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Behnke–Stein Theorem
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G_k \subset \mathbb^n (i.e., G_k \subset G_) of domains of holomorphy is again a domain of holomorphy. It was proved by Heinrich Behnke and Karl Stein in 1938. This is related to the fact that an increasing union of pseudoconvex domains is pseudoconvex and so it can be proven using that fact and the solution of the Levi problem. Though historically this theorem was in fact used to solve the Levi problem, and the theorem itself was proved using the Oka–Weil theorem In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil. Statement The Oka–Weil theorem .... This theorem again holds for Stein manifolds, but it is not known if it holds for Stein space. References * * Several complex variable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Of Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables (and analytic space), that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function f:(z_1,z_2, \ldots, z_n) \rightarrow f(z_1,z_2, \ldots, z_n) is -tuples of complex numbers, classically studied on the complex coordinate space \Complex^n. As in complex analysis of functions of one variable, which is the case , the functions studied are ''holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of Holomorphy
In mathematics, in the theory of functions of Function of several complex variables, several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be analytic continuation, extended to a bigger domain. Formally, an open set \Omega in the ''n''-dimensional complex space ^n is called a ''domain of holomorphy'' if there do not exist non-empty open sets U \subset \Omega and V \subset ^n where V is connected space, connected, V \not\subset \Omega and U \subset \Omega \cap V such that for every holomorphic function f on \Omega there exists a holomorphic function g on V with f = g on U In the n=1 case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulation point, accumulating everywhere on the boundary (topology), boundary of the domain, which must then be a analytic continuation#Natural boundary, natural boundary for a domain of definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinrich Behnke
Heinrich Adolph Louis Behnke (Horn, 9 October 1898 – Münster, 10 October 1979) was a German mathematician and rector at the University of Münster. Life and career He was born into a Lutheran family in Horn, a suburb of Hamburg. He attended the University of Göttingen and submitted his doctoral thesis to the University of Hamburg. He was noted for work on complex analysis with Henri Cartan and Peter Thullen. His first wife, Aenne Albersheim, was Jewish, but she died soon after the birth of their son. He was concerned about his son's ethnicity during the Nazi period. In 1936 he was elected a member of the ''Deutsche Akademie der Naturforscher Leopoldina''. Selected publications *with Peter Thullen: Theorie der Funktionen mehrerer komplexer Veränderlicher, Springer Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete, 1934, 2nd edn. with collaboration by Reinhold Remmert 1970 *with Friedrich Sommer: Theorie der Funktionen einer komplexen Veränderlichen, Spring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Stein (mathematician)
Karl Stein (1 January 1913 in Hamm, Westphalia – 19 October 2000) was a German mathematician. He is well known for complex analysis and cryptography. Stein manifolds and Stein factorization are named after him. Career Karl Stein received his doctorate with his dissertation on the topic ''Zur Theorie der Funktionen mehrerer komplexer Veränderlichen; Die Regularitätshüllen niederdimensionaler Mannigfaltigkeiten'' at the University of Münster under the supervision of Heinrich Behnke in 1937. Karl Stein was conscripted into the Wehrmacht sometime before 1942, and trained as a cryptographer to work at OKW/Chi, the Cipher Department of the High Command of the Wehrmacht. He was assigned to manage the OKW/Chi IV, Subsection a, which was a unit responsible for security of own processes, cipher devices testing, and invention of new cipher devices. He managed a staff of 11 In 1955 he became professor at the Ludwig Maximilian University of Munich and emeritated in 1981. In 1990 ... [...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] |
Pseudoconvex Domain
In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ... in the ''n''-dimensional complex space C''n''. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let :G\subset ^n be a domain, that is, an open set, open connected space, connected subset. One says that G is ''pseudoconvex'' (or ''Friedrich Hartogs, Hartogs pseudoconvex'') if there exists a continuous function, continuous plurisubharmonic function \varphi on G such that the set :\ is a relatively compact subset of G for all real numbers x. In other words, a domain is pseudoconvex if G has a continuous plurisubharmonic bounded exhaustion function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Problem
In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. Definition Suppose X is a complex manifold of complex dimension n and let \mathcal O(X) denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold: * X is holomorphically convex, i.e. for every compact subset K \subset X, the so-called ''holomorphically convex hull'', ::\bar K = \left \, :is also a ''compact'' subset of X. * X is holomorphically separable, i.e. if x \neq y are two points in X, then there exists f \in \mathcal O(X) such that f(x) \neq f(y). Non-compact Riemann surfaces are Stein manifolds Let ''X'' be a connected, non-compact R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oka–Weil Theorem
In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil. Statement The Oka–Weil theorem states that if ''X'' is a Stein space and ''K'' is a compact \mathcal(X)-convex subset of ''X'', then every holomorphic function in an open neighborhood of ''K'' can be approximated uniformly on ''K'' by holomorphic functions on \mathcal(X) (i.e. by polynomials). Applications Since Runge's theorem may not hold for several complex variables, the Oka–Weil theorem is often used as an approximation theorem for several complex variables. The Behnke–Stein theorem was originally proved using the Oka–Weil theorem. See also * Oka coherence theorem References Bibliography * * * * * * Further reading * – An example where Runge's theorem In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables (and analytic space), that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function f:(z_1,z_2, \ldots, z_n) \rightarrow f(z_1,z_2, \ldots, z_n) is -tuples of complex numbers, classically studied on the complex coordinate space \Complex^n. As in complex analysis of functions of one variable, which is the case , the functions studied are ''holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
