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Hadamard Three-circle Theorem
In complex analysis, a branch of mathematics, the Hadamard three-circle theorem is a result about the behavior of holomorphic functions. Statement Hadamard three-circle theorem: Let f(z) be a holomorphic function on the annulus r_1\leq\left, z\ \leq r_3. Let M(r) be the maximum of , f(z), on the circle , z, =r. Then, \log M(r) is a convex function of the logarithm \log (r). Moreover, if f(z) is not of the form cz^\lambda for some constants \lambda and c, then \log M(r) is strictly convex as a function of \log (r). The conclusion of the theorem can be restated as :\log\left(\frac\right)\log M(r_2)\leq \log\left(\frac\right)\log M(r_1) +\log\left(\frac\right)\log M(r_3) for any three concentric circles of radii r_1 Proof The three circles theorem follows from the fact that for any real ''a'', the function Re log(''z''''a''''f''(''z'')) is harmonic between two circles, and therefore takes its maximum value on one of the circles ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist, and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long. In 1895, his paper on scoring chess tournaments is the earliest use of eigenvector centrality. Landau taught at the University of Berlin from 1899 to 1909, after which he held a chair at the University of Göttingen. He married Marianne Ehrlich, the daughter of the Nobel Prize-winning biologist Paul Ehrlich, in 1905. At the 1912 International Congress of Mathematicians Landau listed four problems in number theory about primes that he said were pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Les Comptes Rendus De L'Académie Des Sciences
LES or Les may refer to: People * Les (given name) * Les (surname) * L.E.S. (producer), hip hop producer Space flight * Launch Entry Suit, worn by Space Shuttle crews * Launch escape system, for spacecraft emergencies * Lincoln Experimental Satellite series, 1960s and 1970s Biology and medicine * Lazy eye syndrome, or amblyopia, a disorder in the human optic nerve * The Liverpool epidemic strain of ''Pseudomonas aeruginosa'' * Lower esophageal sphincter * Lupus erythematosus systemicus Places * The Lower East Side neighborhood of Manhattan, New York City * Les, Catalonia, a municipality in Spain * Leş, a village in Nojorid Commune, Bihor County, Romania * ''Les'', the Hungarian name for Leșu Commune, Bistriţa-Năsăud County, Romania * Les, a village in Tejakula district, Buleleng regency, Bali, Indonesia * Lesotho, IOC and UNDP country code * Lès, a word featuring in many French placenames Transport * Leigh-on-Sea railway station, National Rail station cod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phragmén–Lindelöf Principle
In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function f (i.e., , f(z), Background In the theory of complex functions, it is known that the modulus (absolute value) of a |
Borel–Carathéodory Theorem
In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory. Statement of the theorem Let a function f be analytic on a closed disc of radius ''R'' centered at the origin. Suppose that ''r'' 0 for some ''z'' on the circle , z, =R, so we may take A>0. Now ''f'' maps into the half-plane ''P'' to the left of the ''x''=''A'' line. Roughly, our goal is to map this half-plane to a disk, apply Schwarz's lemma there, and make out the stated inequality. w \mapsto w/A - 1 sends ''P'' to the standard left half-plane. w \mapsto R(w+1)/(w-1) sends the left half-plane to the circle of radius ''R'' centered at the origin. The composite, which maps 0 to 0, is the desired map: :w \mapsto \frac. From Schwarz's lemma applied to the composite of this map and ''f'', we have :\frac \leq , z, . Take , ''z'', & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Three-line Theorem
In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard. Statement Define F(z) by : F(z)=f(z) M(a)^M(b)^ where , F(z), \leq 1 on the edges of the strip. The result follows once it is shown that the inequality also holds in the interior of the strip. After an affine transformation in the coordinate z, it can be assumed that a = 0 and b = 1. The function : F_n(z) = F(z) e^e^ tends to 0 as , z, tends to infinity and satisfies , F_n, \leq 1 on the boundary of the strip. The maximum modulus principle can therefore be applied to F_n in the strip. So , F_n(z), \leq 1. Because F_n(z) tends to F(z) as n tends to infinity, it follows that , F(z), \leq 1. ∎ Applications The three-line theorem can be used to prove the Hadamard three-circle theorem for a bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy's Theorem
In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let f be a holomorphic function on the open ball centered at zero and radius R in the complex plane, and assume that f is not a constant function. If one defines :I(r) = \frac \int_0^\! \left, f(r e^) \ \,d\theta for 0< r < R, then this function is strictly increasing and is a convex function of . See also * *Hadamard three-circle theorem In complex analysis, a branch of mathematics, the
Hadamar ...
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Logarithmically Convex Function
In mathematics, a function ''f'' is logarithmically convex or superconvex if \circ f, the composition of the logarithm with ''f'', is itself a convex function. Definition Let be a convex subset of a real vector space, and let be a function taking non-negative values. Then is: * Logarithmically convex if \circ f is convex, and * Strictly logarithmically convex if \circ f is strictly convex. Here we interpret \log 0 as -\infty. Explicitly, is logarithmically convex if and only if, for all and all , the two following equivalent conditions hold: :\begin \log f(tx_1 + (1 - t)x_2) &\le t\log f(x_1) + (1 - t)\log f(x_2), \\ f(tx_1 + (1 - t)x_2) &\le f(x_1)^tf(x_2)^. \end Similarly, is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all . The above definition permits to be zero, but if is logarithmically convex and vanishes anywhere in , then it vanishes everywhere in the interior of . Equivalent conditions If is a di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Principle
In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study. Solutions of a differential inequality in a domain ''D'' satisfy the maximum principle if they achieve their maxima at the boundary of ''D''. The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations. In a simple two-dimensional 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, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat, and Picard. He obtained his doctorate in 1892 and in the same year was awarded the for his essay on the Riemann zeta function. In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the University of Bordeaux, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
