over finite intervals

(0.001 seconds)

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: 1.5 Calculus of Two or More Variables
Finite Integrals
Suppose that $a,b,c$ are finite, $d$ is finite or $+\infty$, and $f(x,y)$, $\ifrac{\partial f}{\partial x}$ 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. …