Hankel transform#Some Hankel transform pairs

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 {{math|F_ν(k)}} 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.

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

An alternative definition says that the Hankel transform of g(r) is{{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]}}

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

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, ∞).

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 $-k^2$.{{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}} 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 $U$.

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

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

If f(r) and g(r) are such that their Hankel transforms {{math|F_ν(k)}} and {{math|G_ν(k)}} 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.

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 {{math|r}}. 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 $Y\_\{l,m\}$:{{Cite book |last=Avery, James Emil |title=Hyperspherical harmonics and their physical applications |isbn=978-981-322-930-3 |oclc=1013827621}}$$e^\{-i\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\}\; =\; (2\; \backslash pi)^\{d/2\}\; (kr)^\{1-d/2\}\backslash sum\_\{l\; =\; 0\}^\{+\backslash infty\}$$

(-i)^{l} J_{d/2-1+l}(kr)\sum_{m}

Y_{l,m}(\Omega_{\mathbf{k}}) Y^{*}_{l,m}(\Omega_{\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:$$F(\backslash mathbf\{k\})\; =\; (2\; \backslash pi)^\{d/2\}\; k^\{1-d/2\}\; \backslash sum\_\{l\; =\; 0\}^\{+\backslash infty\}\; (-i)^\{l\}\; \backslash sum\_\{m\}Y\_\{l,m\}(\backslash Omega\_\{\backslash mathbf\{k\}\})$$

\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}}. 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$$k^\{d/2-1\}F\_\{l,m\}(k)\; =\; (2\; \backslash pi)^\{d/2\}\; (-i)^\{l\}$$

\int_{0}^{+\infty}r^{d/2-1}f_{l,m}(r)J_{d/2-1+l}(kr)r\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 $r^\{d/2-1\}$).

If a two-dimensional function {{math|f(r)}} 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).

If a three-dimensional function {{math|f(r)}} 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$$\backslash sqrt\{k\}\; F\_\{l,m\}(k)\; =$$

\int_{0}^{+\infty}\sqrt{r} f_{l,m}(r)J_{l+1/2}(kr)r\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.

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{{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}}$$k^\{d/2-1\}F(k)\; =\; (2\; \backslash pi)^\{d/2\}$$

\int_{0}^{+\infty}r^{d/2-1}f(r)J_{d/2-1}(kr)r\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 $(2\; \backslash pi)^\{d/2\}$.

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

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

such that the two-dimensional Fourier transform of {{math|f(r)}} 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.{{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}} The expansion coefficients {{math|f_m,t}} are accessible with discrete Fourier transform techniques:{{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 }} 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 {{math|g}} 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 {{math|f_m,t}} expressed as sums of {{math|g_m,j}}.

This is one flavor of fast Hankel transform techniques.

The Hankel transform is one member of the 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

: $FA\; =\; H.$

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.

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{{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}}

$$r\; =\; r\_0\; e^\{-\backslash rho\},\; \backslash quad\; k\; =\; k\_0\; \backslash ,\; e^\{\backslash kappa\}.$$

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 $\backslash tilde\; J\_\backslash nu$:{{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}}

\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.{{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}}

{{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}}

|-

| $\backslash frac\{1\}\{\backslash ,\; z^2\; +\; r^2\; \backslash ,\}$

| $K\_0(kz),\; \backslash quad\; z\; \backslash in\; \backslash mathbb\{C\}$

|-

|rowspan="2"| $\backslash frac\{e^\{iar\}\}\{r\}$

|

|-

| $\backslash frac\{1\}\{\backslash ,\backslash sqrt\{k^2\; -\; a^2\backslash ,\}\backslash ,\},\; \backslash quad\; a\; >\; 0,\; \backslash ;\; k\; >\; a$

|-

| $e^\{-\backslash frac\{1\}\{2\}\; a^2r^2\}$

| $\backslash frac\{1\}\{\backslash ,a^2\backslash ,\}\; \backslash ,\; e^\{-\backslash tfrac\{k^2\}\{2\backslash ,a^2\}\}$

|-

| $\backslash frac\{1\}\{r\}\; J\_0(lr)\; \backslash ,\; e^\{-sr\}$

| $\backslash frac\{2\}\{\backslash ,\; \backslash pi\; \backslash sqrt\{\; (k\; +\; l)^2\; +\; s^2\; \backslash ,\}\; \backslash ,\}\; K\backslash left(\; \backslash sqrt\{\backslash frac\{4kl\}\{(k\; +\; l)^2\; +\; s^2\}\; \backslash ,\}\; \backslash right)$

|-

| $-r^2\; f(r)$

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

|}

{{math|K_n(z)}} is a modified Bessel function of the second kind.

{{math|K(z)}} 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 {{math|( k, θ )}} applied to a spherically symmetric function {{math| F₀(k) .}}

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 {{math|n − m ≥ 0}}.

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

{{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

|last1=Polyanin

|first1=A. D.

|last2=Manzhirov

|first2=A. V.

|title=Handbook of Integral Equations

|publisher=CRC Press

|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=49–67

|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=529–551

| 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=407–412

|year=1967

|doi=10.1090/S0025-5718-67-99149-1

|doi-access=free

}}

• {{cite journal

|first1=Robert F.

|last1=MacKinnon

|title=The asymptotic expansions of Hankel transforms and related integrals

|journal=Mathematics of Computation

|volume=26

|issue=118

|pages=515–527

|year=1972

|doi=10.1090/S0025-5718-1972-0308695-9

| jstor = 2003243|doi-access=free

}}

• {{cite journal

|last1=Linz

|first1=Peter

|last2=Kropp

|first2=T. E.

|title=A note on the computation of integrals involving products of trigonometric and Bessel functions

|journal=Mathematics of Computation

|volume=27

|issue=124

|pages=871–872

|year=1973

| jstor = 2005522

|doi=10.2307/2005522|doi-access=free

}}

• {{cite journal

|first=Robert J

|last=Noll

|title=Zernike polynomials and atmospheric turbulence

|journal=Journal of the Optical Society of America

|volume=66

|issue=3

|year=1976

|pages=207–211

|doi=10.1364/JOSA.66.000207

|bibcode=1976JOSA...66..207N

}}

• {{cite journal

|first1=A. E.

|last1=Siegman

|title=Quasi-fast Hankel transform

|journal=Opt. Lett.

|volume=1

|issue=1

|pages=13–15

|bibcode=1977OptL....1...13S

|doi=10.1364/OL.1.000013

|year=1977

|pmid=19680315

}}

• {{cite journal

|first1=Vittorio

|last1=Magni

|first2=Giulio

|last2=Cerullo

|first3=Sandro

|last3=De Silverstri

|title=High-accuracy fast Hankel transform for optical beam propagation

|journal=J. Opt. Soc. Am. A

|volume=9

|issue=11

|pages=2031–2033

|year=1992

|doi=10.1364/JOSAA.9.002031

|bibcode = 1992JOSAA...9.2031M }}

• {{cite journal

|first1=A.

|last1=Agnesi

|first2=Giancarlo C.

|last2=Reali

|first3=G.

|last3=Patrini

|first4=A.

|last4=Tomaselli

|title=Numerical evaluation of the Hankel transform: remarks

|journal=Journal of the Optical Society of America A

|volume=10

|issue=9

|page=1872

|year=1993

|doi=10.1364/JOSAA.10.001872

|bibcode=1993JOSAA..10.1872A

}}

• {{cite journal

|first1=Richard

|last1=Barakat

|title=Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy

|journal=Applied Mathematics Letters

|volume=9

|issue=5

|pages=21–26

|mr=1415467

|year=1996

|doi=10.1016/0893-9659(96)00067-5

|doi-access=free

}}

• {{cite journal

|first1=José A.

|last1=Ferrari

|first2=Daniel

|last2=Perciante

|first3=Alfredo

|last3=Dubra

|title=Fast Hankel transform of nth order

|journal= J. Opt. Soc. Am. A

|volume=16

|issue=10

|pages=2581–2582

|doi=10.1364/JOSAA.16.002581

|bibcode=1999JOSAA..16.2581F

|year=1999

}}

• {{cite journal

|first1=Thomas

|last1=Wieder

|title=Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL

|journal=ACM Trans. Math. Softw.

|volume=25

|issue=2

|pages=240–250

|year=1999

|doi=10.1145/317275.317284

|doi-access=free

}}

• {{cite journal

|first1=Luc

|last1=Knockaert

|title=Fast Hankel transform by fast sine and cosine transforms: the Mellin connection

|journal=IEEE Trans. Signal Process.

|volume=48

|issue=6

|pages=1695–1701

|year=2000

|hdl=20.500.12860/4476

|doi=10.1109/78.845927

|url=https://scholar.archive.org/work/2klp6tigojbgppayv6tgjxeu5m

|bibcode=2000ITSP...48.1695K

|citeseerx=10.1.1.721.1633

}}

• {{cite journal

|first1=D. W.

|last1=Zhang

|first2=X.-C.

|last2=Yuan

|first3=N. Q.

|last3=Ngo

|first4=P.

|last4=Shum

|title=Fast Hankel transform and its application for studying the propagation of cylindrical electromagnetic fields

|journal=Opt. Express

|volume=10

|issue=12

|pages=521–525

|year=2002

|doi=10.1364/oe.10.000521

|pmid=19436390

|bibcode=2002OExpr..10..521Z

|doi-access=free

}}

• {{cite journal

|first1=Joanne

|last1=Markham

|first2=Jose-Angel

|last2=Conchello

|title=Numerical evaluation of Hankel transforms for oscillating functions

|journal=J. Opt. Soc. Am. A

|volume=20

|issue=4

|pages=621–630

|year=2003

|doi=10.1364/JOSAA.20.000621

|pmid=12683487

|bibcode = 2003JOSAA..20..621M }}

• {{cite journal

|first1=César D.

|last1=Perciante

|first2=José A.

|last2=Ferrari

|title=Fast Hankel transform of nth order with improved performance

|journal=J. Opt. Soc. Am. A

|volume=21

|number=9

|year=2004

|pages=1811–2

|doi=10.1364/JOSAA.21.001811

|pmid=15384449

|bibcode=2004JOSAA..21.1811P

}}

• {{cite journal

|first1=Manuel

|last1=Gizar-Sicairos

|first2=Julio C.

|last2=Guitierrez-Vega

|title=Computation of quasi-discrete Hankel transform of integer order for propagating optical wave fields

|journal=J. Opt. Soc. Am. A

|volume=21

|issue=1

|year=2004

|pages=53–58

|doi=10.1364/JOSAA.21.000053

|pmid=14725397

|bibcode = 2004JOSAA..21...53G }}

• {{cite journal

|first1=Charles

|last1=Cerjan

|title=The Zernike-Bessel representation and its application to Hankel transforms

|journal=J. Opt. Soc. Am. A

|volume=24

|issue=6

|doi=10.1364/JOSAA.24.001609

|pages=1609–1616

|year=2007

|pmid=17491628

|bibcode=2007JOSAA..24.1609C

|url=https://zenodo.org/record/894588

}}

{{div col end}}

{{Authority control}}

Category:Integral transforms