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Wiman-Valiron Theory
Wiman-Valiron theory is a mathematical theory invented by Anders Wiman as a tool to study the behavior of arbitrary entire functions. After the work of Wiman, the theory was developed by other mathematicians, and extended to more general classes of analytic functions. The main result of the theory is an asymptotic formula for the function and its derivatives near the point where the maximum modulus of this function is attained. Maximal term and central index By definition, an entire function can be represented by a power series which is convergent for all complex z: f(z)=\sum_^\infty a_nz^n. The terms of this series tend to 0 as n\to\infty, so for each z there is a term of maximal modulus. This term depends on r:=, z, . Its modulus is called the ''maximal term'' of the series: \mu(r,f)=\max_k , a_k, r^k=:, a_n, r^n,\quad r\geq 0. Here n is the exponent for which the maximum is attained; if there are several maximal terms, we define n as the largest exponent of them. This numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anders Wiman
Anders Wiman (11 February 1865 – 13 August 1959) was a Swedish mathematician. He is known for his work in algebraic geometry and applications of group theory. Life Wiman was born to well-off land-owing farmer family in Hammarlöv, Sweden in 1865. He attended school in Lund, and graduated from secondary school in 1885. In the autumn of the same year, Wiman went to Lund University studying Mathematics, Botany and Latin. He attained Bachelor's degree in 1887 and his Licentiate in 1891. He continued his study in the same university under supervision of Carl Fabian Björling and was awared doctorate in 1892, with thesis ''Klassifikation af regelytorna af sjette graden'' (Classification of regular surfaces of degree 6). In 1892, Wiman was appointed as a docent (equivalent of assistant professor) in Lund University. There, his work on the classification of finite geometrical groups in the last few years of the 19th century was seen impressive. He classified all algebraic curves of ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entire Function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. Properties Every entire function can be represented as a power series f(z) = \sum_^\infty a_n z^n that converges everywhere in the complex plane, hen ... [...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 requires t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emile Borel
Emil or Emile may refer to: Literature *'' Emile, or On Education'' (1762), a treatise on education by Jean-Jacques Rousseau * ''Émile'' (novel) (1827), an autobiographical novel based on Émile de Girardin's early life *'' Emil and the Detectives'' (1929), a children's novel *"Emil", nickname of the Kurt Maschler Award for integrated text and illustration (1982–1999) *'' Emil i Lönneberga'', a series of children's novels by Astrid Lindgren Military * Emil (tank), a Swedish tank developed in the 1950s * Sturer Emil, a German tank destroyer People * Emil (given name), including a list of people with the given name ''Emil'' or ''Emile'' * Aquila Emil (died 2011), Papua New Guinean rugby league footballer Other * ''Emile'' (film), a Canadian film made in 2003 by Carl Bessai * Emil (river), in China and Kazakhstan See also * * * Aemilius (other) *Emilio (other) *Emílio (other) *Emilios (other) Emilios, or Aimilios, (Greek: Αιμίλιο ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Mathematical Bulletin
The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antonio Lei and Javad Mashreghi. The journal publishes short articles in all areas of mathematics that are of sufficient interest to the general mathematical public. Abstracting and indexing The journal is abstracted in: for the Canadian Mathematical Bulletin. * '''' * '' [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. In the simplest case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unless is a constant function, the maximum cannot also be achieved anywhere on itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georges Valiron
Georges Jean Marie Valiron (7 September 1884 – 17 March 1955) was a French mathematician, notable for his contributions to analysis, in particular, the asymptotic behaviour of entire functions of finite order and Tauberian theorems. Biography Valiron obtained his Ph.D. from the University of Paris in 1914, under supervision of Émile Borel. Since 1922 he held a professorship at the University of Strasbourg, and since 1931 a chair at the University of Paris. He gave a plenary speech at the 1932 International Congress of Mathematicians in Zürich and was an invited speaker of the ICM in 1920 in Strasbourg and in 1928 in Bologna. His treatise on mathematical analysis in two volumes (''Théorie des fonctions'' and ''Équations fonctionnelles'') is a classic and has been translated into numerous languages under diverse titles and has gone through many new editions, both French and non-French. He was awarded the title Commander of the Legion of Honour in 1954. One of Valiron's doctor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch's Theorem (complex Variables)
In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists. It is named after André Bloch. Statement Let ''f'' be a holomorphic function in the unit disk , ''z'', ≤ 1 for which :, f'(0), =1 Bloch's Theorem states that there is a disk S ⊂ D on which f is biholomorphic and f(S) contains a disk with radius 1/72. Landau's theorem If ''f'' is a holomorphic function in the unit disk with the property , ''f′''(0), = 1, then let ''Lf'' be the radius of the largest disk contained in the image of ''f''. Landau's theorem states that there is a constant ''L'' defined as the infimum of ''Lf'' over all such functions ''f'', and that ''L'' ≥ ''B''. This theorem is named after Edmund Landau. Valiron's theorem Bloch's theorem was inspired by the following theorem of Georges Valiron: Theorem. If ''f'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Escaping Set
In mathematics, and particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ. That is, a complex number z_0\in\mathbb belongs to the escaping set if and only if the sequence defined by z_ := f(z_n) converges to infinity as n gets large. The escaping set of f is denoted by I(f). For example, for f(z)=e^z, the origin belongs to the escaping set, since the sequence :0,1,e,e^e,e^,\dots tends to infinity. History The iteration of transcendental entire functions was first studied by Pierre Fatou in 1926 The escaping set occurs implicitly in his study of the explicit entire functions f(z)=z+1+\exp(-z) and f(z)=c\sin(z). The first study of the escaping set for a general transcendental entire function is due to Alexandre Eremenko who used Wiman-Valiron theory. He conjectured that every connected component of the escaping set of a transcendental entire function is unbounded. This has b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
