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Hartogs' Principle
In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be compact, therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an isolated singularity is always a removable singularity for any analytic function of complex variables. A first version of this theorem was proved by Friedrich Hartogs,See the original paper of and its description in various historical surveys by , and . In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of , and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula''". and as such it is known also as Hartogs's lemma and Hartogs's principle: in earlie ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Integral Formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis. Theorem Let be an open subset of the complex plane , and suppose the closed disk defined as :D = \bigl\ is completely contained in . Let be a holomorphic function, and let be the circle, oriented counterclockwise, forming the boundary of . Then for every in the interior of , :f(a) = \frac \oint_\gamma \frac\,dz.\, The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bump Function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain \R^n forms a vector space, denoted \mathrm^\infty_0(\R^n) or \mathrm^\infty_\mathrm(\R^n). The dual space of this space endowed with a suitable topology is the space of distributions. Examples The function \Psi:\R \to \R given by \Psi(x) = \begin \exp\left( -\frac\right), & x \in (-1,1) \\ 0, & \text \end is an example of a bump function in one dimension. It is clear from the construction that this function has compact support, since a function of the real line has compact support if and only if it has bounded closed support. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article. This function can be interpreted as the Gaussian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of Holomorphy
In mathematics, in the theory of functions of 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 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, 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 accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal. For n \geq 2 this is no longer true, as it follows from Hartogs' lemma. Equivalent conditions For a domain \Omega the followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenpreis's Fundamental Principle
In mathematical analysis, Ehrenpreis's fundamental principle, introduced by Leon Ehrenpreis, states: :Every solution of a system (in general, overdetermined) of homogeneous partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...s with constant coefficients can be represented as the integral with respect to an appropriate Radon measure over the complex “ characteristic variety” of the system. References Mathematical analysis {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the problem can be stated as follows: :Given a function ''f'' that has values everywhere on the boundary of a region in R''n'', is there a unique continuous function ''u'' twice continuously differentiable in the interior and continuous on the boundary, such that ''u'' is harmonic in the interior and ''u'' = ''f'' on the boundary? This requirement is called the Dirichlet boundary condition. The main issue is to prove the existence of a solution; uniqueness can be proved using the maximum principle. History The Dirichlet problem goes back to George Green, who studied the problem on general domains with general boundary con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome. Biography He was born in Acireale, a town near Catania in Sicily, the elder of the four sons of Giuseppe Fichera and Marianna Abate. His father Giuseppe was a professor of mathematics and influenced the young Gaetano starting his lifelong passion. In his young years he was a talented football player. On 1 February 1943 he was in the Italian Army and during the events of September 1943 he was taken prisoner by the Nazist troops, kept imprisoned in Teramo and then sent to Verona: he succeeded in escaping from there and reached the Italian region of Emilia-Romagna, spending with partisans the last year of war. After the war he was first in Rome and then in Trieste, where he met Matelda Colautti, who became his wife in 1952. Education and aca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Ehrenpreis
Eliezer 'Leon' Ehrenpreis (May 22, 1930 – August 16, 2010, Brooklyn) was a mathematician at Temple University who proved the Malgrange–Ehrenpreis theorem, the fundamental theorem about differential operators with constant coefficients. He previously held tenured positions at Yeshiva University and at the Courant Institute at New York University. Early life and education Leon was born in New York City to a family of Jewish immigrants from Eastern Europe. He graduated from Stuyvesant High School and studied Mathematics as an undergraduate at City College of New York. Afterward, he enrolled as a doctoral student at Columbia University where he studied under mathematician Claude Chevalley, obtaining his PhD in 1953 at the age of 23. His doctoral thesis was entitled "Theory of Distributions in Locally Compact Spaces". Religion Ehrenpreis was also a Rabbi, having received his ordination from the renowned Rabbi Moshe Feinstein. He was the author of a work on the Chumash and othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
