# closure

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## 9 matching pages

##### 1: 1.4 Calculus of One Variable
A generalization of the Riemann integral is the Stieltjes integral $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)$, where $\alpha(x)$ is a nondecreasing function on the closure of $(a,b)$, which may be bounded, or unbounded, and $\,\mathrm{d}\alpha(x)$ is the Stieltjes measure. … For $\alpha(x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $\,\mathrm{d}\alpha$ is absolutely continuous if $\alpha(x)$ is continuous and there exists a weight function $w(x)\geq 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that … 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: 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. …
##### 4: 1.9 Calculus of a Complex Variable
Also, the union of $S$ and its limit points is the closure of $S$. …
##### 6: 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 $\,\mathrm{d}\mu(x)$, where the measure $\mu$ is the Lebesgue–Stieltjes measure $\mu_{\alpha}$ corresponding to a bounded nondecreasing function $\alpha$ on the closure of $(a,b)$ with an infinite number of points of increase, and such that $\int_{a}^{b}|x|^{n}\,\mathrm{d}\mu(x)<\infty$ for all $n$. …
##### 7: 2.10 Sums and Sequences
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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: 1.16 Distributions
The closure of the set of points where $\phi\not=0$ is called the support of $\phi$. …
##### 9: 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). …