Hankel transform

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 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 formula_1 of a function "f"("r") is given by
where formula_3 is the Bessel function of the first kind of order formula_1 with formula_5. The inverse Hankel transform of is defined as
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
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 formula_8.
Alternative definition.
An alternative definition says that the Hankel transform of "g"("r") is
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. 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, ∞).
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 formula_14. In the axisymmetric case, the partial differential equation is transformed as
which is an ordinary differential equation in the transformed variable formula_16.
Orthogonality.
The Bessel functions form an orthogonal basis with respect to the weighting factor "r":
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
Parseval's theorem, which states
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 formula_20 of a formula_21-dimensional vector . Its formula_21-dimensional Fourier transform is defined asformula_23To rewrite it in hyperspherical coordinates, we can use the decomposition of a plane wave into formula_21-dimensional hyperspherical harmonics formula_25:formula_26where formula_27 and formula_28 are the sets of all hyperspherical angles in the formula_29-space and formula_30-space. This gives the following expression for the formula_21-dimensional Fourier transform in hyperspherical coordinates:formula_32If we expand formula_20 and formula_34 in hyperspherical harmonics:formula_35the Fourier transform in hyperspherical coordinates simplifies toformula_36This 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 formula_37).
Special cases.
Fourier transform in two dimensions.
If a two-dimensional function is expanded in a multipole series,
then its two-dimensional Fourier transform is given byformula_39whereformula_40is the formula_41-th order Hankel transform of formula_42 (in this case formula_41 plays the role of the angular momentum, which was denoted by formula_44 in the previous section).
Fourier transform in three dimensions.
If a three-dimensional function is expanded in a multipole series over spherical harmonics,
then its three-dimensional Fourier transform is given byformula_46whereformula_47is the Hankel transform of formula_48 of order formula_49.
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 byformula_50which is the Hankel transform of formula_51 of order formula_52 up to a factor of formula_53.
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 :
such that the two-dimensional Fourier transform of becomes
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
the Fourier-Chebyshev series coefficients emerge as
Using the re-expansion
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 operator, and 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. 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 variablesformula_60In these new variables, the Hankel transform readsformula_61whereformula_62Now the integral can be calculated numerically with formula_63 complexity using fast Fourier transform. The algorithm can be further simplified by using a known analytical expression for the Fourier transform of formula_64:formula_65The optimal choice of parameters formula_66 depends on the properties of formula_67, in particular its asymptotic behavior at formula_68 and formula_69.
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, formula_67 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.
Some Hankel transform pairs.
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 is a modified Bessel function of the second kind.
 is the complete elliptic integral of the first kind.
The expression
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):
for even .