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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] |
Nyquist Stability Criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, is a graphical technique for determining the stability criterion, stability of a linear dynamical system. Because it only looks at the Nyquist plot of the Open-loop controller, open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane Singularity (mathematics), singularities must be known). As a result, it can be applied to systems defined by non-rational functions, such as systems with delays. In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. In addition, there is a natural generalization to more complex systems with M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argument Principle1
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persuasion. Arguments are intended to determine or show the degree of truth or acceptability of another statement called a conclusion. The process of crafting or delivering arguments, argumentation, can be studied from three main perspectives: the logical, the dialectical and the rhetorical perspective. In logic, an argument is usually expressed not in natural language but in a symbolic formal language, and it can be defined as any group of propositions of which one is claimed to follow from the others through deductively valid inferences that preserve truth from the premises to the conclusion. This logical perspective on argument is relevant for scientific fields such as mathematics and computer science. Logic is the study of the forms of reasoni ... [...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] |
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real analysis), pioneered the field complex analysis, and the study of permutation groups in abstract algebra. Cauchy also contributed to a number of topics in mathematical physics, notably continuum mechanics. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: : "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific worker; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. Biography Youth and education Cauchy was the son of Lou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Smithies
Frank Smithies FRSE (10 March 1912 – 16 November 2002) was a British mathematician who worked on integral equations, functional analysis, and the history of mathematics. He was elected as a fellow of the Royal Society of Edinburgh in 1961. He was an alumnus and an academic of Cambridge University The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo .... Publications * * References * * External links * * * {{DEFAULTSORT:Smithies, Frank 20th-century British mathematicians Fellows of the Royal Society of Edinburgh 1912 births 2002 deaths Mathematical analysts Alumni of the University of Cambridge Academics of the University of Cambridge Scientists from Edinburgh British historians of mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control Stability theory, stability; often with the aim to achieve a degree of Optimal control, optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or Setpoint (control system), set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel–Plana Formula
In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that \sum_^f\left(a+n\right)= \frac+\int_^f\left(x\right)dx+i\int_^\fracdt For the case a=0 we have :\sum_^\infty f(n)=\frac + \int_0^\infty f(x) \, dx+ i \int_0^\infty \frac \, dt. It holds for functions ''ƒ'' that are holomorphic in the region Re(''z'') ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that , ''ƒ'', is bounded by ''C''/, ''z'', 1+''ε'' in this region for some constants ''C'', ''ε'' > 0, though the formula also holds under much weaker bounds. . An example is provided by the Hurwitz zeta function, :\zeta(s,\alpha)= \sum_^\infty \frac = \frac + \frac 1 + 2\int_0^\infty\frac\frac, which holds for all s \in \mathbb, . Another powerful example is applying the formula to the function e^n^: we obtain \Gamma(x+1)=\operatorname_\left(e^\right)+\theta(x) where \Gamma(x) is the gamma f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Sum Symmetric Polynomial
In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the ''rationals,'' but not over the ''integers.'' Definition The power sum symmetric polynomial of degree ''k'' in n variables ''x''1, ..., ''x''''n'', written ''p''''k'' for ''k'' = 0, 1, 2, ..., is the sum of all ''k''th powers of the variables. Formally, : p_k (x_1, x_2, \dots,x_n) = \sum_^n x_i^k \, . The first few of these polynomials are :p_0 (x_1, x_2, \dots,x_n) = 1 + 1 + \cdots + 1 = n \, , :p_1 (x_1, x_2, \dots,x_n) = x_1 + x_2 + \cdots + x_n \, , :p_2 (x_1, x_2, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...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 so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
