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Arakelyan's Theorem
In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset. Theorem Let Ω be an open subset of \Complex and ''E'' a relatively closed subset of Ω. By Ω* is denoted the Alexandroff compactification of Ω. Arakelyan's theorem states that for every ''f'' continuous in ''E'' and holomorphic in the interior of ''E'' and for every ''ε'' > 0 there exists ''g'' holomorphic in Ω such that , ''g'' − ''f'', < ''ε'' on ''E'' if and only if Ω* \ ''E'' is connected and locally connected. See also * * |
Mergelyan's Theorem
Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951. Statement :Let ''K'' be a compact subset of the complex plane C such that C∖''K'' is connected. Then, every continuous function ''f'' : ''K''\to C, such that the restriction ''f'' to int(''K'') is holomorphic, can be approximated uniformly on ''K'' with polynomials. Here, int(''K'') denotes the interior of ''K''. Mergelyan's theorem also holds for open Riemann surfaces :If ''K'' is a compact set without holes in an open Riemann surface ''X'', then every function in \mathcal (K) can be approximated uniformly on K by functions in \mathcal(X). Mergelyan's theorem does not always hold in higher dimensions (spaces of several complex variables), but it has some consequences. History Mergelyan's theorem is a generalization of the Weierstrass approximation theorem and Runge's theorem. In the case that C∖''K'' is ''not'' co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandroff Compactification
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let ''X'' be a topological space. Then the Alexandroff extension of ''X'' is a certain compact space ''X''* together with an open embedding ''c'' : ''X'' → ''X''* such that the complement of ''X'' in ''X''* consists of a single point, typically denoted ∞. The map ''c'' is a Hausdorff compactification if and only if ''X'' is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge's Theorem
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following: Denoting by C the set of complex numbers, let ''K'' be a compact set, compact subset of C and let ''f'' be a function (mathematics), function which is holomorphic function, holomorphic on an open set containing ''K''. If ''A'' is a set containing Existential quantification, at least one complex number from every bounded set, bounded connected set, connected component of C\''K'' then there exists a sequence (r_n)_ of rational functions which uniform convergence, converges uniformly to ''f'' on ''K'' and such that all the pole (complex analysis), poles of the functions (r_n)_ are in ''A.'' Note that not every complex number in ''A'' needs to be a pole of every rational function of the sequence (r_n)_. We merely know that for all members of (r_n)_ that do have poles, those poles lie in ''A' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Complex Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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