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44 relations: Analytic function, Atlas (topology), Branch point, Compact space, Complex analysis, Complex manifold, Complex plane, Differentiable function, Disk (mathematics), Essential singularity, Filter (signal processing), Filter design, Function (mathematics), Gamma function, Gauss–Lucas theorem, Holomorphic function, Hurwitz's theorem (complex analysis), Isolated point, Isomorphism, John Wiley & Sons, Laurent series, Logarithm, Marden's theorem, Meromorphic function, Multiplicative inverse, Neighbourhood (mathematics), Nyquist stability criterion, Open set, Point at infinity, Pole–zero plot, Polynomial, Rational function, Residue (complex analysis), Riemann hypothesis, Riemann sphere, Riemann surface, Riemann zeta function, Riemann–Roch theorem, Rouché's theorem, Sendov's conjecture, Singularity (mathematics), Square root, Taylor series, Zero of a function.
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.
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Atlas (topology)
In mathematics, particularly topology, one describes a manifold using an atlas.
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Branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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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 manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.
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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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Differentiable function
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.
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Disk (mathematics)
In geometry, a disk (also spelled disc).
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Essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.
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Filter (signal processing)
In signal processing, a filter is a device or process that removes some unwanted components or features from a signal.
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Filter design
Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which are contradictory.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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Gamma function
In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.
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Gauss–Lucas theorem
In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometrical relation between the roots of a polynomial P and the roots of its derivative P'.
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Holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
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Hurwitz's theorem (complex analysis)
In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit.
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Isolated point
In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton is an open set in the topological space S (considered as a subspace of X).
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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John Wiley & Sons
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
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Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.
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Logarithm
In mathematics, the logarithm is the inverse function to exponentiation.
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Marden's theorem
In mathematics, Marden's theorem, named after Morris Marden but proven much earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative.
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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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Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.
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Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
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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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Point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
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Pole–zero plot
In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as.
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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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Rational function
In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.
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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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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 sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.
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Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.
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Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.
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Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles.
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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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Sendov's conjecture
In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable.
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Singularity (mathematics)
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.
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Square root
In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.
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Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
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Zero of a function
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
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References
[1] https://en.wikipedia.org/wiki/Zeros_and_poles
