# limit points (or limiting points)

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##### 1: 2.1 Definitions and Elementary Properties

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►Let $\mathbf{X}$ be a point set with a limit point
$c$.
As $x\to c$ in $\mathbf{X}$
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►If $c$ is a finite limit point of $\mathbf{X}$, then
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►Similarly for finite limit point
$c$ in place of $\mathrm{\infty}$.
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►where $c$ is a finite, or infinite, limit point of $\mathbf{X}$.
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##### 2: Mathematical Introduction

##### 3: 1.9 Calculus of a Complex Variable

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►A point
${z}_{0}$ is a

*limit point*(*limiting point*or*accumulation point*) of a set of points $S$ in $\u2102$ (or $\u2102\cup \mathrm{\infty}$) if every neighborhood of ${z}_{0}$ contains a point of $S$ distinct from ${z}_{0}$. …As a consequence, every neighborhood of a limit point of $S$ contains an infinite number of points of $S$. Also, the union of $S$ and its limit points is the*closure*of $S$. … ►A function $f(z)$ is*complex differentiable*at a point $z$ if the following limit exists: … ►
1.9.49
$$R=\underset{n\to \mathrm{\infty}}{lim\; inf}{\left|{a}_{n}\right|}^{-1/n}.$$

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##### 4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►By

*Weyl’s alternative*${n}_{1}$ equals either 1 (the*limit point case*) or 2 (the*limit circle case*), and similarly for ${n}_{2}$. … A*boundary value*for the end point $a$ is a linear form $\mathcal{B}$ on $\mathcal{D}({\mathcal{L}}^{\ast})$ of the form … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. … ►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 $51$ 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 $q$

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►

28.6.20
$$\underset{n\to \mathrm{\infty}}{lim\; inf}\frac{{\rho}_{n}^{(j)}}{{n}^{2}}\ge k{k}^{\prime}{(K\left(k\right))}^{2}=\mathrm{2.04183\hspace{0.33em}4}\mathrm{\dots},$$

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##### 7: 1.10 Functions of a Complex Variable

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►If ${f}_{2}(z)$, analytic in ${D}_{2}$, equals ${f}_{1}(z)$ on an arc in $D={D}_{1}\cap {D}_{2}$, or on just an infinite number of points with a limit point in $D$, then they are equal throughout $D$ and ${f}_{2}(z)$ is called an

*analytic continuation*of ${f}_{1}(z)$. … ►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. …##### 8: 18.2 General Orthogonal Polynomials

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►If $d\mu \in \mathbf{M}(a,b)$ then the interval $[b-a,b+a]$ is included in the support of $d\mu $, and outside $[b-a,b+a]$ the measure $d\mu $ only has discrete mass points
${x}_{k}$ such that $b\pm a$ are the only possible limit points of the sequence $\{{x}_{k}\}$, see Máté et al. (1991, Theorem 10).
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##### 9: 3.3 Interpolation

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►For example, for $k+1$ coincident points the limiting form is given by $\left[{z}_{0},{z}_{0},\mathrm{\dots},{z}_{0}\right]f={f}^{(k)}({z}_{0})/k!$.
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