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42 relations: Analytic function, Atlas (topology), Base (topology), Complex analysis, Complex plane, Connected space, Connectedness, Covering space, Disk (mathematics), Domain (mathematical analysis), Domain of a function, Domain of holomorphy, Equivalence relation, Fabry gap theorem, Functional equation, Gamma function, George Pólya, Germ (mathematics), Holomorphic function, Holomorphic functional calculus, Identity theorem, Inverse function theorem, Lacunary function, Lars Ahlfors, Mathematics, Mittag-Leffler star, Multivalued function, Natural logarithm, Open set, Power series, Restriction (mathematics), Riemann surface, Riemann zeta function, Series (mathematics), Several complex variables, Sheaf (mathematics), Sheaf cohomology, Simply connected space, Singularity (mathematics), Topology, Transitive closure, Transitive relation.
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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Base (topology)
In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.We are using a convention that the union of empty collection of sets is the empty set.
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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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Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
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Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece".
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Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.
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Disk (mathematics)
In geometry, a disk (also spelled disc).
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Domain (mathematical analysis)
In mathematical analysis, a domain is any connected open subset of a finite-dimensional vector space.
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Domain of a function
In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.
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Domain of holomorphy
In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a set which is maximal in the sense that there exists a holomorphic function on this set which cannot be extended to a bigger set.
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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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Fabry gap theorem
In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them.
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Functional equation
In mathematics, a functional equation is any equation in which the unknown represents a function.
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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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George Pólya
George Pólya (Pólya György; December 13, 1887 – September 7, 1985) was a Hungarian mathematician.
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Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind which captures their shared local properties.
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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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Holomorphic functional calculus
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions.
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Identity theorem
In complex analysis, a branch of mathematics, the identity theorem for holomorphic functions states: given functions f and g holomorphic on a domain D (open and connected subset), if f.
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Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point.
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Lacunary function
In analysis, a lacunary function, also known as a lacunary series, is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within which it is defined by a power series.
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Lars Ahlfors
Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mittag-Leffler star
In complex analysis, a branch of mathematics, the Mittag-Leffler star of a complex-analytic function is a set in the complex plane obtained by attempting to extend that function along rays emanating from a given point.
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Multivalued function
In mathematics, a multivalued function from a domain to a codomain is a heterogeneous relation.
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Natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.
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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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Power series
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.
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Restriction (mathematics)
In mathematics, the restriction of a function f is a new function f\vert_A obtained by choosing a smaller domain A for the original function f. The notation f is also used.
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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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Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
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Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions on the n-tuples of complex numbers.
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
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Sheaf cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space.
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Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
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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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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Transitive closure
In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive.
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Transitive relation
In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.
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Analytic continuation into a domain of a function given on part of the boundary, Analytic continuations, Analytic extension, Analytical continuation, Analytically continued, Hadamard's gap theorem, Meromorphic continuation, Natural boundary.
References
[1] https://en.wikipedia.org/wiki/Analytic_continuation
