|
Riemann Mapping Theorem
In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane \mathbb which is not all of \mathbb, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U onto the open unit disk :D = \. This mapping is known as a Riemann mapping. Intuitively, the condition that U be simply connected means that U does not contain any “holes”. The fact that f is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Henri Poincaré proved that the map f is unique up to rotation and recentering: if z_0 is an element of U and \phi is an arbitrary angle, then there exists precisely one ''f'' as above such that f(z_0)=0 and such that the argument of the derivative of f at th ... [...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] |
Lars Ahlfors
Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his textbook on complex analysis. Background Ahlfors was born in Helsinki, Finland. His mother, Sievä Helander, died at his birth. His father, Axel Ahlfors, was a professor of engineering at the Helsinki University of Technology. The Ahlfors family was Swedish-speaking, so he first attended the private school Nya svenska samskolan where all classes were taught in Swedish. Ahlfors studied at University of Helsinki from 1924, graduating in 1928 having studied under Ernst Lindelöf and Rolf Nevanlinna. He assisted Nevanlinna in 1929 with his work on Denjoy's conjecture on the number of asymptotic values of an entire function. In 1929 Ahlfors published the first proof of this conjecture, now known as the Denjoy–Carleman–Ahlfors theorem. It states that the number of asymptotic values approached by an entire function of order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipót Fejér
Lipót Fejér (or Leopold Fejér, ; 9 February 1880 – 15 October 1959) was a Hungarian mathematician of Jewish heritage. Fejér was born Leopold Weisz, and changed to the Hungarian name Fejér around 1900. Biography He was born in Pécs, Austria-Hungary, into the Jewish family of Victoria Goldberger and Samuel Weiss. His maternal great-grandfather Samuel Nachod was a doctor and his grandfather was a renowned scholar, author of a Hebrew-Hungarian dictionary. Leopold's father, Samuel Weiss, was a shopkeeper in Pecs. In primary schools Leopold was not doing well, so for a while his father took him away to home schooling. The future scientist developed his interest in mathematics in high school thanks to his teacher Sigismund Maksay. Fejér studied mathematics and physics at the University of Budapest and at the University of Berlin, where he was taught by Hermann Schwarz. In 1902 he earned his doctorate from University of Budapest (today Eötvös Loránd University). From 190 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Koebe
Paul Koebe (15 February 1882 – 6 August 1945) was a 20th-century German mathematician. His work dealt exclusively with the complex numbers, his most important results being on the uniformization of Riemann surfaces in a series of four papers in 1907–1909. He did his thesis at Berlin, where he worked under Hermann Schwarz. He was an extraordinary professor at Leipzig from 1910 to 1914, then an ordinary professor at the University of Jena before returning to Leipzig in 1926 as an ordinary professor. He died in Leipzig. He conjectured the Koebe quarter theorem on the radii of disks in the images of injective functions, in 1907. His conjecture became a theorem when it was proven by Ludwig Bieberbach in 1916, and the function f(z)=z/(1-z)^2 providing a tight example for this theorem became known as the Koebe function. Awards * 1922, Ackermann–Teubner Memorial Award See also * Koebe groups *Midsphere *Riemann mapping theorem In complex analysis, the Riemann mapping theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include Graph of a function, graphs of Multivalued function, multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a Surface (topology), surface: a two-dimensional real manifold, but it contains more structure (specifically a Complex Manifold, complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and Metrizabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Theorem (conformal Mapping)
In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, published by Carathéodory in 1913, states that any conformal mapping sending the unit disk to some region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions. Proofs of Carathéodory's theorem The first proof of Carathéodory's theorem presented here is a summary of the short self-contained account in ; there are related proofs in and . Clearly if ''f'' admits an extension to a homeomorphism, then ''∂U'' must be a Jordan curve. Conversely if ''∂U'' is a Jordan curve, the first step is to prove ''f'' extends continuously to the closure of ''D''. In fact this will hold if and only if ''f'' is uniformly continuo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential Theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which satisfy Poisson's equation—or in the vacuum, Laplace's equation. There is considerable overlap between potential theory and the theory of Poisson's equation to the extent that it is impossible to draw a distinction between these two fields. The difference is more one of emphasis than subject matter and rests on the following distinction: potential theory focuses on the properties of the functions as opposed to the properties of the equation. For example, a result about the Mathematical singularity, singularities of harmonic functions would be said to belong to potential theory whilst a result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constantin Carathéodory
Constantin Carathéodory (; 13 September 1873 – 2 February 1950) was a Greeks, Greek mathematician who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, and measure theory. He also created an axiomatic formulation of thermodynamics. Carathéodory is considered one of the greatest mathematicians of his era and the most renowned Greek mathematics, Greek mathematician since Ancient history, antiquity. Origins Constantin Carathéodory was born in 1873 in Berlin to Greeks, Greek parents and grew up in Brussels. His father , a lawyer, served as the Ottoman Empire, Ottoman ambassador to Belgium, St. Petersburg and Berlin. His mother, Despina, née Petrokokkinos, was from the island of Chios. The Carathéodory family, originally from Bosna, Edirne, Bosna, was well established and respected in Istanbul, Constantinople, and its members held many important governmental positions. His grandfather ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Functions
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L is a linear differential operator, then * the Green's function G is the solution of the equation where \delta is Dirac's delta function; * the solution of the initial-value problem L y = f is the convolution Through the superposition principle, given a linear ordinary differential equation (ODE), one can first solve for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead. Under many- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Fogg Osgood
William Fogg Osgood (March 10, 1864 – July 22, 1943) was an American mathematician. Education and career William Fogg Osgood was born in Boston on March 10, 1864. In 1886, he graduated from Harvard, where, after studying at the universities of Göttingen (1887–1889) and Erlangen ( Ph.D., 1890), he was instructor (1890–1893), assistant professor (1893–1903), and thenceforth professor of mathematics. From 1918 to 1922, he was chairman of the department of mathematics at Harvard. He became professor emeritus in 1933. From 1934 to 1936, he was visiting professor of mathematics at Peking University. From 1899 to 1902, he served as editor of the ''Annals of Mathematics'', and in 1905–1906 was president of the American Mathematical Society, whose ''Transactions'' he edited in 1909–1910. Contributions The works of Osgood dealt with complex analysis, in particular conformal mapping and uniformization of analytic functions, and calculus of variations. He was invited by Feli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain (mathematical Analysis)
In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space or the complex coordinate space . A connected open subset of coordinate space is frequently used for the domain of a function. The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term ''domain'', some use the term ''region'', some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as ''non-empty connected open subset''. Conventions One common convention is to define a ''domain'' as a connected open set but a ''region'' as the union of a domain w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Curve Theorem
In topology, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region Boundary (topology), bounded by the curve (not to be confused with the interior (topology), interior of a set) and an "exterior" region containing all of the nearby and far away exterior points. Every path (topology), continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it." (). More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. The Jordan curve theorem is named after the mathematician Camil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
