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Bieberbach Conjecture
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients a_n of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a_0=0 and a_1=1. That is, we consider a function defined on the open unit disk which is holomorphic and injective ('' univalent'') with Taylor series of the form :f(z)=z+\sum_ a_n z^n. Such functions are called ''schlicht''. The theorem then states that : , a_n, \leq n \quad \textn\geq 2. The Koebe function (see below) is a function in which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient. Schlicht functions The normalizat ... [...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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schramm–Loewner Evolution
In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter ''κ'' and a domain in the complex plane ''U'', it gives a family of random curves in ''U'', with ''κ'' controlling how much the curve turns. There are two main variants of SLE, ''chordal SLE'' which gives a family of random curves from two fixed boundary points, and ''radial SLE'', which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property. It was discovered by as a conjectured scaling limit of the planar uniform spanning tree (UST) and the planar loop-erased random walk (LERW) probabilistic processes, and developed by him together with Greg Lawler and We ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Korevaar
Jacob "Jaap" Korevaar (born 25 January 1923) is a Dutch mathematician. He was part of the faculty of the University of California San Diego and University of Wisconsin–Madison, as well as the University of Amsterdam ( Korteweg-de Vries Institute for Mathematics). Korevaar became a member of the Royal Netherlands Academy of Arts and Sciences in 1975. He won the 1987 Lester R. Ford Award, and the 1989 Chauvenet Prize, for an essay on Louis de Branges de Bourcia's proof of the Bieberbach conjecture. In 2012 he became a fellow of the American Mathematical Society. Korevaar is the older brother of the Olympic water polo player Nijs Korevaarbr>He centenarian, turned 100 on 25 January 2023. References Prof. dr. J. Korevaar, 1923 - at the University of Amsterdam The University of Amsterdam (abbreviated as UvA, nl, Universiteit van Amsterdam) is a public research university located in Amsterdam, Netherlands. The UvA is one of two large, publicly funded research universities in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christian Pommerenke
Christian Pommerenke (born 17 December 1933 in Copenhagen) is a mathematician known for his work in complex analysis. He studied at the University of Göttingen (1954–58), achieving diploma in mathematics (1957), Ph.D. (1959) on the dissertation ''Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden'' (1959) and habilitation (1963). Pommerenke subsequently joined the faculty as Assistant (1958–64) and Privatdozent (1964–66). Around the same time he served as assistant professor at the University of Michigan in Ann Arbor (1961–62), was at Harvard University (1962–63) and was guest lecturer and reader at Imperial College in London (1965–67). Since 1967 he has been professor in complex analysis at the mathematics department of the Technical University of Berlin. from tu-berlin.de He is now an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl FitzGerald
Carl Fitzgerald is a professional Canadian Football fullback who is currently a free agent. He was drafted by the Winnipeg Blue Bombersin the third round of the 2013 CFL Draft. At the conclusion of the 2015 CFL season, Fitzgerald was re-signed for 2016 by the Saskatchewan Roughriders The Saskatchewan Roughriders are a professional Canadian football team based in Regina, Saskatchewan. The Roughriders compete in the Canadian Football League (CFL) as a member club of the league's West Division. The Roughriders were founded in .... References External links Winnipeg Blue Bombers bio Canadian football fullbacks 1990 births Living people Saskatchewan Roughriders players Winnipeg Blue Bombers players {{Canadianfootball-fullback-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Askey
Richard Allen Askey (4 June 1933 – 9 October 2019) was an American mathematician, known for his expertise in the area of special functions. The Askey–Wilson polynomials (introduced by him in 1984 together with James A. Wilson) are on the top level of the (q-)Askey scheme, which organizes orthogonal polynomials of (q-)hypergeometric type into a hierarchy. The Askey–Gasper inequality for Jacobi polynomials is essential in de Brange's famous proof of the Bieberbach conjecture. Askey earned a B.A. at Washington University in 1955, an M.A. at Harvard University in 1956, and a Ph.D. at Princeton University in 1961. After working as an instructor at Washington University (1958–1961) and University of Chicago (1961–1963), he joined the faculty of the University of Wisconsin–Madison in 1963 as an Assistant Professor of Mathematics. He became a full professor at Wisconsin in 1968, and since 2003 was a professor emeritus. Askey was a Guggenheim Fellow, 1969–1970, which acad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Gautschi
Walter Gautschi (born December 11, 1927) is a Swiss- American mathematician, known for his contributions to numerical analysis. He has authored over 200 papers in his area and published four books. Born in Basel, he has a Ph.D. in mathematics from the University of Basel on the thesis ''Analyse graphischer Integrationsmethoden'' advised by Alexander Ostrowski and Andreas Speiser (1953). Since then, he did postdoctoral work as a Janggen-Pöhn Research Fellow at the ''Istituto Nazionale per le Applicazioni del Calcolo'' in Rome (1954) and at the Harvard Computation Laboratory (1955). He had positions at the National Bureau of Standards (1956–59), the American University in Washington D.C., the Oak Ridge National Laboratory (1959–63) before joining Purdue University where he has worked from 1963 to 2000 and now being professor emeritus. He has been a Fulbright Scholar at the Technical University of Munich (1970) and held visiting appointments at the University of Wisconsin� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Polynomial
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The definition is in IV.1; the differential equation – in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi. Definitions Via the hypergeometric function The Jacobi polynomials are defined via the hypergeometric function as follows: :P_n^(z)=\frac\,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac(1-z)\right), where (\alpha+1)_n is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Askey–Gasper Inequality
In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture. Statement It states that if \beta\geq 0, \alpha+\beta\geq -2, and -1\leq x\leq 1 then :\sum_^n \frac \ge 0 where :P_k^(x) is a Jacobi polynomial. The case when \beta=0 can also be written as :_3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right)>0, \qquad 0\leq t-1. In this form, with a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbach conjecture. Proof gave a short proof of this inequality, by combining the identity :\begin \frac &\times _3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right) = \\ &= \frac \times _3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac(\alpha+1);\tfrac(\alpha+2),\alpha+1;t \right ) \end with the Clausen inequality. Generalizations give some generalizations of the Askey–Gasper inequality to basic hypergeometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loewner Equation
In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (conformal mappings of the open disk onto the complex plane with a curve joining 0 to ∞ removed), Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of Carathéodory to the whole plane leads to a one parameter family of conformal mappings, called a Loewner chain, as well as a two parameter family of holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part. The Loewner semigroup generalizes the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Branges Space
In mathematics, a de Branges space (sometimes written De Branges space) is a concept in functional analysis and is constructed from a de Branges function. The concept is named after Louis de Branges who proved numerous results regarding these spaces, especially as Hilbert spaces, and used those results to prove the Bieberbach conjecture. De Branges functions A Hermite-Biehler function, also known as de Branges function is an entire function ''E'' from \Complex to \Complex that satisfies the inequality , E(z), > , E(\bar z), , for all ''z'' in the upper half of the complex plane \Complex^+ = \. Definition 1 Given a Hermite-Biehler function , the de Branges space is defined as the set of all entire functions ''F'' such that F/E,F^/E \in H_2(\Complex^+) where: * \Complex^+ = \ is the open upper half of the complex plane. * F^(z) = \overline. * H_2(\Complex^+) is the usual Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomor ... [...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, hence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |