|
Schwarzian Derivative
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces. It is named after the German mathematician Hermann Schwarz. Definition The Schwarzian derivative of a holomorphic function of one complex variable is defined by : (Sf)(z) = \left( \frac\right)' - \frac\left(\right)^2 = \frac-\frac\left(\right)^2. The same formula also defines the Schwarzian derivative of a function of one real variable. The alternative notation :\ = (Sf)(z) is frequently used. Properties The Schwarzian derivative of any Möbius transformation : g(z) = \frac is zero. Conversely, the Möbius transformations are the only functions with this property. Thus, the Schwarzian d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wronskian
In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian of two differentiable functions and is . More generally, for real- or complex-valued functions , which are times differentiable on an interval , the Wronskian as a function on is defined by W(f_1, \ldots, f_n) (x)= \begin f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n' (x)\\ \vdots & \vdots & \ddots & \vdots \\ f_1^(x)& f_2^(x) & \cdots & f_n^(x) \end,\quad x\in I. That is, it is the determinant of the matrix constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the th derivative, thus forming a square matrix. When the functions are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Triangle Function
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a Schwarz triangle, although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a Möbius triangle, the inverse of the Schwarz triangle function is a single-valued automorphic function for that triangle's triangle group. More specifically, it is a modular function. Formula Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle in radians. Each of ''α'', ''β'', and ''γ'' may take values between 0 and 1 inclusive. Following Nehari, these angles are in clockwise order, with the vertex having angle ''πα'' at the origin and the vertex having angle ''πγ'' lying on the real line. The Schwarz triangle function can be given in terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Reflection Principle
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane. In notation, if F(z) is a function that satisfies the above requirements, then its extension to the rest of the complex plane is given by the formula, F(\bar) = \overline. That is, we make the definition that agrees along the real axis. The result proved by Hermann Schwarz is as follows. Suppose that ''F'' is a continuous function on the closed upper half plane \left\ , holomorphic on the upper half plane \left\ , which takes real values on the real axis. Then the extension formula given above is an analytic continuation to the whole complex plane. In practice it would be better to have a theorem th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lamé Function
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials. The Lamé equation Lamé's equation is :\frac + (A+B\weierp(x))y = 0, where ''A'' and ''B'' are constants, and \wp is the Weierstrass elliptic function. The most important case is when B\weierp(x) = - \kappa^2 \operatorname^2x , where \operatorname is the elliptic sine function, and \kappa^2 = n(n+1)k^2 for an integer ''n'' and k the elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of ''B'' the solutions have branch points. By changing the independent variable to t with t=\operatorname x, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory Of Ordinary Differential Equations
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz–Christoffel Mapping
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction. They were introduced independently by Elwin Christoffel in 1867 and Hermann Schwarz in 1869. Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces, hyperbolic art, and fluid dynamics. Definition Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping ''f'' from the upper half-plane : \ to the interior of the polygon. The function ''f'' maps the real axis to the edges of the polygon. If the polygon has interior angles \alpha,\beta,\gamma, \ldots, then this mapping is given by : f(\zeta) = \int^\zeta \frac \,\mathrmw where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Mapping
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, 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 essentially unique: if ''z''0 is an element of ''U'' and φ 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeev Nehari
Zeev Nehari (born Willi Weissbach; 2 February 1915 – 1978) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ... who worked on Complex Analysis, Univalent Functions Theory and Differential and Integral Equations. He was a student of Michael (Mihály) Fekete. The Nehari manifold is named after him. Selected publications * * *Nehari, Zeev (1952), "Some inequalities in the theory of functions"Transactions of the American Mathematical Society 1953, Vol.75, pp. 256–286. * * References * External links * {{DEFAULTSORT:Nehari, Zeev 1915 births 1978 deaths Jewish emigrants from Nazi Germany to Mandatory Palestine 20th-century Israeli mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
