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Laguerre
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials (see Laguerre polynomials). Laguerre's method is a root-finding algorithm tailored to polynomials. He laid the foundations of a geometry of oriented spheres (Laguerre geometry and Laguerre plane), including the Laguerre transformation or transformation by reciprocal directions. Works Selection * * * * Théorie des équations numériques', Paris: Gauthier-Villars. 1884 on Google Books * * Oeuvres de Laguerrepubl. sous les auspices de l'Académie des sciences par MM. Charles Hermite, Henri Poincaré, et Eugène Rouché.'' (Paris, 1898-1905) (reprint: New York : Chelsea publ., 1972 ) Extensive lists More than 80 articleson Nundam.org.p See also * Isotropic line * ''q''-Laguerre polynomials * Big ''q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Wave Transformation
Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames. They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham, with Bateman giving the transformation its name.Bateman (1908); Bateman (1909); Cunningham (1909) They correspond to the conformal group of "transformations by reciprocal radii" in relation to the framework of Lie sphere geometry, which were already known in the 19th century. Time is used as fourth dimension as in Minkowski space, so spherical wave transformations are connected to the Lorentz transformation of special relativity, and it turns out that the conformal group of spacetime includes the Lorentz group and the Poincaré group as subgroups. However, only the Lorentz/Poincaré groups represent symmetries of all laws of nature including mechanics, whereas the conformal group is related to certain areas such as electrodynamics.Kastrup (2008)Walter (20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of xy'' + (\alpha + 1 - x)y' + ny = 0~. where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , …, are a polynomial sequence which may be defined by the Rodrigues formula, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Plane
In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre. The classical Laguerre plane is an incidence structure that describes the incidence behaviour of the curves y=ax^2+bx+c , i.e. parabolas and lines, in the real affine plane. In order to simplify the structure, to any curve y=ax^2+bx+c the point (\infty,a) is added. A further advantage of this completion is that the plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder (see below). The classical real Laguerre plane Originally the classical Laguerre plane was defined as the geometry of the oriented lines and circles in the real Euclidean plane (see ). Here we prefer the parabola model of the classical Laguerre plane. We define: \mathcal P:=\R^2\cup (\\times\R), \ \infty \notin \R, the set of point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Transformation
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials (see Laguerre polynomials). Laguerre's method is a root-finding algorithm tailored to polynomials. He laid the foundations of a geometry of oriented spheres (Laguerre geometry and Laguerre plane), including the Laguerre transformation or transformation by reciprocal directions. Works Selection * * * * Théorie des équations numériques', Paris: Gauthier-Villars. 1884 on Google Books * * Oeuvres de Laguerrepubl. sous les auspices de l'Académie des sciences par MM. Charles Hermite, Henri Poincaré, et Eugène Rouché.'' (Paris, 1898-1905) (reprint: New York : Chelsea publ., 1972 ) Extensive lists More than 80 articleson Nundam.org.p See also * Isotropic line * ''q''-Laguerre polynomials * Big ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Beam
In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of most (but not all) lasers, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a lens, the transverse ''phase'' dependence is altered; this results in a ''different'' Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam (for a given wavelength and polarization) are determined by a single parameter: the so-called waist . At any position relative to the waist (focus) along a beam having a specified , the field amplitudes and phases are thereby determinedSvelto, pp. 153–5. as detailed below. The equations below assume a beam with a circular cross-section at all va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Laguerre Quadrature
In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind: :\int_^ e^ f(x)\,dx. In this case :\int_^ e^ f(x)\,dx \approx \sum_^n w_i f(x_i) where ''x''''i'' is the ''i''-th root of Laguerre polynomial ''L''''n''(''x'') and the weight ''w''''i'' is given byEquation 25.4.45 in 10th reprint with corrections. :w_i = \frac . The following Python code with the SymPy library will allow for calculation of the values of x_i and w_i to 20 digits of precision: from sympy import * def lag_weights_roots(n): x = Symbol("x") roots = Poly(laguerre(n, x)).all_roots() x_i = t.evalf(20) for rt in roots w_i = rt / ((n + 1) * laguerre(n + 1, rt)) ** 2).evalf(20) for rt in roots return x_i, w_i print(lag_weights_roots(5)) For more general functions To integrate the function f we apply the following transformation : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Transform
In mathematics, Laguerre transform is an integral transform named after the mathematician Edmond Laguerre, which uses generalized Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only ... L_n^\alpha(x) as kernels of the transform.McCully, Joseph. "The Laguerre transform." SIAM Review 2.3 (1960): 185-191. The Laguerre transform of a function f(x) is :L\ = \tilde f_\alpha(n) = \int_^\infty e^ x^\alpha \ L_n^\alpha(x)\ f(x) \ dx The inverse Laguerre transform is given by :L^\ = f(x) = \sum_^\infty \binom^ \frac \tilde f_\alpha(n) L_n^\alpha(x) Some Laguerre transform pairs References {{Reflist, 30em Integral transforms Mathematical physics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre's Method
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 solve the equation for a given polynomial . One of the most useful properties of this method is that it is, from extensive empirical study, very close to being a "sure-fire" method, meaning that it is almost guaranteed to always converge to ''some'' root of the polynomial, no matter what initial guess is chosen. However, for computer computation, more efficient methods are known, with which it is guaranteed to find all roots (see ) or all real roots (see Real-root isolation). This method is named in honour of Edmond Laguerre, a French mathematician. Definition The algorithm of the Laguerre method to find one root of a polynomial of degree is: * Choose an initial guess * For ** If p(x_k) is very small, exit the loop ** Calculate G = \frac ** Calculate H = G^2 - \frac ** Calculate a = \frac , where the sign is chosen to gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Laguerre Polynomials
In mathematics, the ''q''-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials ''P''(''x'';''q'') are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by . give a detailed list of their properties. Definition The ''q''-Laguerre polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called hy ...s and the q-Pochhammer symbol by :\displaystyle L_n^(x;q) = \frac _1\phi_1(q^;q^;q,-q^x). Orthogonality Orthogonality is defined by the unimono nature of the polynomials' convergence at boundaries in integral form. References * * * *{{citation , last=Moak, first=Daniel S., title=The q-analogue of the Laguerre polynomials, journal=J. Math. Anal. Appl., volume=81, issue=1, pages=20†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre–Pólya Class
The Laguerre–Pólya class is the class of entire functions consisting of those functions which are locally the limit of a series of polynomials whose roots are all real. by D. Dryanov and Q. I. Rahman, ''Methods and Applications of Analysis'' 6 (1) 1999, pp. 21–38. Any function of Laguerre–Pólya class is also of Pólya class. The product of two functions in the class is also in the class, so the class constitutes a monoid under the operation of function multiplication. Some properties of a function in the Laguerre–Pólya class are: *All root of a function, roots are real. * for ''x'' and ''y'' real. * is a non-dec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Line
In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form. Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point that depend on the imaginary unit :Edmond Laguerre (1870) "Sur l’emploi des imaginaires en la géométrie" Oeuvres de Laguerre2: 89 : First system: (y - \beta) = (x - \alpha) i, : Second system: (y - \beta) = -i (x - \alpha) . Laguerre then interpreted these lines as geodesics: :An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line ''situated at a finite distance in the plane'' is zero. In other terms, these lines satisfy the differential equation . On an arbitrary surface one can study c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Formula
The Laguerre formula (named after Edmond Laguerre) provides the acute angle \phi between two proper real lines, as follows: :\phi=, \frac \operatorname \operatorname(I_1,I_2,P_1,P_2), where: * \operatorname is the principal value of the complex logarithm * \operatorname is the cross-ratio of four collinear points * P_1 and P_2 are the points at infinity of the lines * I_1 and I_2 are the intersections of the absolute conic, having equations x_0=x_1^2+x_2^2+x_3^2=0, with the line joining P_1 and P_2. The expression between vertical bars is a real number. Laguerre formula can be useful in computer vision, since the absolute conic has an image on the retinal plane which is invariant under camera displacements, and the cross ratio of four collinear points is the same for their images on the retinal plane. Derivation It may be assumed that the lines go through the origin. Any isometry leaves the absolute conic invariant, this allows to take as the first line the ''x'' axis and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
