on a point set

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1: 2.1 Definitions and Elementary Properties

… ►Let $\mathbf{X}$ be a point set with a limit point $c$. … ►

2.1.16 $$f(x)\sim a_0+a_1(x-c)+a_2{(x-c)}^2+\mathrm{\cdots },$$ $x\to c$ in $\mathbf{X}$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $f(x)$: function, $\mathbf{X}$: unbounded set, $c$: limit point and $a_s$: coefficients

Referenced by:

§2.1(iii), §2.1(iii), Erratum (V1.0.12) for Subsection 2.1(iii)

Permalink:

http://dlmf.nist.gov/2.1.E16

Encodings:

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See also:

Annotations for §2.1(iii), §2.1 and Ch.2

… ►If the set $\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha \le \mathrm{ph}x\le \beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty }$. …Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $\mathbf{U}$, and for each nonnegative integer $n$ …

2: 1.5 Calculus of Two or More Variables

… ►A function is continuous on a point set $D$ if it is continuous at all points of $D$. … … ►For $f(x,y)$ defined on a point set $D$ contained in a rectangle $R$, let … ►where $D$ is the image of $D^{*}$ under a mapping $(u,v)\to (x(u,v),y(u,v))$ which is one-to-one except perhaps for a set of points of area zero. …Again the mapping is one-to-one except perhaps for a set of points of volume zero. …

3: 1.9 Calculus of a Complex Variable

… ►An open set in $ℂ$ is one in which each point has a neighborhood that is contained in the set. ►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 $z_0$. … ►A domain $D$, say, is an open set in $ℂ$ that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. … ►

1.9.49 $$R=\underset{n\to \mathrm{\infty }}{lim\; inf}{|a_n|}^{-1/n}.$$

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Defines:

$R$: radius of convergence (locally)

Symbols:

$lim\; inf$: least limit point and $n$: nonnegative integer

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: 18.2 General Orthogonal Polynomials

… ►Let $X$ be a finite set of distinct points on $ℝ$, or a countable infinite set of distinct points on $ℝ$, and $w_x$, $x\in X$, be a set of positive constants. …when $X$ is a finite set of $N+1$ distinct points. …

5: 2.8 Differential Equations with a Parameter

… ►in which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $\mathbf{D}$ in $ℝ$ or $ℂ$. … ►Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi )$, $j=1,2$, that depend on arbitrarily chosen reference points ${\alpha }_j$, are $C^{\mathrm{\infty }}$ or analytic on $\mathbf{\Delta }$, and as $u\to \mathrm{\infty }$ …

6: 21.3 Symmetry and Quasi-Periodicity

… ►The set of points ${\mathbf{m}}_1+\mathbf{\Omega }{\mathbf{m}}_2$ form a $g$-dimensional lattice, the period lattice of the Riemann theta function. …

7: 1.6 Vectors and Vector-Valued Functions

… ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. …

8: 4.13 Lambert $W$-Function

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Figure 4.13.1: Branches $\mathrm{Wp}\left(x\right)$ and $\mathrm{Wm}\left(x\right)$ of the Lambert $W$-function. $A$ and $B$ denote the points $-1/\mathrm{e}$ and $\mathrm{e}$, respectively, on the $x$-axis. Magnify

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9: 26.13 Permutations: Cycle Notation

… ►An element of ${𝔖}_n$ with $a_1$ fixed points, $a_2$ cycles of length $2,\mathrm{\dots },a_n$ cycles of length $n$, where $n=a_1+2a_2+\mathrm{\cdots }+n{a}_n$, is said to have cycle type $(a_1,a_2,\mathrm{\dots },a_n)$. …

10: 22.19 Physical Applications

… ►As $a\to \sqrt{1/\beta }$ from below the period diverges since $a=\pm \sqrt{1/\beta }$ are points of unstable equilibrium. … ►For an initial displacement with , bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta }$. …As $a\to \sqrt{2/\beta }$ from below the period diverges since $x=0$ is a point of unstable equlilibrium. …