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Schlicht Function
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 unit disc, 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 series, 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 function, holomorphic and injective (''Univalent function, 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 for which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute val ... [...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] |
Löwner Equation
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German. Early life and career Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner. Loewner received his Ph.D. from the University of Prague in 1917 under supervision of Georg Pick. One of his central mathematical contributions is the proof of the Bieberbach conjecture in the first highly nontrivial case of the third coefficient. The technique he introduced, the Loewner differential equation, has had far-reaching implications in geometric function theory; it was used in the final solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner worked at the University of Berlin, University of Prague, University of Louisville, Brown University, Syracuse University and eventually at Stanford University. His students include Lipman Bers, Roger Horn, Adrian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christian Pommerenke
Christian Pommerenke (17 December 1933 – 18 August 2024) was a German mathematician known for his work in complex analysis. Life and career Pommerenke studied at the University of Göttingen (1954–1958), 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 (1958–1964) and Privatdozent (1964–1966). Around the same time he served as assistant professor at the University of Michigan in Ann Arbor (1961–1962), was at Harvard University (1962–1963) and was guest lecturer and reader at Imperial College in London (1965–1967). Since 1967 he was professor in complex analysis at the mathematics department of Technische Universität Berlin. He was later an emeritus ''Emeritus/Emerita'' () is an honorary title granted to someone who retires from a position of distinction, most common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl FitzGerald
Carl may refer to: *Carl, Georgia, city in USA *Carl, West Virginia, an unincorporated community *Carl (name), includes info about the name, variations of the name, and a list of people with the name *Carl², a TV series * "Carl", an episode of television series ''Aqua Teen Hunger Force'' * An informal nickname for a student or alum of Carleton College CARL may refer to: *Canadian Association of Research Libraries *Colorado Alliance of Research Libraries See also *Carle (other) *Charles *Carle, a surname *Karl (other) *Karle (other) Karle may refer to: Places * Karle (Svitavy District), a municipality and village in the Czech Republic * Karli, India, a town in Maharashtra, India ** Karla Caves, a complex of Buddhist cave shrines * Karle, Belgaum, a settlement in Belgaum ... {{disambig ja:カール zh:卡尔 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Askey
Richard Allen Askey (June 4, 1933 – October 9, 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. Biography Askey earned a B.A. at Washington University in St. Louis 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 Fel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Gautschi
Walter Gautschi (; ; born December 11, 1927) is a Swiss-born American mathematician, writer and professor emeritus of Computer science and Mathematics at Purdue University in West Lafayette, Indiana. He is primarily known for his contributions to numerical analysis and has authored over 200 papers in his area and published four books. Early life and education Gautschi was born December 11, 1927, in Basel, Switzerland, to Heinrich Gautschi (1901-1975). His paternal family originally hailed from Reinach. His patrilineal uncle, Adolf Eduard Gautschi, was a custodian and landscape painter. He had one twin brother Werner (1927-1959). He completed a Ph.D. in mathematics from the University of Basel on the thesis ''Analyse graphischer Integrationsmethoden'' advised by Alexander Ostrowski and Andreas Speiser (1953). Career 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 H ... [...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 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 following equivalent expression: :P_n^ (z) = \frac \sum_^n \frac \left(\frac\right)^m. Rodrigues' formula An equivalent definition is given by Rodrigues' formula: :P_n^(z) = ... [...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)\\ =&\sum_ \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 hypergeometri ... [...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 on the open upper half plane. Definition 2 A de Branges space can also be defined as all entire f ... [...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 f(z) has a root at w, then f(z)/(z-w), taking the limit value at w, 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. Just as meromorphic functions can be viewed as a generalization of rational fractions, entire functions can be viewed as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
