Cauchy Estimates

	Cauchy Estimates In mathematics, specifically in complex analysis, Cauchy's estimate gives local bounds for the derivatives of a holomorphic function. These bounds are optimal. Cauchy's estimate is also called Cauchy's inequality, but must not be confused with the Cauchy–Schwarz inequality. Statement and consequence Let f be a holomorphic function on the open ball B(a, r) in \mathbb C. If M is the sup of , f, over B(a, r), then Cauchy's estimate says: for each integer n > 0, :, f^(a), \le \frac M where f^ is the ''n''-th complex derivative of f; i.e., f' = \frac and f^ = (f^)^' (see ). Moreover, taking f(z) = z^n, a = 0, r = 1 shows the above estimate cannot be improved. As a corollary, for example, we obtain Liouville's theorem (complex analysis), Liouville's theorem, which says a bounded entire function is constant (indeed, let r \to \infty in the estimate.) Slightly more generally, if f is an entire function bounded by A + B, z, ^k for some constants A, B and some integer k > 0, then f is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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]
	Derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. '' Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Holomorphic Function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is '' analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term '' analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Cauchy–Schwarz Inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics. Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by . The corresponding inequality for integrals was published by and . Schwarz gave the modern proof of the integral version. Statement of the inequality The Cauchy–Schwarz inequality states that for all vectors \mathbf and \mathbf of an inner product space where \langle \cdot, \cdot \rangle is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Complex Derivative In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is ''analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''regu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Liouville's Theorem (complex Analysis) In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that , f(z), \leq M for all z\in\Complex is constant. Equivalently, non-constant holomorphic functions on \Complex have unbounded images. The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits two or more complex numbers must be constant. Statement Liouville's theorem: Every holomorphic function f:\mathbb C \to \mathbb C for which there exists a positive number M such that , f(z), \leq M for all z\in\Complex is constant. More succinctly, Liouville's theorem states that every bounded entire function must be constant. Proof This important theorem has several proofs. A standard analytical proof uses the fact that holomorphic functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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]
picture info	Differentiation Under The Integral Sign In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integrands are functions dependent on $x,$ the derivative of this integral is expressible as $\begin & \frac \left (\int_^ f(x,t)\,dt \right ) \\ &= f\big(x,b(x)\big)\cdot \frac b(x) - f\big(x,a(x)\big)\cdot \frac a(x) + \int_^\frac f(x,t) \,dt \end$ where the partial derivative $\tfrac$ indicates that inside the integral, only the variation of $f(x, t)$ with $x$ is considered in taking the derivative. In the special case where the function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Maximal 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]
	Equicontinuous In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus ''sequences'' of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of ''C''(''X''), the space of continuous functions on a compact Hausdorff space ''X'', is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in ''C''(''X'') is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions ''fn'' on either a metric space or a locally compact space is continuous. If, in addition, ''fn'' are holomorphic, then the limit is also holomorphic. The uniform b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]