outside the interval [0,1]

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1: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
For proofs of these results and further information see Walker (2003).
2: 3.8 Nonlinear Equations
There is no guaranteed convergence: the first approximation $x_{2}$ may be outside $[x_{0},x_{1}]$. …
3: 15.6 Integral Representations
In (15.6.2) the point $\ifrac{1}{z}$ lies outside the integration contour, $t^{b-1}$ and $(t-1)^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\infty)$, and $(1-zt)^{a}=1$ at $t=0$. …
4: 30.15 Signal Analysis
Let $\tau$ $(>0)$ and $\sigma$ $(>0)$ be given. … Equations (30.15.4) and (30.15.6) show that the functions $\phi_{n}$ are $\sigma$-bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma,\sigma]$. … The sequence $\phi_{n}$, $n=0,1,2,\dots$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^{2}(-\tau,\tau)$. … for (fixed) $\Lambda_{0}<\alpha\leq 1$, is given by …If $0<\alpha\leq\Lambda_{0}$, then $\mathrm{B}=1$. …
5: 18.2 General Orthogonal Polynomials
Then $\{r_{n}(x)\}$ are OP’s on $(0,\infty)$ with respect to weight function $x^{-\frac{1}{2}}v(x)$ and $\{s_{n}(x)\}$ are OP’s on $(0,\infty)$ with respect to weight function $x^{\frac{1}{2}}v(x)$. … For OP’s $p_{n}(x)$ with weight function in the class $\mathcal{G}$ there are asymptotic formulas as $n\to\infty$, respectively for $x\in\mathbb{C}$ outside $[-1,1]$ and for $x\in[-1,1]$, see Szegő (1975, Theorems 12.1.2, 12.1.4). … In further generalizations of the class $\mathcal{S}$ discrete mass points $x_{k}$ outside $[-1,1]$ are allowed. If these $x_{k}$ satisfy $\sum_{k}(|x_{k}|-1)^{\ifrac{1}{2}}<\infty$ then Szegő type asymptotics outside $[-1,1]$ can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following). … If $\,\mathrm{d}\mu\in{\mathbf{M}}(a,b)$ then the interval $[b-a,b+a]$ is included in the support of $\,\mathrm{d}\mu$, and outside $[b-a,b+a]$ the measure $\,\mathrm{d}\mu$ only has discrete mass points $x_{k}$ such that $b\pm a$ are the only possible limit points of the sequence $\{x_{k}\}$, see Máté et al. (1991, Theorem 10). …
6: 1.14 Integral Transforms
If $f(t)$ and $f^{\prime}(t)$ are piecewise continuous on $[0,\infty)$ with discontinuities at ($0=$) $t_{0}, then … If $x^{\sigma-1}f(x)$ is integrable on $(0,\infty)$ for all $\sigma$ in $a<\sigma, then the integral (1.14.32) converges and $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ is an analytic function of $s$ in the vertical strip $a<\Re s. … Suppose $x^{-\sigma}f(x)$ and $x^{\sigma-1}g(x)$ are absolutely integrable on $(0,\infty)$ and either $\mathscr{M}\mskip-3.0mug\mskip 3.0mu\left(\sigma+\mathrm{i}t\right)$ or $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-\sigma-\mathrm{i}t\right)$ is absolutely integrable on $(-\infty,\infty)$. … If $x^{\sigma-1}f(x)$ and $x^{\sigma-1}g(x)$ are absolutely integrable on $(0,\infty)$, then for $s=\sigma+\mathrm{i}t$, … Suppose $f(t)$ is continuously differentiable on $(-\infty,\infty)$ and vanishes outside a bounded interval. …
7: 14.28 Sums
When $\Re z_{1}>0$, $\Re z_{2}>0$, $|\operatorname{ph}\left(z_{1}-1\right)|<\pi$, and $|\operatorname{ph}\left(z_{2}-1\right)|<\pi$, …where the branches of the square roots have their principal values when $z_{1},z_{2}\in(1,\infty)$ and are continuous when $z_{1},z_{2}\in\mathbb{C}\setminus(0,1]$. … where $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are ellipses with foci at $\pm 1$, $\mathcal{E}_{2}$ being properly interior to $\mathcal{E}_{1}$. The series converges uniformly for $z_{1}$ outside or on $\mathcal{E}_{1}$, and $z_{2}$ within or on $\mathcal{E}_{2}$. … 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …
8: 36.7 Zeros
The zeros in Table 36.7.1 are points in the $\mathbf{x}=(x,y)$ plane, where $\operatorname{ph}\Psi_{2}\left(\mathbf{x}\right)$ is undetermined. All zeros have $y<0$, and fall into two classes. … Just outside the cusp, that is, for $x^{2}>8|y|^{3}/27$, there is a single row of zeros on each side. With $n=0,1,2,\dots$, they are located approximately at … Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. …
9: 12.14 The Function $W\left(a,x\right)$
For real $\mu$ and $t$ oscillations occur outside the $t$-interval $[-1,1]$. … uniformly for $t\in[1+\delta,\infty)$. … uniformly for $t\in[-1+\delta,1-\delta]$, with $\eta$ given by (12.10.23) and ${\widetilde{\cal A}}_{s}(t)$ given by (12.10.24). … uniformly for $t\in[-1+\delta,\infty)$, with $\zeta$, $\phi(\zeta)$, $A_{s}(\zeta)$, and $B_{s}(\zeta)$ as in §12.10(vii). … In the oscillatory intervals we write …
10: 33.22 Particle Scattering and Atomic and Molecular Spectra
With $e$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_{1}e,Z_{2}e$ and masses $m_{1},m_{2}$ separated by a distance $s$ is $V(s)=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar c/s$, where $Z_{j}$ are atomic numbers, $\varepsilon_{0}$ is the electric constant, $\alpha$ is the fine structure constant, and $\hbar$ is the reduced Planck’s constant. … For $Z_{1}Z_{2}=-1$ and $m=m_{e}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_{0}=\hbar/(m_{e}c\alpha)$, and to a multiple of the Rydberg constant, … Customary variables are $(\epsilon,r)$ in atomic physics and $(\eta,\rho)$ in atomic and nuclear physics. Both variable sets may be used for attractive and repulsive potentials: the $(\epsilon,r)$ set cannot be used for a zero potential because this would imply $r=0$ for all $s$, and the $(\eta,\rho)$ set cannot be used for zero energy $E$ because this would imply $\rho=0$ always. … The Coulomb solutions of the Schrödinger and Klein–Gordon equations are almost always used in the external region, outside the range of any non-Coulomb forces or couplings. …