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Jensen's Formula
In complex analysis, Jensen's formula relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. The formula was introduced by and forms an important statement in the study of entire functions. Formal statement Suppose that f is an analytic function in a region in the complex plane \mathbb which contains the closed disk \mathbb_r of radius r>0 about the origin, a_1,a_2,\ldots,a_n are the zeros of f in the interior of \mathbb_r (repeated according to their respective multiplicity), and that f(0)\neq 0. Jensen's formula states that :\log , f(0), = -\sum_^n \log \left( \frac\right) + \frac \int_0^ \log, f(re^), \, d\theta. This formula establishes a connection between the moduli of the zeros of f in the interior of \mathbb_r and the average of \log, f(z), on the boundary circle , z, =r, and can be seen as a generalisation of the mean value property of harmonic functions. Namely, if f has no zeros in \mathbb_r, then J ... [...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] |
Acta Mathematica
''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics. According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journals".. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 4.273, ranking it 5th out of 330 journals in the category "Mathematics". Publication history The journal was established by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences. The journal was printed and distributed by Springer from 2006 to 2016. Since 2017, Acta Mathematica has been published electronically and in print by International Press. Its electronic version is open access without publishing fees. Poincaré episode The journal's "most famous episode" (according to Villani) concerns Henri Poincaré, who won a prize offered in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paley–Wiener Theorem
In mathematics, a Paley–Wiener theorem is a theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. It is named after Raymond Paley (1907–1933) and Norbert Wiener (1894–1964) who, in 1934, introduced various versions of the theorem. The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration). The original work by Paley and Wiener is also used as a namesake in the fields of control theory and harmonic analysis; introducing the Paley–Wiener condition for spectral factorization and the Paley–Wiener criterion for non-harmonic Fourier series respectively. These are related mathematical concepts that place the decay properties of a function in context of stability pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson. Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to ''n''-dimensional problems. Two-dimensional Poisson kernels On the unit disc In the complex plane, the Poisson kernel for the unit disc is given by P_r(\theta) = \sum_^\infty r^e^ = \frac = \operatorname\left(\frac\right), \ \ \ 0 \le r < 1. This can be thought of in two ways: either as a function of ''r'' and ''θ'', or as a family of functions of ''θ'' indexed by ''r''. If is the open unit disc in C, T is the boundary of the disc, and ''f'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rolf Nevanlinna
Rolf Herman Nevanlinna (né Neovius; 22 October 1895 – 28 May 1980) was a Finnish mathematician who made significant contributions to complex analysis. Background Nevanlinna was born Rolf Herman Neovius, becoming Nevanlinna in 1906 when his father changed the family name. The Neovius-Nevanlinna family contained many mathematicians: Edvard Engelbert Neovius (Rolf's grandfather) taught mathematics and topography at a military academy; Edvard Rudolf Neovius (Rolf's uncle) was a professor of mathematics at the University of Helsinki from 1883 to 1900; Lars Theodor Neovius-Nevanlinna (Rolf's uncle) was an author of mathematical textbooks; and Otto Wilhelm Neovius-Nevanlinna (Rolf's father) was a physicist, astronomer and mathematician. After Otto obtained his Ph.D. in physics from the University of Helsinki, he studied at the Pulkovo Observatory with the German astronomer Herman Romberg, whose daughter, Margarete Henriette Louise Romberg, he married in 1892. Otto and Margarete the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically, a Möbius transformation can be obtained by first applying the inverse stereographic projection from the plane to the unit sphere, moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group (mathematics), group called the Möbius group, which is the projective linear group . Together with its subgroups, it has numerous applications in mathematics and physics. Möbius geometry, Möbius geometries and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Matrix Spectral Factorization
Polynomial Matrix Spectral Factorization or Matrix Fejer–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial matrices are widely studied in the fields of systems theory and control theory and have seen other uses relating to stable polynomials. In stability theory, Spectral Factorization has been used to find determinantal matrix representations for bivariate stable polynomials and real zero polynomials. Given a univariate positive polynomial, i.e., p(t) > 0 for all t \in \mathbb, the Fejer–Riesz Theorem yields the polynomial spectral factorization p(t) = q(t)\bar(t). Results of this form are generically referred to as Positivstellensatz. Likewise, the Polynomial Matrix Spectral Factorization provides a factorization for positive definite polynomial matrices. This decomposition also relates to the Cholesky decomposition for scalar matrices A =LL^* . This result was originally proven by Norbert Wiener in a more general context whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control Stability theory, stability; often with the aim to achieve a degree of Optimal control, optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or Setpoint (control system), set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-analytic Function
In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If ''f'' is an analytic function on an interval 'a'',''b''nbsp;⊂ R, and at some point ''f'' and all of its derivatives are zero, then ''f'' is identically zero on all of 'a'',''b'' Quasi-analytic classes are broader classes of functions for which this statement still holds true. Definitions Let M=\_^\infty be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions ''C''''M''( 'a'',''b'' is defined to be those ''f'' ∈ ''C''∞( 'a'',''b'' which satisfy :\left , \frac(x) \right , \leq A^ k! M_k for all ''x'' ∈ 'a'',''b'' some constant ''A'', and all non-negative integers ''k''. If ''M''''k'' = 1 this is exactly the class of real analytic functions on 'a'',''b'' The class ''C''''M''( 'a'',''b'' is said to be ''quasi-analytic'' if whenever ''f'' ∈&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Factorization Theorem
In mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial. It is named for Jacques Hadamard. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root. It is closely related to Weierstrass factorization theorem, which does not restrict to entire functions with finite orders. Formal statement Define the Hadamard canonical factors E_n(z) := (1-z) \prod_^n e^Entire functions of finite order \rho have Hadamard's canonical representation:f(z)=z^me^\prod_^\infty E_p(z/a_n)where a_k are those roots of f that are not zero (a_k \neq 0), m is the order of the zero of f at z = 0 (the case m = 0 being taken to mean f(0) \neq 0), Q a polynomial (whose degree we shall call q), and p is the smallest n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if for every x_0 in its domain, its Taylor series about x_0 converges to the function in some neighborhood of x_0 . This is stronger than merely being infinitely differentiable at x_0 , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \cdots in which the coefficients a_0, a_1, \dots a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
