# over finite intervals

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## 1—10 of 12 matching pages

##### 1: 10.22 Integrals

##### 2: 10.43 Integrals

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###### §10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty})$

…##### 3: 2.8 Differential Equations with a Parameter

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►and for simplicity $\xi $ is assumed to range over a finite or infinite interval
$({\alpha}_{1},{\alpha}_{2})$ with $$, ${\alpha}_{2}>0$.
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##### 4: 1.4 Calculus of One Variable

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►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. …##### 5: 2.1 Definitions and Elementary Properties

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►If $c$ is a finite limit point of $\mathbf{X}$, then
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►For (2.1.14) $\mathbf{X}$ can be the positive real axis or any unbounded sector in $\u2102$ of finite angle.
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►Similarly for finite limit point $c$ in place of $\mathrm{\infty}$.
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►where $c$ is a finite, or infinite, limit point of $\mathbf{X}$.
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►Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over.
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##### 6: 24.17 Mathematical Applications

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►Let $0\le h\le 1$ and $a,m$, and $n$ be integers such that $n>a$, $m>0$, and ${f}^{(m)}(x)$ is absolutely integrable over
$[a,n]$.
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###### Calculus of Finite Differences

… ►Let ${\mathcal{S}}_{n}$ denote the class of functions that have $n-1$ continuous derivatives on $\mathbb{R}$ and are polynomials of degree at most $n$ in each interval $(k,k+1)$, $k\in \mathbb{Z}$. …##### 7: 2.4 Contour Integrals

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►The result in §2.3(ii) carries over to a complex parameter $z$.
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►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty})$.
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►in which $a$ is finite, $b$ is finite or infinite, and $\omega $ is the angle of slope of $\mathcal{P}$ at $a$, that is, $lim(\mathrm{ph}\left(t-a\right))$ as $t\to a$ along $\mathcal{P}$.
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$z$ ranges along a ray or over an annular sector ${\theta}_{1}\le \theta \le {\theta}_{2}$, $|z|\ge Z$, where $\theta =\mathrm{ph}z$, $$, and $Z>0$. $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$.

Excluding $t=a$, $\mathrm{\Re}\left({\mathrm{e}}^{\mathrm{i}\theta}p(t)-{\mathrm{e}}^{\mathrm{i}\theta}p(a)\right)$ is positive when $t\in \mathcal{P}$, and is bounded away from zero uniformly with respect to $\theta \in [{\theta}_{1},{\theta}_{2}]$ as $t\to b$ along $\mathcal{P}$.

##### 8: 1.5 Calculus of Two or More Variables

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###### Finite Integrals

… ►Suppose that $a,b,c$ are finite, $d$ is finite or $+\mathrm{\infty}$, and $f(x,y)$, $\partial f/\partial x$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times [c,d)$. … ►Then the*double integral*of $f(x,y)$ over $R$ is defined by … ►Moreover, if $a,b,c,d$ are finite or infinite constants and $f(x,y)$ is piecewise continuous on the set $(a,b)\times (c,d)$, then … ►Finite and infinite integrals can be defined in a similar way. …##### 9: 1.8 Fourier Series

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►For $f(x)$ piecewise continuous on $[a,b]$ and real $\lambda $,
…(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda $.
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►Let $f(x)$ be an absolutely integrable function of period $2\pi $, and continuous except at a finite number of points in any bounded interval.
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►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $|{f}^{\prime \prime}(x)|$ are integrable over
$(-\mathrm{\infty},\mathrm{\infty})$.
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►Suppose that $f(x)$ is continuous and of bounded variation on $[0,\mathrm{\infty})$.
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##### 10: 2.3 Integrals of a Real Variable

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►If, in addition, $q(t)$ is infinitely differentiable on $[0,\mathrm{\infty})$ and
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►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$.
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►Since $q(t)$ need not be continuous (as long as the integral converges), the case of a finite integration range is included.
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►oscillate rapidly and cancel themselves over most of the range.
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►If $p(b)$ is finite, then both endpoints contribute:
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