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Jordan's Lemma
In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathematician Camille Jordan. Statement Consider a complex-valued, continuous function , defined on a semicircular contour :C_R = \ of positive radius lying in the upper half-plane, centered at the origin. If the function is of the form :f(z) = e^ g(z) , \quad z \in C , with a positive parameter , then Jordan's lemma states the following upper bound for the contour integral: :\left, \int_ f(z) \, dz \ \le \frac M_R \quad \text \quad M_R := \max_ \left, g \left(R e^\right) \ . with equality when vanishes everywhere, in which case both sides are identically zero. An analogous statement for a semicircular contour in the lower half-plane holds when . Remarks * If is continuous on the semicircular contour for all large and :then by Jordan's lemma \lim_ \int_ f(z)\, dz = 0. ... [...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] |
Jordan's Lemma
In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathematician Camille Jordan. Statement Consider a complex-valued, continuous function , defined on a semicircular contour :C_R = \ of positive radius lying in the upper half-plane, centered at the origin. If the function is of the form :f(z) = e^ g(z) , \quad z \in C , with a positive parameter , then Jordan's lemma states the following upper bound for the contour integral: :\left, \int_ f(z) \, dz \ \le \frac M_R \quad \text \quad M_R := \max_ \left, g \left(R e^\right) \ . with equality when vanishes everywhere, in which case both sides are identically zero. An analogous statement for a semicircular contour in the lower half-plane holds when . Remarks * If is continuous on the semicircular contour for all large and :then by Jordan's lemma \lim_ \int_ f(z)\, dz = 0. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Complex Analysis
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasonin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimation Lemma
In complex analysis, the estimation lemma, also known as the inequality, gives an upper bound for a contour integral. If is a complex-valued, continuous function on the contour and if its absolute value is bounded by a constant for all on , then :\left, \int_\Gamma f(z) \, dz\ \le M\, l(\Gamma), where is the arc length of . In particular, we may take the maximum :M:= \sup_, f(z), as upper bound. Intuitively, the lemma is very simple to understand. If a contour is thought of as many smaller contour segments connected together, then there will be a maximum for each segment. Out of all the maximum s for the segments, there will be an overall largest one. Hence, if the overall largest is summed over the entire path then the integral of over the path must be less than or equal to it. Formally, the inequality can be shown to hold using the definition of contour integral, the absolute value inequality for integrals and the formula for the length of a curve as follows: :\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concave Function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued function f on an interval (or, more generally, a convex set in vector space) is said to be ''concave'' if, for any x and y in the interval and for any \alpha \in ,1/math>, :f((1-\alpha )x+\alpha y)\geq (1-\alpha ) f(x)+\alpha f(y) A function is called ''strictly concave'' if :f((1-\alpha )x+\alpha y) > (1-\alpha ) f(x)+\alpha f(y) for any \alpha \in (0,1) and x \neq y. For a function f: \mathbb \to \mathbb, this second definition merely states that for ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Integral
In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, although that is typically reserved for #Complex line integral, line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the Dot product, scalar product of the vector field with a Differential (infinitesimal), differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on interval (mathematics), intervals. Many simple formulae in physics, such as the definition of Work (physics), work as have natural continuous analogues in terms of l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Pole
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point is a pole of a function if it is a zero of the function and is holomorphic (i.e. complex differentiable) in some neighbourhood of . A function is meromorphic in an open set if for every point of there is a neighborhood of in which at least one of and is holomorphic. If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros. Definitions A function of a complex variable is holo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue (complex Analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f\colon \mathbb \setminus \_k \rightarrow \mathbb that is holomorphic except at the discrete points ''k'', even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem. Definition The residue of a meromorphic function f at an isolated singularity a, often denoted \operatorname(f,a), \operatorname_a(f), \mathop_f(z) or \mathop_f(z), is the unique value R such that f(z)- R/(z-a) has an analytic antiderivative in a punctured disk 0<\vert z-a\vert<\delta. Alternatively, residues can be calculated by finding |
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] |
Estimation Lemma
In complex analysis, the estimation lemma, also known as the inequality, gives an upper bound for a contour integral. If is a complex-valued, continuous function on the contour and if its absolute value is bounded by a constant for all on , then :\left, \int_\Gamma f(z) \, dz\ \le M\, l(\Gamma), where is the arc length of . In particular, we may take the maximum :M:= \sup_, f(z), as upper bound. Intuitively, the lemma is very simple to understand. If a contour is thought of as many smaller contour segments connected together, then there will be a maximum for each segment. Out of all the maximum s for the segments, there will be an overall largest one. Hence, if the overall largest is summed over the entire path then the integral of over the path must be less than or equal to it. Formally, the inequality can be shown to hold using the definition of contour integral, the absolute value inequality for integrals and the formula for the length of a curve as follows: :\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof. Statement of Cauchy's residue theorem The statement is as follows: Residue theorem: Let U be a simply connected open subset of the complex plane containing a finite list of points a_1, \ldots, a_n, U_0 = U \smallsetminus \, and a function f holomorphic function, holomorphic on U_0. Letting \gamma be a closed rectifiable curve in U_0, and denoting the residue (complex analysis), residue of f at each point a_k by \operatorname(f, a_k) and the winding number of \gamma around a_k by \operatorname(\gamma, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants. Affine geometry The affine transformations of the upper half-plane include # shifts (x,y)\mapsto (x+c,y), c\in\mathbb, and # dilations (x,y)\mapsto (\lambda x,\lambda y), \lambda > 0. Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes A to B. :Proof: First shift the center of to Then take \lambda=(\text\ B)/(\text\ A) and dilate. Then shift to the center of Inversive geometry Definition: \mathcal := \left\ . can be recognized as the circle of radius centered at and as the polar plot of \rho(\theta) = \cos \theta. Proposition: in and are collinear points. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
