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1: 1.4 Calculus of One Variable

… ►

1.4.33

\mathcal{V}_{a,b}\left(f\right)=\sup\sum^{n}_{j=1}|f(x_{j})-f(x_{j-1})|,

ⓘ

Defines:: $\mathcal{V}\left(\NVar{f}\right)$ : total variation and $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation
Symbols:: $\sup$ : least upper bound (supremum), $j$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.4.E33
Encodings:: pMML, png, TeX
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

►where the supremum is over all sets of points

x_{0}<x_{1}<\cdots<x_{n}

in the closure of

(a,b)

, that is,

(a,b)

with

a, b

added when they are finite. … ►If

f(x)

is continuous on the closure of

(a,b)

and

f^{\prime}(x)

is continuous on

(a,b)

, then …

2: 13.9 Zeros

… ►Let

P_{\alpha}

denote the closure of the domain that is bounded by the parabola

y^{2}=4\alpha(x+\alpha)

and contains the origin. …

3: 18.2 General Orthogonal Polynomials

… ►It is assumed throughout this chapter that for each polynomial

p_{n}(x)

that is orthogonal on an open interval

(a,b)

the variable

x

is confined to the closure of

(a,b)

unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … ►More generally than (18.2.1)–(18.2.3),

w(x)\,\mathrm{d}x

may be replaced in (18.2.1) by a positive measure

\,\mathrm{d}\alpha(x)

, where

\alpha(x)

is a bounded nondecreasing function on the closure of

(a,b)

with an infinite number of points of increase, and such that

0<\int_{a}^{b}x^{2n}\,\mathrm{d}\alpha(x)<\infty

for all

n

. …

4: 3.7 Ordinary Differential Equations

… ►Let

(a,b)

be a finite or infinite interval and

q(x)

be a real-valued continuous (or piecewise continuous) function on the closure of

(a,b)

. … ►If

q(x)

C^{\infty}

on the closure of

(a,b)

, then the discretized form (3.7.13) of the differential equation can be used. …

5: 1.9 Calculus of a Complex Variable

… ►Also, the union of

S

and its limit points is the closure of

S

. …

6: 13.7 Asymptotic Expansions for Large Argument

… ►

See accompanying text — Figure 13.7.1: Regions ${\textbf{R}}_{1}$ , ${\textbf{R}}_{2}$ , $\overline{\textbf{R}}_{2}$ , ${\textbf{R}}_{3}$ , and $\overline{\textbf{R}}_{3}$ are the closures of the indicated unshaded regions bounded by the straight lines and circular arcs centered at the origin, with $r=|b-2a|$ . Magnify

…

7: 2.10 Sums and Sequences

… ►

(a)

On the strip $a\leq\Re z\leq n$ , $f(z)$ is analytic in its interior, $f^{(2m)}(z)$ is continuous on its closure, and $f(z)=o\left(e^{2\pi|\Im z|}\right)$ as $\Im z\to\pm\infty$ , uniformly with respect to $\Re z\in[a,n]$ .

…

8: 18.18 Sums

… ►In this subsection the variables

x

and

y

are not confined to the closures of the intervals of orthogonality; compare §18.2(i). …

9: 1.16 Distributions

… ►The closure of the set of points where

\phi\not=0

is called the support of

\phi

. …