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Plancherel–Rotach Asymptotics
The Plancherel–Rotach asymptotics are asymptotic results for orthogonal polynomials. They are named after the Swiss mathematicians Michel Plancherel and his PhD student Walter Rotach, who first derived the asymptotics for the Hermite polynomial and Laguerre polynomial. Nowadays asymptotic expansions of this kind for orthogonal polynomials are referred to as ''Plancherel–Rotach asymptotics'' or of ''Plancherel–Rotach type''. The case for the associated Laguerre polynomial was derived by the Swiss mathematician Egon Möcklin, another PhD student of Plancherel and George Pólya at ETH Zurich. Hermite polynomials Let H_n(x) denote the n-th Hermite polynomial. Let \epsilon and \omega be positive and fixed, then * for x =(2n+1)^\cos \varphi and \epsilon \leq \varphi \leq \pi -\epsilon :: e^H_n(x) =2^(n!)^(\pi n)^(\sin \varphi)^ \bigg\ * for x =(2n+1)^\cosh \varphi and \epsilon \leq \varphi \leq \omega :: e^H_n(x) =2^(n!)^(\pi n)^(\sinh \varphi)^ \exp\left left(\tfrac+\tfrac\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. The theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886. The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a '' convergent'' Taylor s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ... to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. These are frequently given by the Rodrigues' formula. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Switzerland
Switzerland, officially the Swiss Confederation, is a landlocked country located in west-central Europe. It is bordered by Italy to the south, France to the west, Germany to the north, and Austria and Liechtenstein to the east. Switzerland is geographically divided among the Swiss Plateau, the Swiss Alps, Alps and the Jura Mountains, Jura; the Alps occupy the greater part of the territory, whereas most of the country's Demographics of Switzerland, 9 million people are concentrated on the plateau, which hosts List of cities in Switzerland, its largest cities and economic centres, including Zurich, Geneva, and Lausanne. Switzerland is a federal republic composed of Cantons of Switzerland, 26 cantons, with federal authorities based in Bern. It has four main linguistic and cultural regions: German, French, Italian and Romansh language, Romansh. Although most Swiss are German-speaking, national identity is fairly cohesive, being rooted in a common historical background, shared ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel Plancherel
Michel Plancherel (; 16 January 1885 – 4 March 1967) was a Swiss people, Swiss mathematician. Biography He was born in Bussy, Fribourg, Bussy (Canton of Fribourg, Switzerland) and obtained his Diplom in mathematics from the University of Fribourg and then his doctoral degree in 1907 with a thesis written under the supervision of Mathias Lerch. Plancherel was a professor in Fribourg (1911), and from 1920 at ETH Zurich. He worked in the areas of mathematical analysis, mathematical physics and algebra, and is known for the Plancherel theorem in harmonic analysis. He was an Invited Speaker of the International Congress of Mathematicians, ICM in 1924 at TorontoPlancherel, Michel (1924" Sur les séries de fonctions orthogonales." In ''Proceedings of the International Mathematical Congress'', Toronto, vol. 1, pp. 619–622. and in 1928 at Bologna. He was married to Cécile Tercier, had nine children, and presided at the ''Mission Catholique Française'' in Zürich. References Ext ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Rotach
Walter may refer to: People and fictional characters * Walter (name), including a list of people and fictional and mythical characters with the given name or surname * Little Walter, American blues harmonica player Marion Walter Jacobs (1930–1968) * Gunther (wrestler), Austrian professional wrestler and trainer Walter Hahn (born 1987), who previously wrestled as "Walter" * Walter, standard author abbreviation for Thomas Walter (botanist) ( – 1789) * "Agent Walter", an early codename of Josip Broz Tito * Walter, pseudonym of the anonymous writer of '' My Secret Life'' * Walter Plinge, British theatre pseudonym used when the original actor's name is unknown or not wished to be included * John Walter (businessman), Canadian business entrepreneur Companies * American Chocolate, later called Walter, an American automobile manufactured from 1902 to 1906 * Walter Energy, a metallurgical coal producer for the global steel industry * Walter Aircraft Engines, Czech manufacturer of aero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomial
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Polynomial
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: xy'' + (1 - x)y' + ny = 0,\ y = y(x) 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 Gauss–Laguerre quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , ..., are a polynomial sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egon Möcklin
Egon is a Danish variant of the male given name Egino. It is most commonly found in Central and Northern Europe. Egon may refer to: People * Egon VIII of Fürstenberg-Heiligenberg (1588–1635), Imperial Count of Fürstenberg-Heiligenberg (1618–1635) and a military leader in the Thirty Years' War * Egon Bahr (1922–2015), German politician * Egon Bittner (1921–2011), American sociologist * Egon Bondy (1930–2007), Czech philosopher * Egon Coordes (born 1944), German footballer and coach * Egon Freiherr von Eickstedt (1892–1965), German physical anthropologist * Egon Eiermann (1904–1970), German architect * Egon Franke (fencer) (1935–2022), Polish Olympic fencer * Egon Franke (politician) (1913–1995), German politician * Egon Frid (born 1957), Swedish politician * Egon Friedell (1878–1938), Austrian writer * Egon Guttman (1927–2021), German-American legal scholar * Egon Hirt (born 1960), German alpine skier * Egon Jensen (politician) (1922–1985), Danish politi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Pólya
George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education. He has been described as one of The Martians (scientists), The Martians, an informal category which included one of his most famous students at ETH Zurich, John von Neumann. Life and works Pólya was born in Budapest, Austria-Hungary, to Anna Deutsch and Jakab Pólya, History of the Jews in Hungary, Hungarian Jews who had converted to Christianity in 1886. Although his parents were religious and he was baptized into the Catholic Church upon birth, George eventually grew up to be an agnostic. He received a PhD under Lipót Fejér in 1912, at Eötvös Loránd University. He was a professor o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ETH Zurich
ETH Zurich (; ) is a public university in Zurich, Switzerland. Founded in 1854 with the stated mission to educate engineers and scientists, the university focuses primarily on science, technology, engineering, and mathematics. ETH Zurich ranks among Europe's best universities. Like its sister institution École Polytechnique Fédérale de Lausanne, EPFL, ETH Zurich is part of the ETH Domain, Swiss Federal Institutes of Technology Domain, a consortium of universities and research institutes under the Swiss Federal Department of Economic Affairs, Education and Research. , ETH Zurich enrolled 25,380 students from over 120 countries, of which 4,425 were pursuing doctoral degrees. Students, faculty, and researchers affiliated with ETH Zurich include 22 Nobel Prize, Nobel laureates, two Fields Medalists, three Pritzker Architecture Prize, Pritzker Prize winners, and one Turing Award, Turing Award recipient, including Albert Einstein and John von Neumann. It is a founding member o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airy Function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear independence, linearly independent solutions to the differential equation \frac - xy = 0 , known as the Airy equation or the Stokes equation. Because the solution of the linear differential equation \frac - ky = 0 is oscillatory for and exponential for , the Airy functions are oscillatory for and exponential for . In fact, the Airy equation is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential). Definitions For real values of , the Airy function of the first kind can be defined by the improper integral, improper Riemann integral: \operatorname(x) = \dfrac\int_0^\infty\cos\left(\dfrac + xt\right)\, dt\equiv \dfrac \lim_ \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 BC), though ''analysis'' as a formal concept is a relatively recent development. The word comes from the Ancient Greek (''analysis'', "a breaking-up" or "an untying" from ''ana-'' "up, throughout" and ''lysis'' "a loosening"). From it also comes the word's plural, ''analyses''. As a formal concept, the method has variously been ascribed to René Descartes ('' Discourse on the Method''), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name). The converse of analysis is synthesis: putting the pieces back together again in a new or different whole. Science and technology Chemistry The field of chemistry uses analysis in three ways: to i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
