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Hankel transform

The Hankel transform, also known as the Bessel transform or Fourier-Bessel transform, is an integral transform that generalizes the Fourier transform for functions exhibiting radial or cylindrical symmetry in one or more dimensions.^[1] It is defined for a function f(r) and order \nu (typically \nu \geq -1/2) as g(y) = \int_0^\infty f(r) J_\nu(yr) r \, dr, where J_\nu denotes the Bessel function of the first kind.^[1] This transform is self-inverse under suitable conditions on f, meaning the original function can be recovered via the same formula applied to g, i.e., f(y) = \int_0^\infty g(r) J_\nu(yr) r \, dr.^[1] It arises naturally from the Fourier transform of radially symmetric functions in higher dimensions, where the angular parts integrate to yield Bessel kernels.^[2] Introduced by the German mathematician Hermann Hankel (1839–1873) in 1869, the transform was developed as part of his broader contributions to analysis, particularly in handling functions dependent on distance from the origin.^[3] Hankel's work built on earlier studies of Bessel functions, which solve certain differential equations relevant to radial problems, and the transform has since been formalized in treatises on special functions.^[4] Key properties include linearity, scaling relations (e.g., H_\nu \{f(ar)\} = a^{-2} H_\nu \{f\}(y/a) for a > 0), and Parseval's theorem, which preserves inner products: \int_0^\infty r f(r) g(r) \, dr = \int_0^\infty y \tilde{f}(y) \tilde{g}(y) \, dy.^[4] These ensure its utility in L^2 spaces with appropriate weights, extending to modified versions for different measures.^[5] The Hankel transform finds extensive applications in physics and engineering for solving partial differential equations with axial symmetry, such as the heat equation, wave equation, and Laplace's equation in cylindrical coordinates.^[4] For instance, it simplifies the analysis of vibrations in circular membranes, temperature distributions in disks, acoustic radiation from cylindrical sources, and electromagnetic field propagation. In optics and imaging, it models beam propagation and wave diffraction in radially symmetric media, while in quantum mechanics, it relates to representations involving Bessel functions.^[6] Numerical implementations, including discrete versions, further enable computations in signal processing and inverse problems.^[7]

Introduction and History

Historical Development

The Hankel transform was introduced by the German mathematician Hermann Hankel in 1869 as a generalization of the Fourier transform applicable to radially symmetric functions, in his paper "Die Cylinderfunctionen erster und zweiter Art" published in Mathematische Annalen.^[8]^[9] Hankel, who served as a professor of mathematics at the universities of Erlangen and Tübingen, advanced several areas of mathematical analysis, including the theory of complex variables and the study of Bessel functions of the third kind—now known as Hankel functions—through a series of papers published in Mathematische Annalen.^[10] His work on integral transforms built upon contemporary developments in function theory, emphasizing representations suitable for cylindrical and spherical geometries.^[10] The transform subsequently evolved into what is commonly referred to as the Fourier-Bessel transform, gaining traction for its utility in addressing radial problems in potential theory, such as boundary value problems involving Laplace's equation in cylindrical coordinates.^[11] Post-Hankel developments in the early 20th century, particularly by G. N. Watson, provided a rigorous formalization of the transform within the comprehensive framework of Bessel function theory in his 1922 treatise, solidifying its role in analytical mathematics.^[12]

Overview and Motivation

The Hankel transform is an integral transform that expresses radially symmetric functions via an integral involving Bessel functions of the first kind as the kernel, providing a means to analyze functions defined on the nonnegative real line. This approach leverages the orthogonality properties of these special functions to represent the input in a transformed domain, facilitating the study of phenomena where dependence on distance from an axis or center is predominant.^[13]^[14] The primary motivation for the Hankel transform arises in problems exhibiting cylindrical or spherical symmetry, such as those in wave propagation, heat conduction, or quantum mechanics in circular geometries, where traditional Cartesian methods become cumbersome. By exploiting radial invariance, it reduces multidimensional integrals—often encountered in such symmetric settings—to more manageable one-dimensional forms, simplifying the solution of partial differential equations and boundary value problems. This dimensional reduction is particularly valuable in mathematical physics, enabling efficient handling of axisymmetric distributions without loss of essential structural information.^[14]^[13] In comparison to the Fourier transform, which decomposes general functions using exponential or trigonometric kernels suited to linear or periodic structures, the Hankel transform specializes in capturing radial dependence through Bessel functions, making it indispensable for circular or spherical domains. While the Fourier transform excels in rectangular coordinates and infinite extents, the Hankel variant addresses the natural basis for rotationally invariant systems, such as vibrations in circular membranes or electromagnetic fields in cylindrical waveguides, thus complementing rather than replacing its more general counterpart.^[13]^[14]

Definition

Standard Form

The standard form of the Hankel transform of order \nu for a function f(r) is given by the integral

H_\nu\{f\}(k) = \int_0^\infty f(r) J_\nu(kr) r \, dr,

where J_\nu denotes the Bessel function of the first kind of order \nu.^[15] The inverse Hankel transform, which recovers the original function, is

f(r) = \int_0^\infty H_\nu\{f\}(k) J_\nu(kr) k \, dk.[15]

In this formulation, r and k serve as radial variables in the spatial and spectral domains, respectively, while \nu specifies the order of the transform and is generally a real number with \nu \geq -1/2; integer values of \nu (such as 0 or 1) are common in physical applications involving cylindrical or spherical symmetry, like wave propagation or diffraction.^[16] Convergence of these integrals requires appropriate conditions on f(r), such as square-integrability in the weighted space L^2((0,\infty), r \, dr) or absolute integrability \int_0^\infty |f(r)| r \, dr < \infty for \nu \geq -1/2, ensuring the transform pair is well-defined.^[13]

Domains of Definition and Variants

The Hankel transform is primarily defined on the Hilbert space L^2((0, \infty); r \, dr), consisting of measurable functions f: (0, \infty) \to \mathbb{C} such that \|f\|^2 = \int_0^\infty |f(r)|^2 r \, dr < \infty. This weighted L^2 space with measure r \, dr ensures that the transform acts as a unitary operator, preserving the norm and facilitating inversion.^[17] Note that different sources use slightly varying normalizations for the Hankel transform; for example, the Digital Library of Mathematical Functions (DLMF) employs the self-inverse form \int_0^\infty f(x) J_\nu(xy) (xy)^{1/2} \, dx, as used in the article introduction.^[13] More specialized Hankel spaces, such as the Bessel-potential spaces B^s(\nu)((0, \infty); \mathbb{C}) for order \nu and smoothness parameter s \geq 0, are constructed as subspaces of L^2((0, \infty); r \, dr) where (1 + k^2)^{s/2} \tilde{f}(k) \in L^2((0, \infty); k \, dk), with \tilde{f} denoting the self-inverse Hankel transform; the norm is then \|f\|_{B^s(\nu)}^2 = \int_0^\infty (1 + k^2)^s |\tilde{f}(k)|^2 k \, dk.^[17] These spaces capture radial functions with controlled decay and smoothness, extending the basic L^2 framework for applications requiring higher regularity. For functions with enhanced smoothness and rapid decay at infinity, the transform is defined on the radial Schwartz space \mathcal{S}(\nu)((0, \infty); \mathbb{C}), the space of infinitely differentiable functions f on (0, \infty) such that \sup_{r > 0} r^{m+1} | \partial_r^m f(r) | < \infty for all integers m \geq 0. This space is dense in the Hankel spaces B^s(\nu), and the self-inverse transform maps it bijectively onto itself, enabling analytic extensions and asymptotic analysis.^[17] An alternative self-inverse formulation of the Hankel transform employs the kernel \sqrt{kr} J_\nu(kr) in the integral \tilde{f}(k) = \int_0^\infty \sqrt{kr} J_\nu(kr) f(r) \, dr, which renders the transform involutory (its own inverse) on suitable dense subspaces like the Schwartz space, differing from the standard form by normalization factors that ensure unitarity. The finite Hankel transform variant restricts the domain to the bounded interval [0, a] for a > 0, defined as H_m = \int_0^a r f(r) J_\nu(j_{m,\nu} r / a) \, dr, where j_{m,\nu} are the positive zeros of the Bessel function J_\nu, ordered increasingly; this discrete analog diagonalizes the Bessel differential operator on [0, a] with boundary conditions f(a) = 0.^[18] Recent q-extensions, such as finite q-Hankel transforms based on big q-Bessel functions, provide discrete analogs for q-deformed settings, with applications in q-harmonic analysis and special function theory.^[19] The case of order \nu = 0 applies specifically to radially symmetric functions in two-dimensional cylindrical coordinates, reducing the multidimensional Fourier transform to a one-dimensional integral suitable for axisymmetric problems like wave propagation in circular domains.^[20]

Mathematical Properties

Orthogonality

The orthogonality of the kernel functions underlying the Hankel transform arises from the continuous spectrum of Bessel functions of the first kind J_\nu, which form an orthogonal basis for suitable radial functions on [0, \infty). Specifically, for fixed order \nu with \operatorname{Re} \nu > -1/2 and wavenumbers k, k' > 0, the functions r \mapsto \sqrt{k r} J_\nu(k r) are orthogonal with respect to the inner product \langle f, g \rangle = \int_0^\infty f(r) g(r) \, dr, yielding the relation

\int_0^\infty J_\nu(k r) J_\nu(k' r) r \, dr = \frac{\delta(k - k')}{k},

where \delta is the Dirac delta function. This holds in the sense of distributions and under suitable decay conditions on the functions to ensure convergence.^[21] The closure property, or completeness of this basis, guarantees that the inverse Hankel transform reconstructs the original function f(r). For a function f(r) in the appropriate space (e.g., L^1(0,\infty) with suitable weight), the forward and inverse transforms satisfy f(r) = \int_0^\infty k \, dk \, \tilde{f}(k) J_\nu(k r), where \tilde{f}(k) is the Hankel transform of f. This follows directly from substituting the forward transform into the inverse and applying the orthogonality relation, which collapses the double integral to \int_0^\infty k \, dk \, \tilde{f}(k) J_\nu(k r) = f(r) via the delta function sifting property.^[21] A proof sketch of the orthogonality relation can be obtained using the Weber–Schafheitlin discontinuous integral, a general formula for \int_0^\infty t^{-\lambda} J_\mu(a t) J_\nu(b t) \, dt with $0 < \operatorname{Re} \lambda < 1 + 2 \operatorname{Re} \min(\mu, \nu). In the special case \mu = \nu, \lambda = 1, and a = k, b = k', the integral evaluates to a form that, in the limit as parameters approach the boundary of convergence, produces the delta function distribution when k = k' and zero otherwise. Alternatively, generating function approaches expand the Bessel functions via their integral representations (e.g., J_\nu(z) = \frac{1}{2\pi} \int_0^{2\pi} e^{i (z \sin \theta - \nu \theta)} \, d\theta) and evaluate the resulting oscillatory integrals to derive the delta function via Fourier analysis. A rigorous non-circular proof avoiding reliance on the transform inversion is provided using asymptotic analysis and regularization techniques.^[13] This orthogonality enables the expansion of radial functions f(r) in continuous Bessel series \tilde{f}(k) = \int_0^\infty f(r) J_\nu(k r) r \, dr, analogous to Fourier series but for unbounded domains, with coefficients given by projections onto the basis. Such expansions are particularly useful for solving radial differential equations in cylindrical or spherical symmetry, where the eigenfunctions J_\nu(k r) diagonalize the radial Laplacian.^[21]

Plancherel and Parseval Theorems

The Plancherel theorem establishes that the Hankel transform is an isometry on the appropriate L² space, preserving the norm of functions under the radial measure. Specifically, for a function f in the suitable domain and its Hankel transform H_\nu\{f\}(k) = \int_0^\infty f(r) J_\nu(kr) r \, dr, the theorem asserts that \int_0^\infty |f(r)|^2 r \, dr = \int_0^\infty |H_\nu\{f\}(k)|^2 k \, dk.^[22] This identity reflects the energy conservation property of the transform, analogous to its Fourier counterpart, and holds for orders \nu \geq -1/2 where the transform is well-defined on L^2((0,\infty), r \, dr).^[23] The Parseval theorem extends this isometry to inner products, stating that for functions f and g in the domain, \int_0^\infty f(r) \overline{g(r)} r \, dr = \int_0^\infty H_\nu\{f\}(k) \overline{H_\nu\{g\}(k)} k \, dk.^[24] This relation underscores the transform's unitarity, enabling the transfer of orthogonality and completeness properties between the spatial and transform domains.^[25] The proofs of these theorems rely on the orthogonality relations of Bessel functions, as established in prior analyses, combined with Fourier-type arguments involving the inversion formula and density of smooth, compactly supported functions in the L² space.^[24] One approaches the result by verifying the identity on a dense subspace and extending via continuity, leveraging the fact that the Hankel kernel induces a resolution of the identity similar to the Fourier exponential.^[23] In the hypergroup framework, the Plancherel measure confirms the isomorphism explicitly.^[25] These theorems find application in deriving Parseval identities for Bessel function expansions, where the continuous Hankel transform serves as the limit of discrete Fourier-Bessel series on finite intervals.^[24] For instance, in the expansion of a function as \sum c_n J_\nu(j_{\nu,n} r / a) over [0, a], the Parseval relation \int_0^a |f(r)|^2 r \, dr = \sum_n |c_n|^2 \frac{a^2}{2} J_{\nu+1}^2(j_{\nu,n}) emerges from the transform's L² preservation as the interval extends to infinity.^[22] This connection facilitates norm computations in series solutions to boundary value problems involving cylindrical symmetry.^[23]

Relations to Other Transforms

Connection to the Fourier Transform

The Hankel transform provides a fundamental connection to the Fourier transform when applied to radially symmetric functions. For a function f(\mathbf{x}) in \mathbb{R}^n that depends only on the radial distance r = |\mathbf{x}|, so f(\mathbf{x}) = f(r), the n-dimensional Fourier transform simplifies to a one-dimensional integral involving a Bessel function, which is precisely the Hankel transform of order \nu = n/2 - 1. This reduction occurs because the rotational symmetry eliminates angular dependence in the transform.^[2] The explicit relation, assuming the Fourier transform is defined as \hat{f}(\mathbf{k}) = \int_{\mathbb{R}^n} f(\mathbf{x}) e^{-i \mathbf{k} \cdot \mathbf{x}} \, d^n \mathbf{x}, is given by

\hat{f}(|\mathbf{k}|) = \frac{(2\pi)^{n/2}}{|\mathbf{k}|^{n/2 - 1}} H_{n/2 - 1}\{f(r) r^{n/2 - 1}\}(|\mathbf{k}|),

where the Hankel transform of order \nu is H_\nu\{g\}(\rho) = \int_0^\infty g(r) J_\nu(\rho r) r \, dr, and J_\nu denotes the Bessel function of the first kind. This formula holds under suitable decay conditions on f to ensure convergence. For instance, in two dimensions (n=2, \nu=0), it yields the zeroth-order Hankel transform directly, while in three dimensions (n=3, \nu=1/2), the Bessel function simplifies to a spherical form involving sine.^[2]^[26] The derivation proceeds by expressing the Fourier integral in hyperspherical coordinates, where d^n \mathbf{x} = r^{n-1} \, dr \, d\sigma with d\sigma the surface measure on the unit sphere S^{n-1}. The phase factor e^{-i \mathbf{k} \cdot \mathbf{x}} = e^{-i |\mathbf{k}| r \cos \theta} depends only on the polar angle \theta between \mathbf{k} and \mathbf{x}. Integrating over the angular variables first gives

\int_{S^{n-1}} e^{-i |\mathbf{k}| r \cos \theta} \, d\sigma = (2\pi)^{n/2} \frac{J_{n/2 - 1}(|\mathbf{k}| r)}{(|\mathbf{k}| r)^{n/2 - 1}},

multiplied by the surface area of the sphere. Substituting back yields the Hankel form after reparameterization. This angular integration leverages the zonal spherical harmonic expansion or generating function properties of Bessel functions.^[2]^[26] A reciprocal perspective views the Hankel transform as a Fourier-like transform in the radial variable, but with respect to the measure incorporating the Bessel kernel and radial weight r \, dr. Specifically, the Hankel transform pair can be interpreted as the Fourier transform on the space of radial functions equipped with the inner product \langle f, g \rangle = \int_0^\infty f(r) \overline{g(r)} r \, dr, where the "Fourier" kernel is J_\nu(k r). This structure preserves the self-adjoint and unitary properties analogous to the standard Fourier transform, facilitating inversion and Plancherel-type identities in the radial setting.^[2]

Relation to the Abel Transform

The Abel transform arises as a projection integral for radially symmetric functions, commonly defined for a function f(r) as

\mathcal{A}\{f\}(p) = 2 \int_p^\infty \frac{f(r) \, dr}{\sqrt{r^2 - p^2}},

where p \geq 0 represents the projection coordinate along a line at distance p from the origin.^[27] This form captures line integrals through a circularly symmetric object, analogous to the Radon transform restricted to radial symmetry.^[28] The zeroth-order Hankel transform relates to the Abel transform through a compositional identity involving the one-dimensional Fourier transform. Specifically, the Fourier transform of the Abel transform of f(r) equals the Hankel transform of order zero of f(r):

\mathcal{F}\{\mathcal{A}\{f\}\}(q) = \int_{-\infty}^\infty \mathcal{A}\{f\}(p) e^{-i 2\pi q p} \, dp = 2\pi \int_0^\infty f(r) J_0(2\pi q r) r \, dr = \mathcal{H}_0\{f\}(q),

where J_0 is the zeroth-order Bessel function of the first kind and \mathcal{F} denotes the Fourier transform.^[28] This connection, part of the Fourier-Hankel-Abel cycle, stems from the radial symmetry and allows the Hankel transform to be computed via Abel projection followed by Fourier transformation.^[27] Inversion formulas further link them, as the inverse Abel transform can be expressed using the inverse Hankel transform after Fourier processing.^[28] A key identity expresses certain Hankel transforms in terms of Abel inverses, particularly useful in computed tomography for reconstructing radial distributions from projections. For an Abel projection I(z), the original function e(r) satisfies

e(r) = 2\pi \int_0^\infty \mathcal{F}\{I\}(q) J_0(2\pi q r) q \, dq,

which is the inverse zeroth-order Hankel transform of the Fourier-transformed projection; this avoids direct evaluation of singular integrals in the standard Abel inversion.^[27] Such relations facilitate efficient reconstruction in applications like plasma diagnostics and optical tomography, where radial symmetry simplifies the inverse problem.^[27] Historically, both transforms emerged in the context of solving integral equations with singular kernels. The Abel transform originated from Niels Henrik Abel's 1826 work on tautochrone problems, formulating the first known integral equation of the form \int_y^\infty f(x) (x - y)^{-1/2} \, dx = g(y).^[29] The Hankel transform, developed by Hermann Hankel in 1869, addressed similar radial integral equations in potential theory, sharing applications in inverting Abel-type equations for axially symmetric problems.^[29] This overlap has persisted in Mellin convolution representations, where the composition of Hankel and Abel operators aligns with multiplicative structures in the Mellin domain for solving fractional integral equations.^[29]

Multidimensional Fourier Cases

In the two-dimensional case, the Fourier transform of a radially symmetric function f(\mathbf{x}) = f(r) with r = |\mathbf{x}| reduces to \hat{f}(\mathbf{k}) = 2\pi H_0 \{f\}(k), where k = |\mathbf{k}| and the zeroth-order Hankel transform is H_0 \{f\}(k) = \int_0^\infty f(r) J_0(kr) r \, dr.^[2] In three dimensions, the Fourier transform of a spherically symmetric function f(\mathbf{x}) = f(r) is \hat{f}(\mathbf{k}) = \frac{4\pi}{k} \int_0^\infty f(r) r \sin(kr) \, dr, where k = |\mathbf{k}|.^[2] More generally, in d dimensions, the Fourier transform of a radially symmetric function involves the Hankel transform of order \nu = d/2 - 1, such that the radial part of the Fourier transform is expressed as \hat{F}_d(s) = (2\pi)^{d/2} s^{-\nu} \int_0^\infty J_\nu(s r) r^{\nu} F(r) r \, dr, with s = |\mathbf{k}| and r = |\mathbf{x}|.^[30]^[2] For functions supported within a disk of finite radius a (i.e., f(r) = 0 for r > a), the infinite Hankel transform integral truncates to a finite domain [0, a], leading to the finite Hankel transform H_{\nu,m} \{f\}(k_m) = \int_0^a f(r) J_\nu(k_m r) r \, dr, where k_m are chosen based on boundary conditions, such as the zeros of the Bessel function J_\nu(k_m a) = 0; this form facilitates series expansions and inversion for bounded radial problems.

Applications

Transforming Laplace's Equation

Laplace's equation, \nabla^2 u = 0, in cylindrical coordinates (r, \phi, z) simplifies under the assumption of azimuthal symmetry, where the solution u(r, z) is independent of \phi. In this case, the equation reduces to

\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{\partial^2 u}{\partial z^2} = 0.

This form is particularly amenable to the Hankel transform of order zero, which acts on the radial variable r.^[31] Applying the Hankel transform \tilde{u}(k, z) = \int_0^\infty r \, u(r, z) \, J_0(kr) \, dr, where J_0 is the Bessel function of the first kind of order zero, transforms the radial part of the Laplacian into -k^2 \tilde{u}(k, z). The partial differential equation thereby converts to the ordinary differential equation

\frac{\partial^2 \tilde{u}}{\partial z^2} - k^2 \tilde{u} = 0.

The general solution is \tilde{u}(k, z) = A(k) e^{k z} + B(k) e^{-k z}. For boundedness in the half-space z > 0, the term e^{k z} is discarded, yielding \tilde{u}(k, z) = A(k) e^{-k z}. In the symmetric case across z = 0, the solution takes the form \tilde{u}(k, z) = A(k) e^{-k |z|}.^[32]^[21] The original solution is recovered via the inverse Hankel transform:

u(r, z) = \int_0^\infty k \, \tilde{u}(k, z) \, J_0(kr) \, dk = \int_0^\infty k \, A(k) \, e^{-k z} \, J_0(kr) \, dk.

The coefficient A(k) is determined from boundary conditions at z = 0, typically u(r, 0) = f(r), giving A(k) = \int_0^\infty r \, f(r) \, J_0(kr) \, dr. For mixed boundary conditions, such as those involving a conducting disk of radius a at potential V_0 for r < a and zero normal derivative (insulated) for r > a at z = 0, the problem leads to a pair of dual integral equations solved using the Hankel transform to find A(k). This approach yields the potential outside the disk as an integral representation ensuring satisfaction of Laplace's equation and the specified boundaries.^[32]^[21]

Uses in Physics and Engineering

In optics, the Hankel transform facilitates the analysis of beam propagation through circular apertures and the evaluation of diffraction integrals, particularly for axially symmetric systems. It enables the decomposition of electromagnetic fields into radial components, allowing efficient modeling of light propagation in lenses and apertures. Recent advancements, such as the fast Hankel transform with high accuracy (FHATHA), have improved numerical simulations for ultrafast optics, enhancing precision in predicting beam behavior under complex conditions. Additionally, in 2024 studies on Bessel beam propagation, the Hankel transform has demonstrated superior accuracy over the fast Fourier transform for axial intensity predictions in non-diffracting beams.^[33]^[34] In acoustics, the Hankel transform solves radial wave equations for cylindrical or spherical symmetry, modeling sound propagation in waveguides and radiation from multipole sources. It provides a framework for matched-mode processing in ocean acoustics, estimating source localization and modal wavenumbers from horizontal wavenumber spectra. In quantum mechanics, the Hankel transform addresses the radial Schrödinger equation for central potentials, reducing the problem to a one-dimensional form and facilitating the transition to momentum space representations. This approach exploits the transform's ability to handle Bessel function solutions, aiding in the analysis of bound states and scattering in spherically symmetric systems. Fast Hankel algorithms further enable time-dependent simulations, preserving radial symmetry while minimizing computational dimensionality. In imaging and tomography, the Hankel transform inverts the circular Radon transform, reconstructing images from projections in circular scan geometries, as applied in synthetic aperture radar (SAR) for high-resolution terrain mapping. It supports efficient Fourier-Hankel inversions for band-limited data, enhancing reconstruction stability in limited-angle tomography. In MRI and radar signal processing, these inversions process radial k-space data and circularly symmetric echoes, improving artifact reduction and resolution in rotational scans. Recent developments include q-Hankel transforms derived from big q-Bessel functions, with 2025 studies exploring their finite variants and zero distributions for applications in deformed quantum systems. Bicomplex Hankel transforms, introduced in 2025 analyses, extend the framework to hyperbolic domains, enabling n-dimensional evaluations with convergence properties suited for multi-variable partial differential equations in advanced signal processing.^[19]^[35]

Numerical Evaluation

Computational Methods

Computing the Hankel transform numerically presents several challenges due to its integral form over the semi-infinite interval [0, ∞), the oscillatory nature of the Bessel functions J_\nu(kr), potential singularities in the integrand at r = 0, and the need for proper handling of the function's decay at infinity to ensure convergence.^[36]^[31] These issues can lead to slow convergence or inaccuracies in standard numerical integration techniques unless adapted appropriately.^[37] Quadrature methods form a foundational approach for numerical evaluation, adapting rules like the trapezoidal or Gaussian quadrature to accommodate the infinite domain and Bessel-induced oscillations. For the infinite interval, transformations such as exponential substitutions (e.g., r = e^{-t}) map [0, ∞) to a finite interval, allowing application of composite trapezoidal rules with weights adjusted for the Jacobian.^[38] Gaussian quadrature, particularly variants using Gauss-Laguerre nodes, is effective for functions with exponential decay, where the nodes and weights are precomputed to integrate against e^{-r} over [0, ∞), and the Bessel factor is incorporated via the integrand.^[39] For oscillatory cases, specialized rules like Filon quadrature interpolate the non-oscillatory part of the integrand (e.g., f(r) r) with polynomials and analytically integrate against the Bessel function, achieving high accuracy for rapidly oscillating kernels at moderate computational cost.^[40] These methods typically require careful truncation of the upper limit based on the function's decay and handling of the near-zero singularity through series expansion of J_\nu(kr) or subtraction techniques.^[41] Series expansions provide an alternative approximation strategy, representing the radial function f(r) in bases orthogonal over [0, ∞), such as Laguerre polynomials, to simplify the transform computation. The Laguerre polynomial expansion f(r) = \sum_{n=0}^\infty c_n L_n(r) e^{-r/2} (or generalized variants) allows the Hankel transform to be expressed as a series of known transform pairs involving Bessel-Laguerre integrals, which can be evaluated term-by-term using pre-tabulated coefficients or recursive formulas.^[42] Discrete Laguerre polynomials offer a finite-sum approximation for even functions with strong decay, reducing the integral to a discrete sum that converges rapidly and avoids direct Bessel evaluations at many points.^[41] Gaussian-Laguerre expansions further leverage sampling at specific nodes, where the transform coefficients are computed via inner products, providing an efficient way to approximate the full transform for radially symmetric functions.^[43] This approach is particularly useful for functions analytic in certain sectors, yielding exponential convergence rates dependent on the expansion's truncation level. A recent advancement employs the sinc quadrature rule following an exponential substitution to address both decay and oscillations robustly. The method transforms the variable via x = \frac{\tau}{\omega} \phi(t - q), where \phi(\xi) = \frac{\xi}{1 - e^{-\xi}} is a single-exponential map, \tau = \frac{\pi}{h}, and q = \frac{\pi}{4\tau}(1 - 2\nu), converting the Hankel integral into a form amenable to sinc approximation on a balanced finite grid.^[44] Error analysis shows discretization error bounded by \mathcal{E}_D \sim 4\pi |\tilde{\rho}| e^{-\frac{2\pi d}{h}}, with truncation errors decaying exponentially (e.g., e^{-\sqrt{2\pi(\nu+2)dM}} on the left and e^{-\bar{\beta}N^k} on the right for suitable parameters), achieving overall exponential convergence like e^{-c \sqrt{m}} for exponentially decaying f.^[44] This technique excels for high-frequency \omega and slowly decaying functions, outperforming traditional quadratures by managing Bessel oscillations without specialized oscillation-handling rules.^[44]

Fast Hankel Transform Algorithms

The fast Hankel transform (FHT) was introduced by Siegman in 1977 as a quasi-fast algorithm approximating the Hankel transform through log-periodic sampling of the radial coordinate and mapping to the zeros of Bessel functions, enabling efficient numerical evaluation akin to the fast Fourier transform (FFT). This method leverages a change of variables to convert the Hankel integral into a form amenable to FFT acceleration, achieving enhanced speed for radially symmetric functions in optics. Refinements to the FHT have continued, particularly for optical applications; a 2025 tutorial proposes a novel numerical scheme that surpasses the accuracy and efficiency of Siegman's original quasi-fast method and the high-accuracy fast Hankel transform (FHATHA) algorithm by optimizing sampling and interpolation strategies for ultrafast optics simulations.^[33] For axially symmetric electromagnetic (EM) propagation, the fast Hankel transform of nth order (FHTn), developed in 1999,^[45] extends the approach to arbitrary integer orders, facilitating direct computation of field propagation in cylindrical systems without explicit Bessel function evaluations. This method has been applied in recent Optica publications from 2023 onward to model vectorial beam propagation and diffractive optics, emphasizing its utility in high-fidelity simulations of laser beams. These algorithms achieve computational complexity of O(N log N), where N is the number of samples, by embedding the Hankel operation within FFT routines. Implementations of FHT and FHTn are available in scientific libraries, such as SciPy's fht and ifht functions in Python for logarithmically spaced data, and MATLAB File Exchange contributions for custom radial beam simulations.^[46] These tools support applications in beam propagation modeling, where FHT enables scalable prediction of Gaussian beam evolution over long distances with minimal numerical dispersion.^[47]

Examples

Common Transform Pairs

Common Hankel transform pairs include several standard forms that arise frequently in applications involving radial symmetry. These pairs are derived from fundamental integrals involving Bessel functions, often using techniques such as differentiation under the integral sign or connections to the Mellin transform. The following examples focus on the order-zero case where applicable, with the Hankel transform defined as H_\nu \{ f \}(k) = \int_0^\infty f(r) J_\nu(kr) r \, dr, assuming suitable conditions for convergence.^[13] One fundamental pair is the transform of the inverse radial function. For f(r) = 1/r with r > 0, the zero-order Hankel transform is H_0 \{ 1/r \}(k) = 1/k for k > 0, interpreted in the Cauchy principal value sense to handle the improper integral. This result stems directly from the known discontinuous Weber-Schafheitlin integral \int_0^\infty J_0(kr) \, dr = 1/k.^[13] Another important pair involves the exponential decay function. For f(r) = e^{-a r} with \operatorname{Re}(a) > 0, the zero-order Hankel transform is given by

H_0 \{ e^{-a r} \}(k) = \frac{a}{(a^2 + k^2)^{3/2}}.

This can be derived using differentiation under the integral sign. Start with the base integral \int_0^\infty e^{-a r} J_0(kr) \, dr = (a^2 + k^2)^{-1/2}, which holds for \operatorname{Re}(a) > 0. The Hankel transform includes an additional factor of r, equivalent to -\frac{d}{da} applied to the base integral, yielding the stated form after simplification. The base integral follows from the general formula for integrals of the form \int_0^\infty t^{\mu-1} e^{-a t} J_\nu(b t) \, dt by setting \mu = 1, \nu = 0. For power-law functions, the Hankel transform connects closely to the Mellin transform via the representation of Bessel functions. For f(r) = r^\mu, the transform of order \nu is

H_\nu \{ r^\mu \}(k) = 2^{\mu + 1} k^{-\mu - 2} \frac{\Gamma\left( \frac{\nu + \mu + 2}{2} \right)}{\Gamma\left( \frac{\nu - \mu}{2} \right)},

under convergence conditions such as -\operatorname{Re}(\nu + 1) < \operatorname{Re} \mu < -1/2. The k^{-\mu-2} scaling arises from the substitution u = k r in the integral, linking it to the Mellin transform of the Bessel function J_\nu(u). The full expression derives from the integral \int_0^\infty t^{\mu + 1} J_\nu(t) \, dt = 2^{\mu + 1} \frac{\Gamma\left( \frac{\nu + \mu + 2}{2} \right)}{\Gamma\left( \frac{\nu - \mu}{2} \right)}. These pairs can be verified using the Plancherel theorem for the Hankel transform, ensuring energy preservation in the transform domain.

Selected Applications of Pairs

One notable application of the zero-order Hankel transform pair involving the step function arises in modeling diffraction from a circular aperture. The aperture function, represented as a rect(r/a) for radius a, transforms under the zero-order Hankel operator to the sombrero function J_1(ka)/k, where k is the radial spatial frequency. This result describes the amplitude distribution in the far-field diffraction pattern, with the intensity given by the square of this transform, yielding the characteristic Airy disk central bright spot surrounded by concentric rings. This pair is fundamental in optical systems for predicting resolution limits in telescopes and microscopes, where the sombrero function quantifies the point spread function due to diffraction.^[48] The Gaussian function pair under the zero-order Hankel transform exhibits self-similarity, where the transform of exp(-r^2 / 2σ^2) yields σ^2 exp(-k^2 σ^2 / 2), a scaled version of the original function. This property is leveraged in optics to model the propagation of Gaussian beams in radially symmetric systems, such as laser beam focusing and ultrafast pulse propagation, as the self-dual nature preserves beam profile integrity under Fourier-Hankel operations in paraxial approximations. In computational optics, this pair facilitates efficient simulations of beam evolution through lenses and free space, enabling analysis of mode stability in fiber optics and resonator designs. Power-law pairs in the Hankel transform of order ν=1/2 are essential in potential theory for three-dimensional problems, particularly for the 1/r potential common in electrostatics and gravitation. The transform relates the radial potential φ(r) ∝ 1/r to its Fourier counterpart ∝ 1/k^2 in reciprocal space, facilitating solutions to Poisson's equation in cylindrical coordinates via the ν=1/2 kernel, which corresponds to the three-dimensional radial Fourier transform. This application is used to compute field distributions around point charges or masses, with the pair enabling analytical inversion for boundary value problems in multipole expansions.^[49] A recent advancement employs Bessel kernel pairs in the Hankel transform to address Doppler line shapes in nuclear physics cross-section calculations. By reformulating the Doppler broadening kernel as a Hankel transform convolution, this method enhances numerical stability for temperature-dependent neutron interactions, particularly on GPU-accelerated platforms for reactor simulations. The approach, detailed in a 2024 study, improves accuracy in resolving broadened resonances without traditional quadrature instabilities, impacting fuel cycle analysis and criticality safety assessments.^[50]

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