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Open Mapping Theorem (complex Analysis)
In complex analysis, the open mapping theorem states that if ''U'' is a domain of the complex plane C and ''f'' : ''U'' → C is a non-constant holomorphic function, then ''f'' is an open map (i.e. it sends open subsets of ''U'' to open subsets of C, and we have invariance of domain.). The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function ''f''(''x'') = ''x''2 is not an open map, as the image of the open interval (−1, 1) is the half-open interval [0, 1). The theorem for example implies that a non-constant holomorphic function cannot map an open disk ''onto'' a portion of any line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1. Proof Assume ''f'' : ''U'' → C is a non-constant holomorphic function and ''U'' is a domain of the complex plane. We have to show that ... [...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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Mapping Theorem (functional Analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. Classical (Banach space) form This proof uses the Baire category theorem, and completeness of both X and Y is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if X and Y are taken to be Fréchet spaces. Suppose A : X \to Y is a surjective continuous linear operator. In order to prove that A is an open map, it is sufficient to show that A maps the open unit ball in X to a neighborhood of the origin of Y. Let U = B_1^X(0), V = B_1^Y(0). Then X = \bigcup_ k U. Since A is surjective: Y = A(X) = A\left(\bigcup_ k U\right) = \bigcup_ A(kU). But Y is Banach so by Baire's category t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions. Statement Let \mathbf = \ be the open unit disk in the complex plane \mathbb centered at the 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), = , z, for some non-zero z or , f'(0), = 1, then f(z) = az for some a \in \mathbb with , a, = 1.Theorem 5.34 in Proof The proof is a straightforward application of the maximum modulus principle on the function :g(z) = \begin \frac\, & \mbox z \neq 0 \\ f'(0) & \mbox z = 0, \end which is holomorphic on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 local maximum that is properly 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 connected open subset ''D'' of the complex plane ℂ 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'' ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rouché's Theorem
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions and holomorphic inside some region K with closed contour \partial K, if on \partial K, then and have the same number of zeros inside K, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour \partial K is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below. Usage The theorem is usually used to simplify the problem of locating zeros, as follows. Given an analytic function, we write it as the sum of two parts, one of which is simpler and grows faster than (thus dominates) the other part. We can then locate the zeros by looking at only the dominating part. For example, the polynomial z^5 + 3z^3 + 7 has exactly 5 zeros in the disk , z, b > 0). By the quadratic formula it has two zeros at -a \pm \sqrt. Rouché's theorem can be used to obtain mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the spoke of a chariot wheel. as a function of axial position ../nowiki>" Spherical coordinates In a spherical coordinate system, the radius describes the distance of a point from a fixed origin. Its position if further defined by the polar angle measured between the radial direction and a fixed zenith direction, and the azimuth angle, the angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and a fixed reference direction in that plane. See also *Bend radius *Filling radius in Riemannian geometry *Radius of convergence * Radius of convexity *Radius of curvature *Radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extreme Value Theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> such that: f(c) \ge f(x) \ge f(d)\quad \forall x\in ,b/math> The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function f on the closed interval ,b/math> is bounded on that interval; that is, there exist real numbers m and M such that: m \le f(x) \le M\quad \forall x \in , b This does not say that M and m are necessarily the maximum and minimum values of f on the interval ,b which is what the extreme value theorem stipulates must also be the case. The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a non-empty compact space to a subset of the real numbers attains a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''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 topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Theorem
In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions ''f'' and ''g'' analytic on a domain ''D'' (open and connected subset of \mathbb or \mathbb), if ''f'' = ''g'' on some S \subseteq D, where S has an accumulation point, then ''f'' = ''g'' on ''D''. Thus an analytic function is completely determined by its values on a single open neighborhood in ''D'', or even a countable subset of ''D'' (provided this contains a converging sequence). This is not true in general for real-differentiable functions, even infinitely real-differentiable functions. In comparison, analytic functions are a much more rigid notion. Informally, one sometimes summarizes the theorem by saying analytic functions are "hard" (as opposed to, say, continuous functions which are "soft"). The underpinning fact from which the theorem is established is the expandability of a holomorphic function into its Taylor series. The connectednes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Point
In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the closure of the complement of . In this sense interior and closure are dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). Definitions Interior point If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is illustrated in the introductory section to this article.) This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists r > 0, such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
