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mathematics, Laguerre transform is an integral transform named after the mathematician Edmond Laguerre, which uses generalized Laguerre polynomials L
Laguerre_transform
Mathematical integral transform
_{0}^{\infty }g(y)K_{iy}(x)\sinh(\pi y)y\,dy.} Laguerre previously studied a similar transform regarding Laguerre function as: g ( y ) = ∫ 0 ∞ f ( x ) e − x
Kontorovich–Lebedev_transform
Hankel transform Hartley transform Hermite transform Hilbert transform Hilbert–Schmidt integral operator Jacobi transform Laguerre transform Laplace
List_of_transforms
French mathematician (1834–1886)
Laguerre form Laguerre formula Laguerre group Laguerre's method Laguerre–Pólya class Laguerre plane Laguerre polynomials Laguerre transform Laguerre transformations
Edmond_Laguerre
Sequence of differential equation solutions
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: x y ″
Laguerre_polynomials
Monochrome light beam whose amplitude envelope is a Gaussian function
solved using the Laguerre–Gaussian modal decomposition. These functions are written in cylindrical coordinates using generalized Laguerre polynomials. Each
Gaussian_beam
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
Joseph Courtney; Churchill, Ruel Vance (1953), Hermite and Laguerre integral transforms : preliminary report Feldheim, Ervin (1938). "Quelques nouvelles
Hermite_transform
Mathematical analysis of frequency content of signals
Laplace Transform based on Laguerre function expansion. The Laguerre method can be used to simulate a weakly nonlinear circuit and the Laguerre method
Multidimensional_transform
Mathematical sequence
coefficients. The Lah and Laguerre transforms naturally arise in the perturbative description of the chromatic dispersion. In Lah-Laguerre optics, such an approach
Lah_number
Complex-valued function
functions Laguerre polynomials § Hardy–Hille formula Hardy, G. H. (1932-07-01). "Addendum: Summation of a Series of Polynomials of Laguerre*". Journal
Mehler_kernel
Mathematical transformation
transformation by reciprocal directions or Laguerre inversion, being a generator of the Laguerre group. It transforms not only spheres into spheres but also
Spherical_wave_transformation
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
Polynomial root-finding algorithm
In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to numerically
Laguerre's_method
Type of orthogonal polynomials
the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer
Classical orthogonal polynomials
Classical_orthogonal_polynomials
William H. Kautz, is a fixed-pole traversal filter, published in 1954. Like Laguerre filters, Kautz filters can be implemented using a cascade of all-pass filters
Kautz_filter
The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and
Fokas_method
Solution of a confluent hypergeometric equation
non-positive integer, then Kummer's function (if it is defined) is a generalized Laguerre polynomial. Just as the confluent differential equation is a limit of the
Confluent hypergeometric function
Confluent_hypergeometric_function
Möbius transformation generalized to rings other than the complex numbers
0 , − 1. {\displaystyle \exp(yb)\mapsto \exp(-yb),\quad b^{2}=1,0,-1.} Laguerre transformations Linear-fractional programming H-infinity methods in control
Linear fractional transformation
Linear_fractional_transformation
Quantum mechanics concept for systems with central potentials, such as atoms
{\displaystyle \gamma =2\nu } . First we transform the radial equation by a few successive substitutions to the generalized Laguerre differential equation, which
Particle in a spherically symmetric potential
Particle_in_a_spherically_symmetric_potential
House museum located in Moca, Puerto Rico
writer Enrique Laguerre to write La Llamarada. The property was restored as a museum and renamed the Palacete Los Moreau in honor of Laguerre's novel. It's
Palacete_Los_Moreau
FX/Hulu series (2022–2026)
One Last Episode Before the Show Ended". TheWrap. Retrieved 2026-07-06. Laguerre-Lewis, Kayla (June 24, 2023). "Paul Rudd's Secret The Bear Role Likely
List_of_The_Bear_characters
American mathematician (1915–1994)
MR 0007832. Loomis, Lynn H. (1944). "A short proof of the completeness of the Laguerre functions". Bull. Amer. Math. Soc. 50 (6): 386–387. doi:10.1090/s0002-9904-1944-08151-2
Lynn_Harold_Loomis
1090/S0002-9947-1969-0249918-0. MR 0249918. —— (1969). "Poisson integrals for Hermite and Laguerre expansions". Transactions of the American Mathematical Society. 139: 231–242
Benjamin_Muckenhoupt
Family of solutions to related differential equations
Bessel functions in terms of the Bessel–Clifford function. In terms of the Laguerre polynomials Lk and arbitrarily chosen parameter t, the Bessel function
Bessel_function
Generalization of beta distribution
1017/9781108768900. ISBN 978-1-108-76890-0. Forrester, Peter (2010). "3. Laguerre and Jacobi ensembles". Log-gases and random matrices. London Mathematical
Matrix variate beta distribution
Matrix_variate_beta_distribution
American mathematician and philosopher (1894–1964)
integral equations and Fourier transforms. Wiener studied polynomial chaos, a key piece of which is the Hermite-Laguerre expansion. This was developed
Norbert_Wiener
Wigner distribution function in physics as opposed to in signal processing
L m ( x ) {\displaystyle L_{m}(x)} denotes the m {\displaystyle m} -th Laguerre polynomial. This may follow from the expression for the static eigenstate
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Atom of the element hydrogen
In other places, the Laguerre polynomial includes a factor of ( n + ℓ ) ! {\displaystyle (n+\ell )!} , or the generalized Laguerre polynomial appearing
Hydrogen_atom
Type of function
) d x . {\displaystyle \langle f,g\rangle =\int w(x)f(x)g(x)\,dx.} For Laguerre polynomials on ( 0 , ∞ ) {\displaystyle (0,\infty )} the weight function
Orthogonal_functions
Quantum mechanical model
,} where Ln are the Laguerre polynomials. This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner
Quantum_harmonic_oscillator
System of complete and orthogonal polynomials
the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line [ 0 , ∞ ) {\displaystyle
Legendre_polynomials
Chinese-American engineer and professor (1904-1989)
Fourier transforms of Laguerre functions" in 1932. Lee, Y. W. (1930). Synthesis of electrical networks by means of the Fourier transforms of Laguerre's functions
Lee_Yuk-wing
Mathematical description of quantum state
Hydrogen atom are expressed in terms of spherical harmonics and generalized Laguerre polynomials (these are defined differently by different authors—see main
Wave_function
Approximation of the definite integral of a function
{1-x^{2}}}} . One may also want to integrate over semi-infinite (Gauss–Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature). It can be
Gaussian_quadrature
Book by Mary L. Boas
Legendre, Bessel, Hermite, and Laguerre functions Partial differential equations Functions of a complex variable Integral transforms Probability and statistics
Mathematical Methods in the Physical Sciences
Mathematical_Methods_in_the_Physical_Sciences
Optical phenomenon
holograms (CGHs) are the calculated interferogram between a plane wave and a Laguerre-Gaussian beam which is transferred to film. The CGH resembles a common
Optical_vortex
Mathematical constant
ISSN 0945-3245. Csordas, G.; Ruttan, A.; Varga, R. S. (1991-06-01). "The Laguerre inequalities with applications to a problem associated with the Riemann
De_Bruijn–Newman_constant
2022 film by Steven Spielberg
Spielberg's parents in real life had a large impact on his work. Kayla Laguerre-Lewis of Screen Rant called the scene the one that drives home why Spielberg's
The_Fabelmans
American Jewish mathematician
176...49K, doi:10.1016/0003-4916(87)90178-3 "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected
Bertram_Kostant
Model for the potential energy of a diatomic molecule
α ) ( z ) {\displaystyle \ L_{n}^{(\alpha )}(z)\ } is a generalized Laguerre polynomial: L n ( α ) ( z ) = z − α e z n ! d n d z n ( z
Morse_potential
Geometry founded on spheres
transformations which fix the ideal point at infinity are the Laguerre transformations of Laguerre geometry. These two subgroups generate the group of Lie transformations
Lie_sphere_geometry
Generalization of perpendicularity
{\displaystyle [-1,1]} . The Laguerre polynomials are orthogonal with respect to the exponential distribution. Somewhat more general Laguerre polynomial sequences
Orthogonality_(mathematics)
Americans of Haitian birth or descent
Online free to borrow Laguerre, Michel S. Diasporic citizenship: Haitian Americans in transnational America (Springer, 2016). Laguerre, Michel S. American
Haitian_Americans
Mathematical model combining space and time
group in four dimensions. The Lorentz group is isomorphic to the Laguerre group transforming planes into planes, it is isomorphic to the Möbius group of the
Spacetime
Country house in Derbyshire, England
decorated with murals of scenes from the life of Julius Caesar by Louis Laguerre, and ascend by the cantilevered Great Stairs to an enfilade of rooms that
Chatsworth_House
American rapper (born 1979)
the team to narrow the search. The winner of the competition was Pierre Laguerre and it was announced in January 2020. His startup idea was "Fleeting Pro"
Chamillionaire
Real numbers adjoined with a nil-squaring element
analysis Perturbation theory Infinitesimal Screw theory Dual-complex number Laguerre transformations Grassmann number Automatic differentiation Alexander J
Dual_number
French science award
ensemble of his mathematical work." (1876) Xavier Kretz (1877) Edmond Laguerre, "for his mathematical works." (1878) Maurice Lévy (1879) Théodore Moutard
Poncelet_Prize
Formulation of quantum mechanics
equation in the form of a modified Laguerre equation (not Hermite's equation!), the solution of which involves the Laguerre polynomials as F n ( u ) = ( −
Phase-space_formulation
Operation on formal power series
( β ) ( x ) {\displaystyle L_{n}^{(\beta )}(x)} denotes an associated Laguerre polynomial, then we have that the h t h {\displaystyle h^{th}} convergent
Generating function transformation
Generating_function_transformation
Physical quantity carried in photons
OAM can also be generated by converting a Hermite-Gaussian beam into a Laguerre-Gaussian one by using an astigmatic system with two well-aligned cylindrical
Angular_momentum_of_light
Development of linear transformations forming the Lorentz group
hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group. In physics, Lorentz transformations
History of Lorentz transformations
History_of_Lorentz_transformations
Formal power series
1 + z𝓑t(z)t, so-termed tree polynomials, the Bell numbers, B(n), the Laguerre polynomials, and the Stirling convolution polynomials. Polynomials are
Generating_function
Description of a quantum-mechanical system
{\displaystyle L_{n-\ell -1}^{2\ell +1}(\cdots )} are the generalized Laguerre polynomials of degree n − ℓ − 1 {\displaystyle n-\ell -1} , n , ℓ , m {\displaystyle
Schrödinger_equation
Mathematics concept
{b}{x}}\right).} The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial: θ n ( x ) = n ! ( − 2 ) n L n − 2 n − 1 ( 2 x ) {\displaystyle
Bessel_polynomials
Device that emits light via optical amplification
with the transverse modes often approximated using Hermite–Gaussian or Laguerre-Gaussian functions. Some high-power lasers use a flat-topped profile known
Laser
Function that is holomorphic on the whole complex plane
then this limit is an entire function. Such entire functions form the Laguerre–Pólya class, which can also be characterized in terms of the Hadamard product
Entire_function
Plane curve: conic section
Moebius-, Laguerre- and Minkowski-planes, p. 36. E. Hartmann, Lecture Note Planar Circle Geometries, an Introduction to Möbius-, Laguerre- and Minkowski
Parabola
second stage takes these potential zeros and “polishes” them by applying Laguerre's method to bring them close to the actual zeros of the polynomial. The
Lindsey–Fox_algorithm
Representation theory of the symplectic group
1007/bf01168709, S2CID 119837229 Thangavelu, S. (1993), Lectures on Hermite and Laguerre expansions, Mathematical Notes, vol. 42, Princeton University Press, ISBN 0-691-00048-4
Oscillator_representation
Mathematical approach to quantum optics
concise formal expression for the P function of Fock states using the Laguerre polynomials is P n ( α , α ∗ ) = L n ( − 1 4 ∂ α ∂ α ∗ ) δ 2 ( α ) . {\displaystyle
Glauber–Sudarshan P representation
Glauber–Sudarshan_P_representation
the roots. Closely related to Newton's method are Halley's method and Laguerre's method. Both use the polynomial and its two first derivations for an iterative
Polynomial_root-finding
Family of power series in mathematics
{}_{1}F_{1}(-n;b;z)} is a polynomial. Up to constant factors, these are the Laguerre polynomials. This implies Hermite polynomials can be expressed in terms
Generalized hypergeometric function
Generalized_hypergeometric_function
Curve from a cone intersecting a plane
Hartmann, Erich, Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes (PDF), retrieved 20 September 2014 (PDF; 891 kB)
Conic_section
Quasi-arithmetic mean Classical orthogonal polynomials Hermite polynomials Laguerre polynomials Jacobi polynomials Gegenbauer polynomials Legendre polynomials
List_of_real_analysis_topics
Generating polynomial of the number of ways to place non-attacking rooks on a chessboard
polynomial of a rectangular chessboard is closely related to the generalized Laguerre polynomial Lnα(x) by the identity R m , n ( x ) = n ! x n L n ( m − n )
Rook_polynomial
Methods of calculating definite integrals
Gauss-Hermite quadrature for integrals on the whole real line and Gauss-Laguerre quadrature for integrals on the positive reals. Monte Carlo methods can
Numerical_integration
Mathematical metric in geometry
the measure on a line. Another important insight was the Laguerre formula by Edmond Laguerre (1853), who showed that the Euclidean angle between two lines
Cayley–Klein_metric
(1905–1968), British advertising designer who drew Times New Roman. Louis Laguerre (1663–1721), decorative painter. Marcellus Laroon (1653–1702), artist.
List of people with Huguenot ancestry
List_of_people_with_Huguenot_ancestry
Plane curve: conic section
Moebius-, Laguerre- and Minkowski Planes, S. 33, (PDF; 757 kB) Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski
Hyperbola
Archived from the original on June 17, 2024. Retrieved September 17, 2024. Laguerre-Lewis, Kayla (June 25, 2024). "All The Evidence Riley Is LGBTQ+ In The
LGBTQ themes in Western animation
LGBTQ_themes_in_Western_animation
Concept in statistics
transparent as one proceeds with the Gaussian convolutions. If Ln is the n-th Laguerre polynomial, W is W ( α , α ∗ ) = ( − 1 ) n 2 π e − 2 | α | 2 L n ( 4 |
Quasiprobability_distribution
Overseas department and region of France
vice-president of CACEM and member of the Executive Council of Martinique Didier Laguerre, Mayor of Fort-de-France, CACEM and Councillor to the Assembly of Martinique
Martinique
Country house in Petworth, West Sussex
Flaxman and John Edward Carew), and wall and ceiling paintings by Louis Laguerre. There is also a terrestrial globe by Emery Molyneux, believed to be the
Petworth_House
French politician (born 1953)
1996) ISBN 2-234-04648-3, subject(s): Pratiques politiques—France—1970–. Laguerre, Christian École, informatique et nouveaux comportements préf. de Ségolène
Ségolène_Royal
City and municipality of Puerto Rico
called a "more profoundly Puerto Rican city than San Juan". See Enrique A. Laguerre's "Mito de la ponceñidad", El Mundo, 8 November 1980, page 9-A. Ayuntamiento
Ponce,_Puerto_Rico
Chemical compound
doi:10.1007/s00436-008-1137-7. PMID 18690475. S2CID 37480828. Vivas N, Laguerre M, Pianet de Boissel I, Vivas de Gaulejac N, Nonier MF (April 2004). "Conformational
Castalagin
Hôtel particulier in Paris, France
the celebrated architect Alexandre-Théodore Brongniart to enlarge and transform his residence on the Rue de Grenelle. Among Brongniart's additions were
Hôtel_de_Besenval
Type of angular momentum in light
beams that are in a helical mode carry nonzero OAM. As an example, any Laguerre-Gaussian mode with rotational mode number l ≠ 0 {\displaystyle l\neq 0}
Orbital angular momentum of light
Orbital_angular_momentum_of_light
Imaging technique based on fluorescence
Fourier transformation along with Laguerre gauss expansion have been used to estimate the lifetime in transformed space. These approaches are faster
Fluorescence-lifetime imaging microscopy
Fluorescence-lifetime_imaging_microscopy
Commune and capital city of Martinique
the wider abolition of slavery in the French colonies, fundamentally transformed Martinique's social structure, accelerating urban migration and altering
Fort-de-France
Scientific instruments
Hermite-Gaussian beams (TEMxy), Laguerre-Gaussian (LG) beams (TEMpl) and Bessel beams. Optical tweezers based on Laguerre-Gaussian beams have the unique
Optical_tweezers
Art museum in London, England
display include Gian Lorenzo Bernini, Grinling Gibbons, Daniel Marot, Louis Laguerre, Antonio Verrio, Sir James Thornhill, William Kent, Robert Adam, Josiah
Victoria_and_Albert_Museum
Geometry of the location of polynomial roots
fractions for real-root isolation. If all roots of a polynomial are real, Laguerre proved the following lower and upper bounds of the roots, by using what
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
quadrature for integrals with weight (1 − x)α (1 + x)β on [−1, 1] Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight
List of numerical analysis topics
List_of_numerical_analysis_topics
Harassment occurring in a public setting
about after many people were outraged at a man attacking a woman (Marie Laguerre) due to her response to his harassment of her. In 2017, the Dutch cities
Street_harassment
Romanian Swedish mathematician
property of having only real zeros, a problem that goes back to Edmond Laguerre and to George Pólya and Issai Schur. These results were subsequently extended
Julius_Borcea
Private research institute in Old San Juan, Puerto Rico
moment. By this time, figures like Juan Manuel García Passalaqua, Enrique Laguerre, Luis De la Rosa, Guillermo Baralt, Blanca Silvestrini, José Ferrer Canales
Center for Advanced Studies on Puerto Rico and the Caribbean
Center_for_Advanced_Studies_on_Puerto_Rico_and_the_Caribbean
Numerical methods for matrix eigenvalue calculation
114–139, doi:10.1016/j.laa.2005.06.022 Li, T. Y.; Zeng, Zhonggang (1992), "Laguerre's Iteration In Solving The Symmetric Tridiagonal Eigenproblem - Revisited"
Eigenvalue_algorithm
2022 single by Idina Menzel
Menken and Schwartz, "Love Power" is a power ballad. According to Kayla Laguerre-Lewis of Screen Rant, it is reminiscent of Broadway musicals, such as Schwartz
Love Power (Disenchanted song)
Love_Power_(Disenchanted_song)
leading foreign painters of allegoric schemes, Antonio Verrio and Louis Laguerre, had died in 1707 and 1721 respectively, Thornhill had the field to himself
Art_of_the_United_Kingdom
Function in statistics
ν > 0 {\displaystyle \nu >0} can also be represented using generalized Laguerre polynomials as Q ν ( a , b ) = 1 − e − a 2 / 2 ∑ k = 0 ∞ ( − 1 ) k L k
Marcum_Q-function
unitary operator (e.g., the transformation of Hermite-Gauss optical modes to Laguerre-Gauss optical modes). It has been shown that any discrete modal unitary
Universal multiport interferometer
Universal_multiport_interferometer
Locus of the zeros of a polynomial of degree two
Interactive Java 3D models of all quadric surfaces Lecture Note Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes, p. 117
Quadric
Quantum mechanics with supersymmetry
differential equations, the analysis produces a recursion relation for the Laguerre polynomials. The outcome is the spectrum of hydrogen-atom energy states
Supersymmetric quantum mechanics
Supersymmetric_quantum_mechanics
2 respectively. Bochner's theorem states that, up to a complex-affine transform of x {\displaystyle x} (that is, of form x ↦ a x + b {\displaystyle x\mapsto
Bochner's theorem (orthogonal polynomials)
Bochner's_theorem_(orthogonal_polynomials)
building to his baroque taste. St Lawrence still contains artwork by Louis Laguerre and Bellucci (who also worked on the decorative scheme of the house). At
Handel_at_Cannons
Quantum explanation of electromagnetic polarization
(June 1992). "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes". Physical Review A. 45 (11): 8186–9. Bibcode:1992PhRvA
Photon_polarization
picture editor Jerry Astor on the very morning that managing editor Andre Laguerre had decided to do a Chamberlain story. Instead of sending a photographer
Rich_Clarkson
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