limit points (or limiting points)

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1: 2.1 Definitions and Elementary Properties

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

2: Mathematical Introduction

… ► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$lim\; inf$	least limit point.
…

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3: 1.9 Calculus of a Complex Variable

… ►A point $z_0$ is a limit point (limiting point or accumulation point) of a set of points $S$ in $ℂ$ (or $ℂ\cup \mathrm{\infty }$) 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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Defines:

$R$: radius of convergence (locally)

Symbols:

$lim\; inf$: least limit point, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.9.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(vi), §1.9 and Ch.1

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4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►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 $ℬ$ on $𝒟({ℒ}^{\ast })$ 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

… ►The number of branch points is infinite, but countable, and there are no finite limit points. …

6: 28.6 Expansions for Small $q$

… ►

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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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $lim\; inf$: least limit point, $n$: integer, $j$: integer, ${\rho }_n^{(j)}$: radii of convergence and $k$: root

Referenced by:

§28.6(i), §28.6(ii)

Permalink:

http://dlmf.nist.gov/28.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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7: 1.10 Functions of a Complex Variable

… ►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

… ►If $d\mu \in 𝐌(a,b)$ 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). …

9: 3.3 Interpolation

… ►For example, for $k+1$ coincident points the limiting form is given by $\left[z_0,z_0,\mathrm{\dots },z_0\right]f=f^{(k)}(z_0)/k!$. …

10: 1.15 Summability Methods

… ►at every point $\theta$ where both limits exist. …