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Maximum Modulus Principle
In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus , f, cannot exhibit a strict maximum that is strictly within the domain of f. In other words, either f is locally a constant function, or, for any point z_0 inside the domain of f there exist other points arbitrarily close to z_0 at which , f, takes larger values. Formal statement Let f be a holomorphic function on some bounded and connected open subset D of the complex plane \mathbb and taking complex values. If z_0 is a point in D such that :, f(z_0), \ge , f(z), for all z in some neighborhood of z_0, then f is constant on D. This statement can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets: If , f, attains a local maximum at z, then the image of a sufficiently small open neighborhood of z cannot be open, so f is constant. Related statement Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Three-lines Theorem
In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard. Statement Define F(z) by : F(z)=f(z) M(a)^M(b)^ where , F(z), \leq 1 on the edges of the strip. The result follows once it is shown that the inequality also holds in the interior of the strip. After an affine transformation in the coordinate z, it can be assumed that a = 0 and b = 1. The function : F_n(z) = F(z) e^e^ tends to 0 as , z, tends to infinity and satisfies , F_n, \leq 1 on the boundary of the strip. The maximum modulus principle can therefore be applied to F_n in the strip. So , F_n(z), \leq 1. Because F_n(z) tends to F(z) as n tends to infinity, it follows that , F(z), \leq 1. ∎ Applications The three-line theorem can be used to prove the Hadamard three-circle theorem for a bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel–Carathéodory Theorem
In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory. Statement of the theorem Let a function f be analytic on a closed disc of radius ''R'' centered at the origin. Suppose that ''r'' 0 for some ''z'' on the circle , z, =R, so we may take A>0. Now ''f'' maps into the half-plane ''P'' to the left of the ''x''=''A'' line. Roughly, our goal is to map this half-plane to a disk, apply Schwarz's lemma there, and make out the stated inequality. w \mapsto w/A - 1 sends ''P'' to the standard left half-plane. w \mapsto R(w+1)/(w-1) sends the left half-plane to the circle of radius ''R'' centered at the origin. The composite, which maps 0 to 0, is the desired map: :w \mapsto \frac. From Schwarz's lemma applied to the composite of this map and ''f'', we have :\frac \leq , z, . Take , ''z'', & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phragmén–Lindelöf Principle
In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function f (i.e., , f(z), Background In the theory of complex functions, it is known that the modulus (absolute value) of a |
Schwarz's Lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise norm , \partial f , ^2 of a holomorphic map f:(X,g_X) \to (Y,g_Y) between Hermitian manifolds under curvature assumptions on g_X and g_Y. The classical Schwarz lemma is a result in complex analysis typically viewed to be about holomorphic functions from the open set, open unit disk \mathbb := \to itself. The Schwarz lemma has opened several branches of complex geometry, and become an essential tool in the use of geometric PDE methods in complex geometry. Statement of the classical Schwarz Lemma Let \mathbf = \ be the open unit disk in the complex number, complex plane \mathbb centered at the origin (mathematics), origin, and let f : \mathbf\rightarrow \mathbb be a holomorphic map such that f(0) = 0 and , f(z), \leq 1 on \mathbf. Then , f(z), \leq , z, for all z \in \mathbf, and , f'(0), \leq 1. Moreover, if , f(z), = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Algebra
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one complex Zero of a function, root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field (mathematics), field of complex numbers is Algebraically closed field, algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, Degree of a polynomial, degree ''n'' polynomial with complex coefficients has, counted with Multiplicity (mathematics)#Multiplicity of a root of a polynomial, multiplicity, exactly ''n'' complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, it is not fundamental for modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Equation
In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the heat equation and its variants have been found to be fundamental in many parts of both pure and applied mathematics. Definition Given an open subset of and a subinterval of , one says that a function is a solution of the heat equation if : \frac = \frac + \cdots + \frac, where denotes a general point of the domain. It is typical to refer to as time and as spatial variables, even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as . For any given value of , the right-hand side of the equation is the Laplace operator, Laplacian of the function . As such, the heat equation is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Integral Formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis. Theorem Let be an open subset of the complex plane , and suppose the closed disk defined as D = \bigl\ is completely contained in . Let be a holomorphic function, and let be the circle, oriented counterclockwise, forming the boundary of . Then for every in the interior of , f(a) = \frac \oint_\gamma \frac\,dz.\, The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Riemann Equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable. These equations are and where and are real differentiable function#Differentiability in higher dimensions, bivariate differentiable functions. Typically, and are respectively the real part, real and imaginary parts of a complex number, complex-valued function of a single complex variable where and are real variables; and are real differentiable functions of the real variables. Then is complex differentiable at a complex point if and only if the partial derivatives of and satisfy the Cauchy–Riemann equations at that point. A holomorphic function is a complex function that is differentiable at eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Principle
In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study. Solutions of a differential inequality in a domain ''D'' satisfy the maximum principle if they achieve their maxima at the boundary of ''D''. The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations. In a simple two-dimensional case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as \nabla^2 f = 0 or \Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
