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Gustav Ferdinand Mehler
Gustav Ferdinand Mehler, or Ferdinand Gustav Mehler (13 December 1835, in Schönlanke, Kingdom of Prussia – 13 July 1895, in Elbing, German Empire) was a German mathematician. He is credited with introducing Mehler's formula; the Mehler–Fock transform; the Mehler–Heine formula; and Mehler functions (conical functions), in connection with his utilization of Zonal spherical function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vect ...s in Electromagnetic theory. References * 19th-century German mathematicians 1835 births 1895 deaths {{Germany-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Das Fotoalbum Für Weierstraß 024 (Gustav Ferdinand Mehler)
Das or DAS may refer to: Organizations * Dame Allan's Schools, Fenham, Newcastle upon Tyne, England * Danish Aviation Systems, a supplier and developer of unmanned aerial vehicles * Departamento Administrativo de Seguridad, a former Colombian intelligence agency * Department of Applied Science, UC Davis * ''Debt Arrangement Scheme'', Scotland, see Accountant in Bankruptcy Places * Das (crater), a lunar impact crater on the far side of the Moon * Das (island), an Emirati island in the Persian Gulf ** Das Island Airport * Das, Catalonia, a village in the Cerdanya, Spain * Das, Iran, a village in Razavi Khorasan Province * Great Bear Lake Airport, Northwest Territories, Canada (IATA code) Science * 1,2-Bis(dimethylarsino)benzene, a chemical compound * DAS28, Disease Activity Score of 28 joints, rheumatoid arthritis measure * Differential Ability Scales, cognitive and achievement tests Technology * Data acquisition system * Defensive aids system, an aircraft defensive syst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schönlanke
Trzcianka (; german: Schönlanke) is a town in the Greater Poland region in northwestern Poland. Since 1999, it has been part of the Greater Poland Voivodeship and Czarnków-Trzcianka County. From 1975 to 1998, it was located in the Piła Voivodeship. In May 2007, Trzcianka had 17,131 inhabitants. Trzcianka is located on the , and three lakes, Sarcze, Okunie and Długie, are located within the town limits. History The settlement, initially named Rozdróżka, was probably founded in the 13th century. It was located on a trade route which connected Poznań and Kołobrzeg. Rozdróżka was mentioned in a document from 1245, when Duke Boleslaus V of Poland gave the land in the Noteć river valley, along with three villages (Biała, Gulcz, and Rozdróżka) to a Polish nobleman named Sędziwój of Czarnków. The new name of these three combined villages was Trzciana Łąka, as it appeared for the first time in 1565, and it was subsequently changed to Trzcianka in the 17th century. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kingdom Of Prussia
The Kingdom of Prussia (german: Königreich Preußen, ) was a German kingdom that constituted the state of Prussia between 1701 and 1918.Marriott, J. A. R., and Charles Grant Robertson. ''The Evolution of Prussia, the Making of an Empire''. Rev. ed. Oxford: Clarendon Press, 1946. It was the driving force behind the unification of Germany in 1871 and was the leading state of the German Empire until its dissolution in 1918. Although it took its name from the region called Prussia, it was based in the Margraviate of Brandenburg. Its capital was Berlin. The kings of Prussia were from the House of Hohenzollern. Brandenburg-Prussia, predecessor of the kingdom, became a military power under Frederick William, Elector of Brandenburg, known as "The Great Elector". As a kingdom, Prussia continued its rise to power, especially during the reign of Frederick II, more commonly known as Frederick the Great, who was the third son of Frederick William I.Horn, D. B. "The Youth of Frederick ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German Empire
The German Empire (),Herbert Tuttle wrote in September 1881 that the term "Reich" does not literally connote an empire as has been commonly assumed by English-speaking people. The term literally denotes an empire – particularly a hereditary empire led by an emperor, although has been used in German to denote the Roman Empire because it had a weak hereditary tradition. In the case of the German Empire, the official name was , which is properly translated as "German Empire" because the official position of head of state in the constitution of the German Empire was officially a "presidency" of a confederation of German states led by the King of Prussia who would assume "the title of German Emperor" as referring to the German people, but was not emperor of Germany as in an emperor of a state. –The German Empire" ''Harper's New Monthly Magazine''. vol. 63, issue 376, pp. 591–603; here p. 593. also referred to as Imperial Germany, the Second Reich, as well as simply Germany, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mehler's Formula
The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. Mehler's formula defined a function and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials (.) based on weight function exp(−²) as :E(x,y) = \sum_^\infty \frac ~ \mathit_n(x)\mathit_n(y) ~. This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis. Physics version In physics, the fundamental solution, ( Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel. It provides the fundamental solution---the most general solution to :\frac = \frac-x^2\varphi \equiv D_x \varphi ~. The orthonormal eigenfunctions of the operator are the Hermite functions, :\psi_n = \frac, with corresponding eigenvalues (2+1), furnishing particular solutions : \varphi_n(x, t)= e^ ~H_n(x) \exp(-x^2/2) ~. The general solution is then a linear c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mehler–Fock Transform
In mathematics, the Mehler–Fock transform is an integral transform introduced by and rediscovered by . It is given by :F(x) =\int_0^\infty P_(x)f(t) dt,\quad (1 \leq x \leq \infty), where ''P'' is a Legendre function In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated L ... of the first kind. Under appropriate conditions, the following inversion formula holds: :f(t) = t \tanh(\pi t) \int_1^\infty P_(x)F(x) dx ,\quad (0 \leq t \leq \infty). References * * * * {{DEFAULTSORT:Mehler-Fock transform Integral transforms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mehler–Heine Formula
In mathematics, the Mehler–Heine formula introduced by Gustav Ferdinand Mehler and Eduard Heine describes the asymptotic behavior of the Legendre polynomials as the index tends to infinity, near the edges of the support of the weight. There are generalizations to other classical orthogonal polynomials, which are also called the Mehler–Heine formula. The formula complements the Darboux formulae which describe the asymptotics in the interior and outside the support. Legendre polynomials The simplest case of the Mehler–Heine formula states that :\lim _P_n\left(\cos\right) = \lim _P_n\left(1-\frac\right) = J_0(z), where is the Legendre polynomial of order , and the Bessel function of order 0. The limit is uniform over in an arbitrary bounded domain in the complex plane. Jacobi polynomials The generalization to Jacobi polynomials is given by Gábor Szegő as follows :\lim_ n^P_n^\left(\cos \frac\right) = \lim_ n^P_n^\left(1-\frac\right) = \left(\frac\right)^ J_\alpha( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mehler Function
In mathematics, conical functions or Mehler functions are functions which can be expressed in terms of Legendre functions of the first and second kind, P^\mu_(x) and Q^\mu_(x). The functions P^\mu_(x) were introduced by Gustav Ferdinand Mehler, in 1868, when expanding in series the distance of a point on the axis of a cone to a point located on the surface of the cone. Mehler used the notation K^\mu(x) to represent these functions. He obtained integral representation and series of functions representations for them. He also established an addition theorem for the conical functions. Carl Neumann obtained an expansion of the functions K^\mu(x) in terms of the Legendre polynomials in 1881. Leonhardt introduced for the conical functions the equivalent of the spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conical Function
In mathematics, conical functions or Mehler functions are functions which can be expressed in terms of Legendre functions of the first and second kind, P^\mu_(x) and Q^\mu_(x). The functions P^\mu_(x) were introduced by Gustav Ferdinand Mehler, in 1868, when expanding in series the distance of a point on the axis of a cone to a point located on the surface of the cone. Mehler used the notation K^\mu(x) to represent these functions. He obtained integral representation and series of functions representations for them. He also established an addition theorem for the conical functions. Carl Neumann obtained an expansion of the functions K^\mu(x) in terms of the Legendre polynomials in 1881. Leonhardt introduced for the conical functions the equivalent of the spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zonal Spherical Function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vector in an irreducible representation of ''G''. The key examples are the matrix coefficients of the '' spherical principal series'', the irreducible representations appearing in the decomposition of the unitary representation of ''G'' on ''L''2(''G''/''K''). In this case the commutant of ''G'' is generated by the algebra of biinvariant functions on ''G'' with respect to ''K'' acting by right convolution. It is commutative if in addition ''G''/''K'' is a symmetric space, for example when ''G'' is a connected semisimple Lie group with finite centre and ''K'' is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
