over finite intervals

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1: 10.22 Integrals

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§10.22(ii) Integrals over Finite Intervals

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2: 10.43 Integrals

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

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3: 2.8 Differential Equations with a Parameter

… ►and for simplicity $\xi$ is assumed to range over a finite or infinite interval $({\alpha }_1,{\alpha }_2)$ with , ${\alpha }_2>0$. …

4: 18.39 Applications in the Physical Sciences

… ►where $x$ is a spatial coordinate, $m$ the mass of the particle with potential energy $V(x)$, $\mathrm{\hslash }=h/(2\pi )$ is the reduced Planck’s constant, and $(a,b)$ a finite or infinite interval. …

5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Two elements $u$ and $v$ in $V$ are orthogonal if $⟨u,v⟩=0$. … ►Consider the second order differential operator acting on real functions of $x$ in the finite interval $[a,b]\subset ℝ$ … ►Let $X=(a,b)$ be a finite or infinite open interval in $ℝ$. … ►More generally, continuous spectra may occur in sets of disjoint finite intervals $[{\lambda }_a,{\lambda }_b]\in (0,\mathrm{\infty })$, often called bands, when $q(x)$ is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … ►We assume a continuous spectrum $\lambda \in {𝝈}_c=[0,\mathrm{\infty })$, and a finite or countably infinite point spectrum ${𝝈}_p$ with elements ${\lambda }_n$. …

6: 1.4 Calculus of One Variable

… ►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. …

7: 2.1 Definitions and Elementary Properties

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

8: 24.17 Mathematical Applications

… ►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 $[a,n]$. … ►

Calculus of Finite Differences

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

9: 2.4 Contour Integrals

… ►The result in §2.3(ii) carries over to a complex parameter $z$. … ►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty })$. … ►in which $a$ is finite, $b$ is finite or infinite, and $\omega$ is the angle of slope of $𝒫$ at $a$, that is, $lim(\mathrm{ph}\left(t-a\right))$ as $t\to a$ along $𝒫$. … ►

$z$ 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 $z$.

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Excluding $t=a$, $\mathrm{\Re }\left({\mathrm{e}}^{\mathrm{i}\theta }p(t)-{\mathrm{e}}^{\mathrm{i}\theta }p(a)\right)$ is positive when $t\in 𝒫$, and is bounded away from zero uniformly with respect to $\theta \in [{\theta }_1,{\theta }_2]$ as $t\to b$ along $𝒫$.

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10: 1.8 Fourier Series

… ►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 $\lambda$. … ►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. … ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime \prime }(x)\right|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. … ►Suppose that $f(x)$ is continuous and of bounded variation on $[0,\mathrm{\infty })$. …