# closure

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## 9 matching pages

##### 1: 1.4 Calculus of One Variable

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►A generalization of the Riemann integral is the

*Stieltjes integral*${\int}_{a}^{b}f(x)d\alpha (x)$, where $\alpha (x)$ is a nondecreasing function on the closure of $(a,b)$, which may be bounded, or unbounded, and $d\alpha (x)$ is the*Stieltjes measure*. … ►For $\alpha (x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $d\alpha $ is*absolutely continuous*if $\alpha (x)$ is continuous and there exists a*weight function*$w(x)\ge 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that … ►where the supremum is over all sets of points $$ 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

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►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.
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##### 3: 3.7 Ordinary Differential Equations

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►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 $(a,b)$.
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►If $q(x)$ is ${C}^{\mathrm{\infty}}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used.
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##### 4: 1.9 Calculus of a Complex Variable

##### 5: 13.7 Asymptotic Expansions for Large Argument

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##### 6: 18.2 General Orthogonal Polynomials

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►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), $w(x)dx$ may be replaced in (18.2.1) by $d\mu (x)$, 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 $$ for all $n$. …##### 7: 2.10 Sums and Sequences

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