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 Jν(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.
is the Bessel function of the first kind of order
The inverse Hankel transform of Fν(k) is defined as 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
An alternative definition says that the Hankel transform of g(r) is[1] The two definitions are related: This means that, as with the previous definition, the Hankel transform defined this way is also its own inverse: The obvious domain now has the condition but this can be extended.
Under the Hankel transform, the Bessel operator becomes a multiplication by
[2] In the axisymmetric case, the partial differential equation is transformed as where
Therefore, the Laplacian in cylindrical coordinates becomes an ordinary differential equation in the transformed function
The Bessel functions form an orthogonal basis with respect to the weighting factor r:[3] If f(r) and g(r) are such that their Hankel transforms Fν(k) and Gν(k) are well defined, then the Plancherel theorem states Parseval's theorem, which states is a special case of the Plancherel theorem.
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.
To rewrite it in hyperspherical coordinates, we can use the decomposition of a plane wave into
-dimensional Fourier transform in hyperspherical coordinates:
the Fourier transform in hyperspherical coordinates simplifies to
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
plays the role of the angular momentum, which was denoted by
If a three-dimensional function f(r) is expanded in a multipole series over spherical harmonics, then its three-dimensional Fourier transform is given by
If a d-dimensional function f(r) does not depend on angular coordinates, then its d-dimensional Fourier transform F(k) also does not depend on angular coordinates and is given by[5]
If a two-dimensional function f(r) is expanded in a multipole series and the expansion coefficients fm are sufficiently smooth near the origin and zero outside a radius R, the radial part f(r)/rm may be expanded into a power series of 1 − (r/R)^2: such that the two-dimensional Fourier transform of f(r) becomes where the last equality follows from §6.567.1 of.
[6] The expansion coefficients fm,t are accessible with discrete Fourier transform techniques:[7] if the radial distance is scaled with the Fourier-Chebyshev series coefficients g emerge as Using the re-expansion yields fm,t expressed as sums of gm,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 A as the Abel transform operator, F as the Fourier transform operator, and H as the zeroth-order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that 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.
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[8]
The algorithm can be further simplified by using a known analytical expression for the Fourier transform of
Since it is based on fast Fourier transform in logarithmic variables,
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.
[10] [11] Kn(z) is a modified Bessel function of the second kind.
The expression coincides with the expression for the Laplace operator in polar coordinates ( k, θ ) applied to a spherically symmetric function F0(k) .
The Hankel transform of Zernike polynomials are essentially Bessel Functions (Noll 1976): for even n − m ≥ 0.