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22 relations: Abel–Plana formula, Argument (complex analysis), Augustin-Louis Cauchy, Complex analysis, Complex plane, Contour integration, Contractible space, Frank Smithies, Logarithmic derivative, Meromorphic function, Multiplicity (mathematics), Nyquist stability criterion, Open set, Polynomial, Power sum symmetric polynomial, Residue (complex analysis), Residue theorem, Riemann hypothesis, Riemann Xi function, Rouché's theorem, Winding number, Zeros and poles.
Abel–Plana formula
In mathematics, the Abel–Plana formula is a summation formula discovered independently by and.
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Argument (complex analysis)
In mathematics, the argument is a multi-valued function operating on the nonzero complex numbers.
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Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy FRS FRSE (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.
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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.
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Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
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Contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
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Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.
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Frank Smithies
Frank Smithies FRSE (10 March 1912, Edinburgh, Scotland – 16 November 2002, Cambridge, England) was a British mathematician who worked on integral equations, functional analysis, and the history of mathematics.
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Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where f' is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely f', scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f. This follows directly from the chain rule.
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Meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.
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Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.
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Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion, discovered by Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, on is a graphical technique for determining the stability of a dynamical system.
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Open set
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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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.
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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.
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Residue theorem
In complex analysis, a discipline within mathematics, 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 as well.
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Riemann hypothesis
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.
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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.
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Rouché's theorem
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K with closed contour \partial K, if |g(z)| \partial K, then f and f + g have the same number of zeros inside K, where each zero is counted as many times as its multiplicity.
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Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point.
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Zeros and poles
In mathematics, a zero of a function is a value such that.
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Redirects here:
Argument variation, Cauchy argument principle, Cauchy's argument principle, Principle of the argument, Variation of argument.
References
[1] https://en.wikipedia.org/wiki/Argument_principle
