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Mathematical operation
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
Hankel_transform
Square matrix in which each ascending skew-diagonal from left to right is constant
the approximation is a Hankel matrix, which can be shown with AAK theory. The Hankel matrix transform, or simply Hankel transform, of a sequence b k {\displaystyle
Hankel_matrix
German mathematician (1839–1873)
introducing the Hankel transform and the Hankel matrix. Hankel was born on 14 February 1839 in Halle, Germany. His father, Wilhelm Gottlieb Hankel, was a physicist
Hermann_Hankel
Family of solutions to related differential equations
the Hankel transform of it (of any given order α > −1/2), gε(k), approaches Jα(k) as ε approaches zero, for any given k. Conversely, the Hankel transform
Bessel_function
Integral transform used in various branches of mathematics
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
Abel_transform
Fourier transform Short-time Fourier transform Rectangular mask short-time Fourier transform Fourier–Bros–Iagolnitzer transform Gabor transform Hankel transform
List_of_transforms
Mathematical integral transform
transforms, such as the Hankel transform, this transform involves integrating over the index of the function rather than its argument. The transform of
Kontorovich–Lebedev_transform
Mathematical transform that expresses a function of time as a function of frequency
Fourier transform – Mathematical operation Indirect Fourier transform Integral transform – Mapping involving integration between function spaces Hankel transform –
Fourier_transform
Fourier transform Chirplet transform Fractional Fourier transform (FRFT) Hankel transform: related to the Fourier Transform of radial functions. Fourier–Bros–Iagolnitzer
List of Fourier-related transforms
List_of_Fourier-related_transforms
and applied mathematics, the Hankel, Y, H transforms all may appear in problems having axial symmetry. Hankel transforms are however much more commonly
Y_and_H_transforms
Mathematical explanation of far field diffraction
on ω. In this case, we have U(ρ,z) equal to the Fourier–Bessel or Hankel transform of the aperture function, A(ρ) Here are given examples of Fraunhofer
Fraunhofer diffraction equation
Fraunhofer_diffraction_equation
Generalization of the hypergeometric function
integral transforms like the Hankel transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels
Meijer_G-function
Mapping involving integration between function spaces
In mathematics, an integral transform is a type of transformation that maps a function from its original function space into another function space via
Integral_transform
Transformation of a mathematical sequence
are homomorphisms of the kernel of the Hankel transform of a series. In the case where the binomial transform is defined as ∑ i = 0 n ( − 1 ) n − i (
Binomial_transform
Infinite series of Bessel functions
Fourier transform over an infinite interval, so the Fourier–Bessel series has a counterpart over an infinite interval, namely the Hankel transform. As said
Fourier–Bessel_series
Theorem in mathematics
transform of f(r). The two-dimensional Fourier transform of f(r) will be a circularly symmetric function given by the zeroth-order Hankel transform of
Projection-slice_theorem
Result used in the theory of propagation of waves
the two-dimensional Fourier transform with cylindrical symmetry, i.e., the Hankel transform. It is found by transforming the spherical wave along the
Sommerfeld_identity
Laplace transform Fourier transform Fractional Fourier Transform Linear canonical transformation Wavelet transform Hankel transform Joukowsky transform Mellin
Transform_theory
transform expression which is called as Hankel transform. Hence the Hankel transform is used to compute the Fourier transform of the 2-D window functions. If
Two-dimensional_window_design
Mathematical formula in harmonic analysis
a multiplicative inverse of a modulo c and Ω is a certain integral Hankel transform of ω. (see Good (1984)) Good, Anton (1984), "Cusp forms and eigenfunctions
Voronoi_formula
Polynomial sequence
(2007). "The Zernike-Bessel representation and its application to Hankel transforms". J. Opt. Soc. Am. A. 24 (6): 1609–16. Bibcode:2007JOSAA..24.1609C
Zernike_polynomials
Library for numerical analysis in C and C++
differentiation Chebyshev approximations Series acceleration Discrete Hankel transform Root finding in one and multiple dimensions Minimization in one and
GNU_Scientific_Library
Lab technique for measuring thermal properties through analyzing reflective properties
the co-aligned laser beams have cylindrical symmetry, therefore the Hankel transform can be used to simplify the computation of the convolution of the equation
Time-domain_thermoreflectance
Numerical method in computational electromagnetics
use of spatial-domain Green's functions. This involves the inverse Hankel transform of the spectral-domain Green's function, which is defined on the Sommerfeld
Method of moments (electromagnetics)
Method_of_moments_(electromagnetics)
American Jewish mathematician
1016/0003-4916(87)90178-3 "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group
Bertram_Kostant
Operation on formal power series
Transform". MathWorld. Solution to exercise 5.71 in Concrete Mathematics. Spivey, M. Z. (2006). "The k-binomial transforms and the Hankel transform"
Generating function transformation
Generating_function_transformation
University in Leipzig, Germany
known for the Mollweide projection Hermann Hankel, German mathematician, known for the Hankel transform Felix Hausdorff, German mathematician, one of
Leipzig_University
Physical phenomenon
a non-radiating guided-wave mode, can be derived by employing the Hankel transform of a radial ground current associated with a realistic terrestrial
Surface_wave
Regions of an electromagnetic field
patterns, often involving trigonometric functions or at worst Fourier or Hankel transform relationships between the antenna current distributions and the observed
Near_and_far_field
Mathematics concept
Bessel function Neumann polynomial Lommel polynomial Hankel transform Fourier–Bessel series Krall, H. L.; Frink, O. (1948). "A New Class
Bessel_polynomials
Topics referred to by the same term
used in audio crossover systems Bessel transform, also known as Fourier-Bessel transform or Hankel transform Bessel window, in signal processing Besselian
Bessel
Indian electrical engineer (born 1979)
non-stationary signal analysis method based on eigenvalue decomposition of Hankel matrix (EVDHM) for the first time in literature. Pachori has also developed
Ram_Bilas_Pachori
"On fractional integration and its application to the theory of Hankel transforms", The Quarterly Journal of Mathematics, Second Series, 11: 293–303
Erdelyi–Kober_operator
Malaysian-American electrical engineer
doi:10.1109/8.7216. Ali, S.; Weng Chew; Kong, J. (July 1982). "Vector Hankel transform analysis of annular-ring microstrip antenna". IEEE Transactions on
Weng_Cho_Chew
expansion coefficients that are related to the intensity function via a Hankel transform I l m ( q ) = ∫ 0 ∞ γ l m ( r ) j l ( q r ) r 2 d r {\displaystyle
Fluctuation_X-ray_scattering
Recursive integer sequence
dominating, each of which corresponds to exactly one Dyck word. The n × n Hankel matrix whose (i, j) entry is the Catalan number Ci+j−2 has determinant 1
Catalan_number
Probability problem
turn leads to a tridiagonal model of positive Hankel kernels. An explicit calculation of the Cayley transform of T shows the connection with what is called
Hamburger_moment_problem
Special function in mathematics
_{C}{\frac {(-z)^{s-1}e^{-az}}{1-e^{-z}}}dz} where C {\displaystyle C} is a Hankel contour counterclockwise around the positive real axis, and the principal
Hurwitz_zeta_function
American engineer & educator (1931–2011)
Electron. QE—13, 955—962 (December 1977). A. E. Siegman, "Quasi fast Hankel transform," Opt. Lett. 1, 13-15 (March 1977). A. E. Siegman, "Effects of small-scale
Anthony_E._Siegman
Matrix equal to its conjugate-transpose
dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer
Hermitian_matrix
American mathematician
doi:10.1215/S0012-7094-59-02623-7. —— (1960). "Variation diminishing Hankel transforms". Journal d'Analyse Mathématique. 8: 307–336. doi:10.1007/BF02786854
Isidore_Isaac_Hirschman_Jr.
the same type. Bessel function Bessel polynomial Lommel polynomial Hankel transform Fourier–Bessel series Abramowitz and Stegun, p. 363, 9.1.82 ff. Erdélyi
Neumann_polynomial
Trying to map moments to a measure that generates them
extends to complex analysis as the trigonometric moment problem in which the Hankel matrices are replaced by Toeplitz matrices and the support of μ is the complex
Moment_problem
Polynomial sequence
polynomials arise in: signal processing as Hermitian wavelets for wavelet transform analysis probability, such as the Edgeworth series, as well as in connection
Hermite_polynomials
German mathematician and astronomer (1790–1868)
Möbius transformations, important in projective geometry, and the Möbius transform of number theory. His interest in number theory led to the important Möbius
August_Ferdinand_Möbius
Matrix used in complex analysis
(2002), Operators, functions, and systems: an easy reading, Vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, vol. 92, American Mathematical
Grunsky_matrix
algorithm uses a method called ALOHA (Annihilating filter-based low rank Hankel structured matrix completion approach). The data of the k-space matrix is
Ghosting_(medical_imaging)
American mathematician (born 1941)
1213–1227. doi:10.4153/CJM-1991-069-8. Dunkl, Charles F. (1992). "Hankel transforms associated to finite reflection groups". Contemp. Math. Contemporary
Charles_F._Dunkl
French mathematician (1924–2003)
supervision of Henri Villat; her dissertation applied the Laplace transform and Hankel transform to the heat equation and Maxwell's equations, using cylindrical
Huguette_Delavault
Analytic function in mathematics
_{H}{\frac {(-x)^{s-1}}{e^{x}-1}}\,\mathrm {d} x} for all s (where H denotes the Hankel contour). We can also find expressions which relate to prime numbers and
Riemann_zeta_function
Hungarian mathematician
{\displaystyle \mathbb {Z} } . Fenyő also uses Fourier transforms of the functions for his other work "On the Hankel-transformation of Schwartz distributions", which
István_Fenyő
American-Canadian mathematician (1930-1995)
developer. Herz, C. S. (1954). "On the mean inversion of Fourier and Hankel transforms". Proc Natl Acad Sci U S A. 40 (10): 996–999. Bibcode:1954PNAS...40
Carl_S._Herz
Branch of engineering and mathematics
information about current status is used to influence future status H infinity Hankel singular value Krener's theorem Lead-lag compensator – Control system componentPages
Control_theory
American-Israeli mathematician
include Erez Lapid. Harmonics on Stiefel manifolds and generalized Hankel transforms. Bull. Amer. Math. Soc. 78 (1972) 451–455. MR 0480872 A theory of
Stephen_Gelbart
Overview of and topical guide to control engineering
parameter systems Fractional-order control Fuzzy logic H-infinity loop-shaping Hankel singular value Kalman filter Krener's theorem Least squares Lyapunov stability
Outline of control engineering
Outline_of_control_engineering
Partial differential equation used in physics
\mathbf {E} _{l,m}^{(M)}\,,\end{aligned}}} where hl(1,2)(x) are the spherical Hankel functions, El(1,2) and Bl(1,2) are determined by boundary conditions, and
Electromagnetic_wave_equation
Electric charge generated in certain solids due to mechanical stress
(Piezoelektrizität) was coined in 1881 by the German physicist Wilhelm Gottlieb Hankel; the English word was derived from German in 1883. The pyroelectric effect
Piezoelectricity
Statement about integration on manifolds
which led to the result bearing his name. It was first published by Hermann Hankel in 1861. This classical case relates the surface integral of the curl of
Generalized_Stokes_theorem
Eigenvalue problem for the Laplace operator
\left(k\left|\mathbf {x} -\mathbf {x'} \right|\right)} for n = 2, where H(1) 0 is a Hankel function, and G ( x , x ′ ) = e i k | x − x ′ | 4 π | x − x ′ | {\displaystyle
Helmholtz_equation
Difference between logarithm and harmonic series
and Hankel Functions ‣ Chapter 10 Bessel Functions". dlmf.nist.gov. Retrieved 2024-11-01. "DLMF: §10.8 Power Series ‣ Bessel Functions and Hankel Functions
Euler's_constant
the SMM method transforms the domain into a cylindrical problem. In this domain total field is written in terms of Bessel and Hankel function solutions
Scattering-matrix_method
Special mathematical function
of the latter. The polylogarithm may be quite generally represented by a Hankel contour integral (Whittaker & Watson 1927, § 12.22, § 13.13), which extends
Polylogarithm
Estonian-born Swedish mathematician (1935–2019)
Besov spaces, differential geometry, Clifford analysis, Fock space and Hankel operators, Fourier and harmonic analysis. Bernard Malgrange, Jacques-Louis
Jaak_Peetre
Argentine mathematician
work on moments theory and lifting theorems for measures, Toeplitz forms, Hankel operators, and scattering systems, as well as their applications using weighted
Cora_Sadosky
Extension of the factorial function
)+2\int _{0}^{\infty }{\frac {\arctan(t/z)}{e^{2\pi t}-1}}\,dt.} Let C be a Hankel contour, meaning a path that begins and ends at the point ∞ on the Riemann
Gamma_function
Matrix with shifting rows
the additional property that a i = a i + n {\displaystyle a_{i}=a_{i+n}} Hankel matrix, an "upside down" (i.e., row-reversed) Toeplitz matrix Szegő limit
Toeplitz_matrix
Extension of the scalar spherical harmonics for use with vector fields
{r} }d\Omega _{k}} In case, when z n {\displaystyle z_{n}} are spherical Hankel functions, one should use the different formulae. For vector spherical harmonics
Vector_spherical_harmonics
Scattering of an electromagnetic plane wave by a sphere
generating functions ψ o e m n {\displaystyle \psi _{^{e}_{o}mn}} spherical Hankel functions of the first kind are chosen). Scattered fields are written in
Mie_scattering
} then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values: | λ ( W X ) | = λ ( W C W O ) . {\displaystyle |\lambda
Cross_Gramian
Linear operator
characteristic polynomials for the principal submatrices of the shift operator. Hankel matrix Meurant, Gérard; Sommariva, Alvise (2014). "Fast variants of the
Jacobi_operator
Method of evaluating certain integrals along paths in the complex plane
_{H}{\dfrac {(-t)^{s-1}}{e^{t}-1}}dt,} where the integration is done over the Hankel contour H {\displaystyle H} , is valid for all complex s {\displaystyle
Contour_integration
Operator Theory, pp. 199–214 Aleman, Alexandru; Constantin, Olivia (2004). "Hankel operators on Bergman spaces and similarity to contractions". International
Bergman_space
Quantum mechanics concept for systems with central potentials, such as atoms
at infinity. The first constraint comes from the fact that Neumann and Hankel functions are singular at the origin. The physical requirement that ψ {\displaystyle
Particle in a spherically symmetric potential
Particle_in_a_spherically_symmetric_potential
Graphical method of determining the stability of a dynamical system
margin Barkhausen stability criterion Circle criterion Control engineering Hankel singular value Reinschke, Kurt (2014). "Chapter 4.3. Das Stabilitätskriterium
Nyquist_stability_criterion
Function in quantum field theory showing probability amplitudes of moving particles
{-\tau _{xy}^{2}}})&\tau _{xy}^{2}<0.\end{cases}}} Here, H1(1) is a Hankel function and K1 is a modified Bessel function. This expression can be derived
Propagator
Number with a real and an imaginary part
expression direction coefficient, often used for cos φ + i sin φ, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass. Later classical
Complex_number
In mathematics, invariant of square matrices
Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants
Determinant
Nonparametric spectral estimation method
the obtained Hankel matrix is transformed into a new series of length N {\displaystyle N} using the one-to-one correspondence between Hankel matrices and
Singular_spectrum_analysis
Special mathematical function
(1-s)}{2\pi i}}\int _{C}{\frac {(-t)^{s-1}e^{-at}}{1-ze^{-t}}}\,dt} where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any
Lerch_transcendent
Model in statistical mechanics
Hucht, Alfred (2021). "The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants". Journal of Physics A: Mathematical and Theoretical
Square_lattice_Ising_model
Solution of a confluent hypergeometric equation
functions and many related functions such as Airy functions, Kelvin functions, Hankel functions. For example, in the special case b = 2a the function reduces
Confluent hypergeometric function
Confluent_hypergeometric_function
15. Mierzejewski-Voznyak 2018, pp. 610–611. Umland 2020, p. 4. Kersten & Hankel 2013, pp. 93―94. Mierzejewski-Voznyak 2018, p. 878. Mierzejewski-Voznyak
Far-right_politics_in_Ukraine
Mathematical approach to quantum physics
r^{2}=(x-x')^{2}+(y-y')^{2}} and H 0 ( 1 ) {\displaystyle H_{0}^{(1)}} is the Hankel function of the first kind. In the one-dimensional case, the solution is
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Relation between sides of a right triangle
proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof. Heath himself favors a different
Pythagorean_theorem
German-born American mathematician
the orthogonal group and the representation of functions as Hankel-Stieltjes transforms". Transactions of the American Mathematical Society. 175: 389–408
A._Edward_Nussbaum
Study of groundwater's movement and distribution
Laplace, Hankel and Fourier transforms (to reduce the number of dimensions of the PDE), similarity transform (also called the Boltzmann transform) is commonly
Hydrogeology
Method of solution to differential equations
the zeros of P N ( z ) {\displaystyle P_{N}(z)} . Taking the Fourier transform of L G ( x , s ) = δ ( x − s ) {\displaystyle LG(x,s)=\delta (x-s)} with
Green's_function
History of physical crystallography to 1895
priority. In 1844 Wilhelm Gottlieb Hankel investigated thermoelectricity in cobalt and iron sulfide crystals. Hankel showed that when certain external
Physical crystallography before X-rays
Physical_crystallography_before_X-rays
Study of the structure and dynamics of the Sun through its oscillation
analysis methods to make inferences from the observational data. The Fourier–Hankel spectral method was originally used to search for wave absorption by sunspots
Helioseismology
symmetrically and positively as approximation behavior aij = (i + j − 1)−1. A Hankel matrix. Identity matrix A square diagonal matrix, with all entries on the
List_of_named_matrices
of a form of infinite order. persymmetrical A persymmetrical matrix is a Hankel matrix. See Sylvester (1853, Glossary p. 543–548). Archaic. Pfaffian A square
Glossary_of_invariant_theory
United States historic place
Landmarks 10 Most Endangered Landmarks list. In 2011, the school was transformed into apartments and the auditorium has now been reopened and renamed
Versailles School and Tyson Auditorium
Versailles_School_and_Tyson_Auditorium
HANKEL TRANSFORM
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