients are accessible with discrete Fourier transform techniques: if the radial distance is scaled with

$r/R\backslash equiv\; \backslash sin\backslash theta,\backslash quad\; 1-(r/R)^2\; =\; \backslash cos^2\backslash theta,$

the Fourier-Chebyshev series coefficients emerge as

$f(r)\backslash equiv\; r^m\; \backslash sum\_j\; g\_\{m,j\}\; \backslash cos(j\backslash theta)=\; r^m\backslash sum\_jg\_\{m,j\}\; T\_j(\backslash cos\backslash theta).$

Using the re-expansion

$$

\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 yields expressed as sums of .

This is one flavor of fast Hankel transform techniques.

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Relation to the Fourier and Abel transforms The Hankel transform is one member of the FHA cycle of integral operators. In two dimensions, if we define as the Abel transform operator, as the Fourier transform operator, and as the zeroth-order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that

$FA\; =\; H.$

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.

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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 $$r\; =\; r\_0\; e^\{-\backslash rho\},\; \backslash quad\; k\; =\; k\_0\; \backslash ,\; e^\{\backslash kappa\}.$$ In these new variables, the Hankel transform reads $$\backslash tilde\; F\_\backslash nu(\backslash kappa)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash tilde\; f(\backslash rho)\; \backslash tilde\; J\_\backslash nu(\backslash kappa\; -\; \backslash rho)\; \backslash ,\backslash mathrm\{d\}\backslash rho,$$ where $$\backslash tilde\; f(\backslash rho)\; =\; \backslash left(r\_0\; \backslash ,\; e^\{-\backslash rho\}\; \backslash right)^\{1-n\}\; \backslash ,\; f(r\_0\; e^\{-\backslash rho\}),$$ $$\backslash tilde\; F\_\backslash nu(\backslash kappa)\; =\; \backslash left(k\_0\; \backslash ,\; e^\{\backslash kappa\}\; \backslash right)^\{1+n\}\; \backslash ,\; F\_\backslash nu(k\_0\; e^\backslash kappa),$$ $$\backslash tilde\; J\_\backslash nu(\backslash kappa-\backslash rho)\; =\; \backslash left(k\_0\; \backslash ,\; r\_0\; \backslash ,\; e^\{\backslash kappa-\backslash rho\}\; \backslash right)^\{1+n\}\; \backslash ,\; J\_\backslash nu(k\_0\; r\_0\; e^\{\backslash kappa-\backslash rho\}).$$

Now the integral can be calculated numerically with $O(N\; \backslash log\; N)$ complexity using fast Fourier transform. The algorithm can be further simplified by using a known analytical expression for the Fourier transform of $\backslash tilde\; J\_\backslash nu$: $$$$

\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)}.
     

The optimal choice of parameters $r\_0,\; k\_0,\; n$ depends on the properties of $f(r),$ in particular its asymptotic behavior at $r\; \backslash to\; 0$ and $r\; \backslash to\; \backslash infty.$

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, $f(r)$ has to be defined on a logarithmic grid. For functions defined on a uniform grid, a number of other algorithms exist, including straightforward quadrature, methods based on the projection-slice theorem, and methods using the asymptotic expansion of Bessel functions.

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Some Hankel transform pairs

Athanasios Papoulis (1981). 9780898743586, Krieger Publishing Company. ISBN 9780898743586

$1$	$\backslash frac\{\backslash delta(k)\}\{k\}$
$\backslash frac\{1\}\{r\}$	$\backslash frac\{1\}\{k\}$
$r$	$-\backslash frac\{1\}\{k^3\}$
$r^3$	$\backslash frac\{9\}\{k^5\}$
$r^m$
$\backslash frac\{1\}\{\backslash sqrt\{r^2\; +\; z^2\backslash ,\}\}$

\,}{k} |- | $\backslash frac\{1\}\{\backslash ,\; z^2\; +\; r^2\; \backslash ,\}$ | $K\_0(kz),\; \backslash quad\; z\; \backslash in\; \backslash mathbb\{C\}$ |- |rowspan="2"| $\backslash frac\{e^\{iar\}\}\{r\}$ | |- | $\backslash frac\{1\}\{\backslash ,\backslash sqrt\{k^2\; -\; a^2\backslash ,\}\backslash ,\},\; \backslash quad\; a\; >\; 0,\; \backslash ;\; k\; >\; a$ |- | $e^\{-\backslash frac\{1\}\{2\}\; a^2r^2\}$ | $\backslash frac\{1\}\{\backslash ,a^2\backslash ,\}\; \backslash ,\; e^\{-\backslash tfrac\{k^2\}\{2\backslash ,a^2\}\}$ |- | $\backslash frac\{1\}\{r\}\; J\_0(lr)\; \backslash ,\; e^\{-sr\}$ | $\backslash frac\{2\}\{\backslash ,\; \backslash pi\; \backslash sqrt\{\; (k\; +\; l)^2\; +\; s^2\; \backslash ,\}\; \backslash ,\}\; K\backslash left(\; \backslash sqrt\{\backslash frac\{4kl\}\{(k\; +\; l)^2\; +\; s^2\}\; \backslash ,\}\; \backslash right)$ |- | $-r^2\; f(r)$ | $\backslash frac\{\backslash ,\; \backslash mathrm\{d\}^2\; F\_0\; \backslash ,\}\{\backslash mathrm\{d\}k^2\}\; +\; \backslash frac\{1\}\{k\}\; \backslash frac\{\backslash ,\; \backslash mathrm\{d\}\; F\_0\; \backslash ,\}\{\backslash mathrm\{d\}k\}$ |}

$r^s$	$\backslash frac\{2^\{s+1\}\}\{\backslash ,\; k^\{s+2\}\; \backslash ,\}\; \backslash ,\; \backslash frac\{\backslash Gamma\backslash left(\backslash tfrac\{1\}\{2\}(2\; +\; \backslash nu\; +\; s)\backslash right)\}\{\backslash Gamma(\backslash tfrac\{1\}\{2\}\; (\backslash nu\; -\; s))\}$
$r^\{\backslash nu-2s\}\; \backslash Gamma(s,\; r^2\; h)$	$\backslash tfrac\{1\}\{2\}\; \backslash left(\backslash tfrac\; k\; 2\backslash right)^\{2s-\backslash nu-2\}\; \backslash gamma\backslash left(1\; -\; s\; +\; \backslash nu,\; \backslash tfrac\{k^2\}\{4h\}\; \backslash right)$
$e^\{-r^2\}\; r^\backslash nu\; \backslash ,\; U(a,\; b,\; r^2)$	$\backslash frac\{\backslash Gamma(2\; +\; \backslash nu\; -\; b)\}\{\backslash ,\; 2\backslash ,\; \backslash Gamma(2\; +\; \backslash nu\; -\; b\; +\; a)\}\; \backslash left(\backslash tfrac\; k\; 2\backslash right)^\backslash nu\; \backslash ,\; e^\{-\backslash frac\{k^2\}\{4\}\; \backslash ,\}\; \backslash ,\; \_1F\_1\backslash left(\; a,\; 2\; +\; a\; -\; b\; +\; \backslash nu,\; \backslash tfrac\{k^2\}\{4\}\; \backslash right)$
$r^n\; J\_\backslash mu(lr)\; \backslash ,\; e^\{-sr\}$	Expressable in terms of elliptic integrals.
$-r^2\; f(r)$	$\backslash frac\{\backslash mathrm\{d\}^2\; F\_\backslash nu\}\{\backslash mathrm\{d\}k^2\}\; +\; \backslash frac\{1\}\{k\}\; \backslash frac\{\backslash ,\; \backslash mathrm\{d\}\; F\_\backslash nu\; \backslash ,\}\{\backslash mathrm\{d\}k\}\; -\; \backslash frac\{\backslash nu^2\}\{k^2\}\; \backslash ,\; F\_\backslash nu$

is a modified Bessel function of the second kind.
is the complete elliptic integral of the first kind.
     

The expression

$\backslash frac\{\backslash ,\; \backslash mathrm\{d\}^2\; F\_0\; \backslash ,\}\{\backslash mathrm\{d\}k^2\}\; +\; \backslash frac\{1\}\{k\}\; \backslash frac\{\backslash ,\; \backslash mathrm\{d\}\; F\_0\; \backslash ,\}\{\backslash mathrm\{d\}k\}$

coincides with the expression for the Laplace operator in polar coordinates applied to a spherically symmetric function

The Hankel transform of Zernike polynomials are essentially Bessel Functions (Noll 1976):

$R\_n^m(r)\; =\; (-1)^\{\backslash frac\{n-m\}\{2\}\}\; \backslash int\_0^\backslash infty\; J\_\{n+1\}(k)\; J\_m(kr)\; \backslash ,\backslash mathrm\{d\}k$

for even .

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See also

• Fourier transform
• Integral transform
• Abel transform
• Fourier–Bessel series
• Neumann polynomial
• Y and H transforms

• Jack D. Gaskill (1978). 9780471292883, John Wiley & Sons. ISBN 9780471292883
• (1998). 9780849328763, CRC Press. ISBN 9780849328763

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