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}+\cdots,

x\to c

\mathbf{X}

ⓘ

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:: pMML, png, TeX
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\leq\operatorname{ph}x\leq\beta

, then by definition the asymptotic property (2.1.13) holds uniformly with respect to

\operatorname{ph}x\in[\alpha,\beta]

|x|\to\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: 1.6 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

. …

4: 1.9 Calculus of a Complex Variable

… ►An open set in

\mathbb{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

\mathbb{C}

(or

\mathbb{C}\cup\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

\mathbb{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

\ifrac{\partial u}{\partial x}

\ifrac{\partial u}{\partial y}

\ifrac{\partial v}{\partial x}

, and

\ifrac{\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=\liminf_{n\to\infty}{\left|a_{n}\right|}^{-1/n}.

ⓘ

Defines:: $R$ : radius of convergence (locally)
Symbols:: $\liminf$ : least limit point, $n$ : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.9.E49
Encodings:: pMML, png, TeX
See also:: Annotations for §1.9(vi), §1.9 and Ch.1

…

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}

\mathbb{R}

\mathbb{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^{\infty}

or analytic on

\mathbf{\Delta}

, and as

u\to\infty

…

6: 21.3 Symmetry and Quasi-Periodicity

… ►The set of points

\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}

form a

g

-dimensional lattice, the period lattice of the Riemann theta function. …

7: 18.2 General Orthogonal Polynomials

… ►Let

X

be a finite set of distinct points on

\mathbb{R}

, or a countable infinite set of distinct points on

\mathbb{R}

, and

w_{x}

x\in X

, be a set of positive constants. …when

X

is a finite set of

N+1

distinct points. … ►If the polynomials

p_{n}(x)

(

n=0,1,\ldots,N

) are orthogonal on a finite set

X

N+1

distinct points as in (18.2.3), then the polynomial

p_{N+1}(x)

of degree

N+1

, up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on

X

. …

8: 26.13 Permutations: Cycle Notation

… ►An element of

\mathfrak{S}_{n}

with

a_{1}

fixed points,

a_{2}

cycles of length

2,\ldots,a_{n}

cycles of length

n

, where

n=a_{1}+2a_{2}+\cdots+na_{n}

, is said to have cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

. …

9: 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

\sqrt{1/\beta}\leq|a|<\sqrt{2/\beta}

, 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. …

10: 3.11 Approximation Techniques

… ►Here

x_{j}

j=1,2,\dots,J

, is a given set of distinct real points and

J\geq n+1

. …