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List Of Things Named After Eduard Heine
{{Short description, none Eduard Heine (16 March 1821, Berlin – October 1881, Halle) was a German mathematician in Prussia. His name is given to several mathematical concepts that he was instrumental in developing: * Andréief–Heine identity * Heine–Borel theorem * Heine–Cantor theorem * Heine–Stieltjes polynomials * Heine definition of continuity * Heine functions * Heine's identity In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as \frac = \frac\sum_^\infty Q_(z) e^ where Q_ is a Legendre function of the second kind, whic ... * Mehler–Heine formula Heine, Eduard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduard Heine
Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician. Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legendre functions (''Handbuch der Kugelfunctionen''). He also investigated basic hypergeometric series. He introduced the Mehler–Heine formula. Biography Heinrich Eduard Heine was born on 16 March 1821 in Berlin, as the eighth child of banker Karl Heine and his wife Henriette Märtens. Eduard was initially home schooled, then studied at the Friedrichswerdersche Gymnasium and Köllnische Gymnasium in Berlin. In 1838, after graduating from gymnasium, he enrolled at the University of Berlin, but transferred to the University of Göttingen to attend the mathematics lectures of Carl Friedrich Gauss and Moritz Stern. In 1840 Heine returned to Berlin, where he studied mathematics under Peter Gustav Lejeune Dirichlet, while also attending c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an emergency decree transferring powers of the Prussian government to German Chancellor Franz von Papen in 1932 and ''de jure'' by an Allied decree in 1947. For centuries, the House of Hohenzollern ruled Prussia, expanding its size with the Prussian Army. Prussia, with its capital at Königsberg and then, when it became the Kingdom of Prussia in 1701, Berlin, decisively shaped the history of Germany. In 1871, Prussian Minister-President Otto von Bismarck united most German principalities into the German Empire under his leadership, although this was considered to be a " Lesser Germany" because Austria and Switzerland were not included. In November 1918, the monarchies were abolished and the nobility lost its political power durin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy%E2%80%93Binet Formula
In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with the entries from any commutative ring. Statement Let ''A'' be an ''m''×''n'' matrix and ''B'' an ''n''×''m'' matrix. Write 'n''for the set , and \tbinomm for the set of ''m''- combinations of 'n''(i.e., subsets of 'n''of size ''m''; there are \tbinom nm of them). For S\in\tbinomm, write ''A'' 'm''''S'' for the ''m''×''m'' matrix whose columns are the columns of ''A'' at indices from ''S'', and ''B''''S'', 'm''/sub> for the ''m''×''m'' matrix whose rows are the rows of ''B'' at indices from ''S''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heine–Cantor Theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if f \colon M \to N is a continuous function between two metric spaces M and N, and M is compact, then f is uniformly continuous. An important special case is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous. Proof Suppose that M and N are two metric spaces with metrics d_M and d_N, respectively. Suppose further that a function f: M \to N is continuous and M is compact. We want to show that f is uniformly continuous, that is, for every positive real number \varepsilon > 0 there exists a positive real number \delta > 0 such that for all points x, y in the function domain M, d_M(x,y) 0 such that d_N(f(x),f(y)) < \varepsilon/2 when , i.e., a fact that is within of implies that is within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heine–Stieltjes Polynomials
In mathematics, the Heine–Stieltjes polynomials or Stieltjes polynomials, introduced by , are polynomial solutions of a second-order Fuchsian equation, a differential equation all of whose singularities are regular. The Fuchsian equation has the form :\frac+\left(\sum _^N \frac \right) \frac + \fracS = 0 for some polynomial ''V''(''z'') of degree at most ''N'' − 2, and if this has a polynomial solution ''S'' then ''V'' is called a Van Vleck polynomial (after Edward Burr Van Vleck) and ''S'' is called a Heine–Stieltjes polynomial. Heun polynomial In mathematics, the local Heun function H \ell (a,q;\alpha ,\beta, \gamma, \delta ; z) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point ''z'' = 0. The local Heun function is called a Heun ...s are the special cases of Stieltjes polynomials when the differential equation has four singular points. References * * * Polynomials {{polynomial-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heine Functions
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 hypergeometric if the ratio of successive terms ''x''''n''+1/''x''''n'' is a rational function of ''n''. If the ratio of successive terms is a rational function of ''q''''n'', then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series _2\phi_1(q^,q^;q^;q,x) was first considered by . It becomes the hypergeometric series F(\alpha,\beta;\gamma;x) in the limit when base q =1. Definition There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as :\;_\phi_k \left begin a_1 & a_2 & \ldots & a_ \\ b_1 & b_2 & \ldots & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heine's Identity
In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as \frac = \frac\sum_^\infty Q_(z) e^ where Q_ is a Legendre function of the second kind, which has degree, ''m'' − , a half-integer, and argument, ''z'', real and greater than one. This expression can be generalized for arbitrary half-integer powers as follows (z-\cos\psi)^ = \sqrt\frac \sum_^ \fracQ_^n(z)e^{im\psi}, where \scriptstyle\,\Gamma is the Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except .... References Special functions Mathematical identities ... [...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] |
