limit point
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1: 2.1 Definitions and Elementary Properties
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►Let be a point set with a limit point
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As in
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►If is a finite limit point of , then
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►Similarly for finite limit point
in place of .
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►where is a finite, or infinite, limit point of .
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2: Mathematical Introduction
3: 1.9 Calculus of a Complex Variable
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►A point
is a limit point (limiting point or accumulation point) of a set of points
in (or ) if every neighborhood of contains a point of distinct from .
…As a consequence, every neighborhood of a limit point of contains an infinite number of points of .
Also, the union of and its limit points is the closure of .
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►A function is complex differentiable at a point
if the following limit exists:
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1.9.49
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4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►By Weyl’s alternative
equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for .
… A boundary value for the end point
is a linear form on of the form
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►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications.
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►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases.
…See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.
5: 28.7 Analytic Continuation of Eigenvalues
6: 28.6 Expansions for Small
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28.6.20
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7: 1.10 Functions of a Complex Variable
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►If , analytic in , equals on an arc in , or on just an infinite number of points with a limit point in , then they are equal throughout and is called an analytic continuation of .
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►If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles.
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8: 18.2 General Orthogonal Polynomials
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►If then the interval is included in the support of , and outside the measure only has discrete mass points
such that are the only possible limit points of the sequence , see Máté et al. (1991, Theorem 10).
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