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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 so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugène Rouché
Eugène Rouché (18 August 1832 – 19 August 1910) was a French mathematician. Career He was an alumnus of the École Polytechnique, which he entered in 1852. He went on to become professor of mathematics at the Charlemagne lyceum then at the École Centrale, and admissions examiner at his alma mater. He is best known for Rouché's theorem in complex analysis, which he published in his alma mater's institutional journal in 1862, and for the Rouché–Capelli theorem in linear algebra. His son, Jacques, was a noted patron of the arts who managed the Paris Opera for thirty years (1914–1944). See also * Rouché's theorem * Rouché–Capelli theorem Rouché–Capelli theorem is a theorem in linear algebra that determines the number of solutions of a system of linear equations, given the ranks of its augmented matrix and coefficient matrix. The theorem is variously known as the: * Rouché� ... References * Rouché et Comberousse (de), Traité de géométrie, tomes I et ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, ''double roots'' counted twice). Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of ''distinct'' elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct". Multiplicity of a prime factor In prime factorization, the multiplicity of a prime factor is its p-adic valuation. For example, the prime factorization of the integer is : the multiplicity of the prime factor is , while the multiplicity of each of the prime factors and is . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argument Principle
In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative. Formulation If ''f'' is a meromorphic function inside and on some closed contour ''C'', and ''f'' has no zeros or poles on ''C'', then : \frac\oint_ \, dz=Z-P where ''Z'' and ''P'' denote respectively the number of zeros and poles of ''f'' inside the contour ''C'', with each zero and pole counted as many times as its multiplicity and order, respectively, indicate. This statement of the theorem assumes that the contour ''C'' is simple, that is, without self-intersections, and that it is oriented counter-clockwise. More generally, suppose that ''f'' is a meromorphic function on an open set Ω in the complex plane and that ''C'' is a closed curve in Ω which avoids all zeros and poles of ''f'' and is contractible to a point inside Ω. For e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Formula
In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadratic equation of the form , with representing an unknown, and coefficients , , and representing known real number, real or complex number, complex numbers with , the values of satisfying the equation, called the Zero of a function, ''roots'' or ''zeros'', can be found using the quadratic formula, x = \frac, where the plus–minus sign, plus–minus symbol "" indicates that the equation has two roots. Written separately, these are: x_1 = \frac, \qquad x_2 = \frac. The quantity is known as the discriminant of the quadratic equation. If the coefficients , , and are real numbers then when , the equation has two distinct real number, real roots; when , the equation has one repeated root, repeated real root; and when , the equation h ... [...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] |
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] |
Open Mapping Theorem (complex Analysis)
In complex analysis, the open mapping theorem states that if U is a domain of the complex plane \mathbb and f: U\to \mathbb is a non-constant holomorphic function, then f is an open map (i.e. it sends open subsets of U to open subsets of \mathbb, 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\to \mathbb is a non-constant holomorphic function and U is a domain of the complex plane. We have to show that every point in f(U) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodor Estermann
Theodor Estermann (5 February 1902 – 29 November 1991) was a German-born British mathematician, working in the field of analytic number theory. The Estermann measure, a measure of the central symmetry of a convex set in the Euclidean plane, is named after him. He was born in Neubrandenburg, Germany, "to keen Zionists who named him in honour of Herzl." His doctorate, completed in 1925, was supervised by Hans Rademacher. He spent most of his career at University College London, eventually as a professor. Heini Halberstam, Klaus Roth and Robert Charles Vaughan were Ph.D. students of his. Though Estermann left Germany in 1929, before the Nazis seized power in 1933, some historians count him among the early emigrants who fled Nazi Germany. The physicist Immanuel Estermann Immanuel Estermann (; March 31, 1900 – March 30, 1973) was a Jewish German-born nuclear physicist and was professor at Carnegie Mellon University, University of Hamburg and Technion. Estermann is known fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
