|
Laguerre Transformations
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] |
Bar-le-Duc
Bar-le-Duc (), formerly known as Bar, is a commune in the Meuse département, of which it is the capital. The department is in Grand Est in northeastern France. The lower, more modern and busier part of the town extends along a narrow valley, shut in by wooded or vine-clad hills, and is traversed throughout its length by the Ornain, which is crossed by several bridges. It is limited towards the north-east by the Marne–Rhine Canal, on the south-west by a small arm of the Ornain, called the ''Canal des Usines'', on the left bank of which the upper town (''Ville Haute'') is situated. The highly rarefied Bar-le-duc jelly, also known as Lorraine jelly, is a spreadable preparation of white currant or red currant fruit preserves, hailing from this town. First referenced in the historical record in 1344, it is also colloquially referred to as "Bar caviar". History Bar-le-Duc was at one time the seat of the county, from 1354 the Duchy of Bar. Though probably of ancient origin, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nouvelles Annales De Mathématiques
The ''Nouvelles Annales de Mathématiques'' (subtitled ''Journal des candidats aux écoles polytechnique et normale'') was a French scientific journal in mathematics. It was established in 1842 by Olry Terquem and Camille-Christophe Gerono, and continued publication until 1927, with later editors including Charles-Ange Laisant and Raoul Bricard. , retrieved 2014-07-14. Initially published by Carilian-Goeury, it was taken over after several years by a different publisher, Bachelier. Although competing in subject matter with |
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] |
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] |
Laguerre Form
In mathematics, the Laguerre form is generally given as a third degree tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...-valued form, that can be written as, :\mathfrak = (w^)^ D a_ + 2 w^w^ D a_ + (w^)^ D a_. Tensors {{differential-geometry-stub ... [...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] |
Meixner Polynomials
In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by . They are given in terms of binomial coefficients and the (rising) Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ... by :M_n(x,\beta,\gamma) = \sum_^n (-1)^kk!(x+\beta)_\gamma^ See also * Kravchuk polynomials References * * * * * * * * * * * * * *{{cite journal , first1= Xiang-Sheng , last1=Wang , first2=Roderick , last2=Wong , title= Global asymptotics of the Meixner polynomials , journal = Asymptot. Anal. , year=2011 , volume=75 , number=3–4 , pages=211–231 , doi=10.3233/ASY-2011-1060 , arxiv=1101.4370 Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big Q-Laguerre Polynomials
In mathematics, the big ''q''-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The 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 P_n(x;a,b;q)=\frac_2\phi_1\left(q^,aqx^;aq;q,\frac\right) Relation to other polynomials Big q-Laguerre polynomials→Laguerre polynomials References * * *{{dlmf, id=18, title=Chapter 18: Orthogonal Polynomials, first=Tom H. , last=Koornwinder, first2=Roderick S. C., last2= Wong, first3=Roelof , last3=Koekoek, , first4=René F. , last4=Swarttouw Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...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] |
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] |
Eugène Rouché
Eugène Rouché (18 August 1832 – 19 August 1910) was a French mathematician. Career He was an alumnus of the École Polytechnique, which he entered in 1852. He went on to become professor of mathematics at the Charlemagne lyceum then at the École Centrale, and admissions examiner at his alma mater. He is best known for Rouché's theorem in complex analysis, which he published in his alma mater's institutional journal in 1862, and for the Rouché–Capelli theorem in linear algebra. His son, Jacques, was a noted patron of the arts who managed the Paris Opera for thirty years (1914–1944). See also * Rouché's theorem * Rouché–Capelli theorem In linear algebra, the Rouché–Capelli theorem determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix. The theorem is variously known as the: * Rouché–Capelli theore ... References * Rouché et Comberousse (de), Traité de géométrie, tomes I et I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
