# over finite intervals

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## 1—10 of 14 matching pages

##### 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 $\alpha_{1}<0$, $\alpha_{2}>0$. …
##### 4: 18.39 Applications in the Physical Sciences
where $x$ is a spatial coordinate, $m$ the mass of the particle with potential energy $V(x)$, $\hbar=h/(2\pi)$ 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 $\left\langle u,v\right\rangle=0$. … Consider the second order differential operator acting on real functions of $x$ in the finite interval $[a,b]\subset\mathbb{R}$Let $X=(a,b)$ be a finite or infinite open interval in $\mathbb{R}$. … More generally, continuous spectra may occur in sets of disjoint finite intervals $[\lambda_{a},\lambda_{b}]\in(0,\infty)$, 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 $\lambda\in\boldsymbol{\sigma}_{c}=[0,\infty)$, and a finite or countably infinite point spectrum $\boldsymbol{\sigma}_{p}$ with elements $\lambda_{n}$. …
##### 6: 1.4 Calculus of One Variable
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. …
##### 7: 2.1 Definitions and Elementary Properties
If $c$ is a finite limit point of $\mathbf{X}$, then … For (2.1.14) $\mathbf{X}$ can be the positive real axis or any unbounded sector in $\mathbb{C}$ of finite angle. … Similarly for finite limit point $c$ in place of $\infty$. … where $c$ is a finite, or infinite, limit point of $\mathbf{X}$. … Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. …
##### 8: 24.17 Mathematical Applications
Let $0\leq h\leq 1$ 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 $\mathcal{S}_{n}$ 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 $(k,k+1)$, $k\in\mathbb{Z}$. …
##### 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 $\sigma\in[c,\infty)$. … in which $a$ is finite, $b$ is finite or infinite, and $\omega$ is the angle of slope of $\mathscr{P}$ at $a$, that is, $\lim(\operatorname{ph}\left(t-a\right))$ as $t\to a$ along $\mathscr{P}$. …
• (b)

$z$ ranges along a ray or over an annular sector $\theta_{1}\leq\theta\leq\theta_{2}$, $|z|\geq Z$, where $\theta=\operatorname{ph}z$, $\theta_{2}-\theta_{1}<\pi$, and $Z>0$. $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$.

• (c)

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

• ##### 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 $(-\infty,\infty)$. … Suppose that $f(x)$ is continuous and of bounded variation on $[0,\infty)$. …