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Argument Principle
In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative. Specifically, if ''f''(''z'') 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''(''z'') 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''(''z'') 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argument Principle1
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments 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 reasoning in arguments and the development of standards and criteria to evaluate arguments. Deductive arguments can be valid, and the valid ones can be sound: in a valid argument, premisses necessitate the conclusion, even if one or more of the premises is false and ... [...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 |
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. 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 writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Smithies
Frank Smithies FRSE (1912–2002) 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 , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola .... 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] |
Abel–Plana Formula
In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that :\sum_^\infty f(n)=\frac 1 2 f(0)+ \int_0^\infty f(x) \, dx+ i \int_0^\infty \frac \, dt. It holds for functions ''f'' 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 , ''f'', 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, . Abel also gave the following variation for alternating sums: :\sum_^\infty (-1)^nf(n)= \frac f(0)+i \int_0^\infty \frac \, dt. Which is related to the Lindelöf summation formula \sum_^(-1)^kf(k)=(-1)^m\int_^f(m-1/2+ix)\frac Proof Let f be holomorphic on \Re(z ... [...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, \d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. 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 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; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...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 of a dynamical system. Because it only looks at the Nyquist plot of the 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 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 multiple inputs and multiple outputs, such as control systems for airplan ... [...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] |
Riemann Xi Function
In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. Definition Riemann's original lower-case "xi"-function, \xi was renamed with an upper-case ~\Xi~ ( Greek letter "Xi") by Edmund Landau. Landau's lower-case ~\xi~ ("xi") is defined as :\xi(s) = \frac s(s-1) \pi^ \Gamma\left(\frac\right) \zeta(s) for s \in \mathbb. Here \zeta(s) denotes the Riemann zeta function and \Gamma(s) is the Gamma function. The functional equation (or reflection formula) for Landau's ~\xi~ is :\xi(1-s) = \xi(s)~. Riemann's original function, rebaptised upper-case ~\Xi~ by Landau, satisfies :\Xi(z) = \xi \left(\tfrac + z i \right), and obeys the functional equation :\Xi(-z) = \Xi(z)~. Both functions are entire and purely real for real arguments. Values The general form for positive even integers is :\xi(2n) = (-1)^\fracB_2^\pi^(2n-1) where ''Bn'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ζ(''s'') is a function whose argument ''s'' may be any complex number ... [...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. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem. Statement The statement is as follows: Let be a simply connected open subset of the complex plane containing a finite list of points , , and a function defined and holomorphic on . Let be a closed rectifiable curve in , and denote the winding number of around by . The line integral of around is equal to times the sum of residues of at the points, each counted as many times as winds around the point: \oint_\gamma f(z)\, dz = 2\pi i \sum_^n \operatorname(\gamma, a_k) \operatorname( f, a_k ). If is a positively oriented simple closed curve, if i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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