# closure

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## 9 matching pages

##### 1: 1.4 Calculus of One Variable
1.4.33 $\mathcal{V}_{a,b}\left(f\right)=\sup\sum^{n}_{j=1}|f(x_{j})-f(x_{j-1})|,$
where the supremum is over all sets of points $x_{0} 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 $q(x)$ is $C^{\infty}$ 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 Figure 13.7.1: Regions 𝐑 1 , 𝐑 2 , 𝐑 ¯ 2 , 𝐑 3 , and 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 − 2 ⁢ a | . 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 $\phi$. …