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In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . 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 In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
and was first developed by the mathematician
Hermann Hankel Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix. Biography Hankel was born on ...
. It is also known as the Fourier–Bessel transform. Just as the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
for an infinite interval is related to the
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
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 \nu of a function ''f''(''r'') is given by : F_\nu(k) = \int_0^\infty f(r) J_\nu(kr) \,r\,\mathrmr, where J_\nu is the Bessel function of the first kind of order \nu with \nu \geq -1/2. The inverse Hankel transform of is defined as : f(r) = \int_0^\infty F_\nu(k) J_\nu(kr) \,k\,\mathrmk, 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, ∞), is piecewise continuous and of bounded variation in every finite subinterval in (0, ∞), and : \int_0^\infty , f(r), \,r^ \,\mathrmr < \infty. 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)^.


Alternative definition

An alternative definition says that the Hankel transform of ''g''(''r'') is : h_\nu(k) = \int_0^\infty g(r) J_\nu(kr) \,\sqrt\,\mathrmr. The two definitions are related: : If g(r) = f(r) \sqrt r, then h_\nu(k) = F_\nu(k) \sqrt k. This means that, as with the previous definition, the Hankel transform defined this way is also its own inverse: : g(r) = \int_0^\infty h_\nu(k) J_\nu(kr) \,\sqrt\,\mathrmk. The obvious domain now has the condition : \int_0^\infty , g(r), \,\mathrmr < \infty, 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 In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpo ...
rather than a
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
), and in this way the Hankel transform and its inverse work for all functions in L2(0, ∞).


Transforming Laplace's equation

The Hankel transform can be used to transform and solve
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...
expressed in cylindrical coordinates. Under the Hankel transform, the Bessel operator becomes a multiplication by -k^2. In the axisymmetric case, the partial differential equation is transformed as : \mathcal_0 \left\ = -k^2 U + \frac U, which is an ordinary differential equation in the transformed variable U.


Orthogonality

The Bessel functions form an
orthogonal basis In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal bas ...
with respect to the weighting factor ''r'': : \int_0^\infty J_\nu(kr) J_\nu(k'r) \,r\,\mathrmr = \frac, \quad k, k' > 0.


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 In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integ ...
states : \int_0^\infty f(r) g(r) \,r\,\mathrmr = \int_0^\infty F_\nu(k) G_\nu(k) \,k\,\mathrmk.
Parseval's theorem In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originate ...
, which states : \int_0^\infty , f(r), ^2 \,r\,\mathrmr = \int_0^\infty , F_\nu(k), ^2 \,k\,\mathrmk, 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 f(\mathbf) of a d-dimensional vector . Its d-dimensional Fourier transform is defined asF(\mathbf) = \int_ f(\mathbf) e^ \,\mathrm\mathbf.To rewrite it in hyperspherical coordinates, we can use the decomposition of a plane wave into d-dimensional hyperspherical harmonics Y_:e^ = (2 \pi)^ (kr)^\sum_^ (-i)^ J_(kr)\sum_ Y_(\Omega_) Y^_(\Omega_),where \Omega_ and \Omega_ are the sets of all hyperspherical angles in the \mathbf-space and \mathbf-space. This gives the following expression for the d-dimensional Fourier transform in hyperspherical coordinates:F(\mathbf) = (2 \pi)^ k^ \sum_^ (-i)^ \sum_Y_(\Omega_) \int_^J_(kr)r^\mathrmr \int f(\mathbf) Y_^(\Omega_) \mathrm\Omega_. If we expand f(\mathbf) and F(\mathbf) in hyperspherical harmonics:f(\mathbf) = \sum_^ \sum_f_(r)Y_(\Omega_),\quad F(\mathbf) = \sum_^ \sum_ F_(k) Y_(\Omega_), the Fourier transform in hyperspherical coordinates simplifies tok^F_(k) = (2 \pi)^ (-i)^ \int_^r^f_(r)J_(kr)r\mathrmr. 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^).


Special cases


Fourier transform in two dimensions

If a two-dimensional function is expanded in a multipole series, :f(r, \theta) = \sum_^\infty f_m(r) e^, then its two-dimensional Fourier transform is given byF(\mathbf k) = 2\pi \sum_m i^ e^ F_m(k),whereF_m(k) = \int_0^\infty f_m(r) J_m(kr) \,r\,\mathrmris 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).


Fourier transform in three dimensions

If a three-dimensional function is expanded in a multipole series over
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...
, :f(r,\theta_,\varphi_) = \sum_^ \sum_^f_(r)Y_(\theta_,\varphi_), then its three-dimensional Fourier transform is given byF(k,\theta_,\varphi_) = (2 \pi)^ \sum_^ (-i)^ \sum_^ F_(k) Y_(\theta_,\varphi_),where\sqrt F_(k) = \int_^\sqrt f_(r)J_(kr)r\mathrmr.is the Hankel transform of \sqrt f_(r) of order (l+1/2). This kind of Hankel transform of half-integer order is also known as the spherical Bessel transform.


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 byk^F(k) = (2 \pi)^ \int_^r^f(r)J_(kr)r\mathrmr.which is the Hankel transform of r^f(r) of order (d/2-1) up to a factor of (2 \pi)^ .


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 \sum_ f_ \left(1 - \left(\tfrac\right)^2 \right)^t, \quad 0 \le r \le R, such that the two-dimensional Fourier transform of becomes :\begin F(\mathbf k) &= 2\pi\sum_m i^ e^ \sum_t f_ \int_0^R r^m \left(1 - \left(\tfrac\right)^2 \right)^t J_m(kr) r\,\mathrmr && \\ &= 2\pi\sum_m i^ e^ R^ \sum_t f_ \int_0^1 x^ (1-x^2)^t J_m(kxR) \,\mathrmx && (x = \tfrac)\\ &= 2\pi\sum_m i^ e^ R^ \sum_t f_ \frac J_(kR), \end where the last equality follows from §6.567.1 of. The expansion coefficients are accessible with
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comp ...
techniques: if the radial distance is scaled with :r/R\equiv \sin\theta,\quad 1-(r/R)^2 = \cos^2\theta, the Fourier-Chebyshev series coefficients emerge as :f(r)\equiv r^m \sum_j g_ \cos(j\theta)= r^m\sum_jg_ T_j(\cos\theta). Using the re-expansion : \cos(j\theta) = 2^\cos^j\theta-\frac2^\cos^\theta +\frac\binom2^\cos^\theta - \frac\binom2^\cos^\theta + \cdots yields expressed as sums of . This is one flavor of fast Hankel transform techniques.


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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
operator, and as the zeroth-order Hankel transform operator, then the special case of the
projection-slice theorem In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: * Take a two-dimensional function ''f''(r), project (e.g. using the ...
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.


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 In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
by a logarithmic change of variables r = r_0 e^, \quad k = k_0 \, e^. In these new variables, the Hankel transform reads \tilde F_\nu(\kappa) = \int_^\infty \tilde f(\rho) \tilde J_\nu(\kappa - \rho) \,\mathrm\rho, where \tilde f(\rho) = \left(r_0 \, e^ \right)^ \, f(r_0 e^), \tilde F_\nu(\kappa) = \left(k_0 \, e^ \right)^ \, F_\nu(k_0 e^\kappa), \tilde J_\nu(\kappa-\rho) = \left(k_0 \, r_0 \, e^ \right)^ \, J_\nu(k_0 r_0 e^). Now the integral can be calculated numerically with O(N \log N) complexity using
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the ...
. The algorithm can be further simplified by using a known analytical expression for the Fourier transform of \tilde J_\nu: \int_^ \tilde J_\nu(x) e^ \,\mathrmx = \frac \, 2^e^. The optimal choice of parameters r_0, k_0, n depends on the properties of f(r), in particular its asymptotic behavior at r \to 0 and r \to \infty. This algorithm is known as the "quasi-fast Hankel transform", or simply "fast Hankel transform". Since it is based on
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the ...
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 In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: * Take a two-dimensional function ''f''(r), project (e.g. using the ...
, and methods using the asymptotic expansion of Bessel functions.


Some Hankel transform pairs

is a
modified Bessel function of the second kind Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
. is the complete elliptic integral of the first kind. The expression : \frac + \frac \frac coincides with the expression for the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
applied to a spherically symmetric function The Hankel transform of
Zernike polynomial In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, winner of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, t ...
s are essentially Bessel Functions (Noll 1976): : R_n^m(r) = (-1)^ \int_0^\infty J_(k) J_m(kr) \,\mathrmk for even .


See also

*
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
*
Integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
* Abel transform * Fourier–Bessel series *
Neumann polynomial In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case \alpha=0, are a sequence of polynomials in 1/t used to expand functions in term of Bessel functions. The first few polynomials are :O_0^(t)=\frac 1 t, :O_1^(t) ...
* Y and H transforms


References

* * * * * * * * * * * * * * * * * * * * * {{Authority control Integral transforms