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 $1/z$ lies outside the integration contour, $t^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty })$, 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 ${\varphi }_n$ are $\sigma$-bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma ,\sigma ]$. … ►The sequence ${\varphi }_n$, $n=0,1,2,\mathrm{\dots }$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^2(-\tau ,\tau )$. … ►for (fixed) , is given by …If , then $\mathrm{B}=1$. …

5: 18.2 General Orthogonal Polynomials

… ►Then $\{r_n(x)\}$ are OP’s on $(0,\mathrm{\infty })$ with respect to weight function $x^{-\frac{1}{2}}v(x)$ and $\{s_n(x)\}$ are OP’s on $(0,\mathrm{\infty })$ with respect to weight function $x^{\frac{1}{2}}v(x)$. … ►For OP’s $p_n(x)$ with weight function in the class $𝒢$ there are asymptotic formulas as $n\to \mathrm{\infty }$, respectively for $x\in ℂ$ 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 $𝒮$ discrete mass points $x_k$ outside $[-1,1]$ are allowed. If these $x_k$ satisfy 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 $d\mu \in 𝐌(a,b)$ then the interval $[b-a,b+a]$ is included in the support of $d\mu$, and outside $[b-a,b+a]$ the measure $d\mu$ 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 $[0,\mathrm{\infty })$ with discontinuities at ($0=$) , then … ►If $x^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty })$ for all $\sigma$ in , then the integral (1.14.32) converges and $ℳf\left(s\right)$ is an analytic function of $s$ in the vertical strip . … ►Suppose $x^{-\sigma }f(x)$ and $x^{\sigma -1}g(x)$ are absolutely integrable on $(0,\mathrm{\infty })$ and either $ℳg\left(\sigma +\mathrm{i}t\right)$ or $ℳf\left(1-\sigma -\mathrm{i}t\right)$ is absolutely integrable on $(-\mathrm{\infty },\mathrm{\infty })$. … ►If $x^{\sigma -1}f(x)$ and $x^{\sigma -1}g(x)$ are absolutely integrable on $(0,\mathrm{\infty })$, then for $s=\sigma +\mathrm{i}t$, … ►Suppose $f(t)$ is continuously differentiable on $(-\mathrm{\infty },\mathrm{\infty })$ and vanishes outside a bounded interval. …

7: 14.28 Sums

… ►When $\mathrm{\Re }{z}_1>0$, $\mathrm{\Re }{z}_2>0$, , and , …where the branches of the square roots have their principal values when $z_1,z_2\in (1,\mathrm{\infty })$ and are continuous when $z_1,z_2\in ℂ\setminus (0,1]$. … ►where ${ℰ}_1$ and ${ℰ}_2$ are ellipses with foci at $\pm 1$, ${ℰ}_2$ being properly interior to ${ℰ}_1$. The series converges uniformly for $z_1$ outside or on ${ℰ}_1$, and $z_2$ within or on ${ℰ}_2$. … ►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 $𝐱=(x,y)$ plane, where $\mathrm{ph}{\mathrm{\Psi }}_2\left(𝐱\right)$ is undetermined. All zeros have , 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 $n=0,1,2,\mathrm{\dots }$, 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 $W(a,x)$

… ►For real $\mu$ and $t$ oscillations occur outside the $t$-interval $[-1,1]$. … ►uniformly for $t\in [1+\delta ,\mathrm{\infty })$. … ►uniformly for $t\in [-1+\delta ,1-\delta ]$, with $\eta$ given by (12.10.23) and ${\widetilde{𝒜}}_s(t)$ given by (12.10.24). … ►uniformly for $t\in [-1+\delta ,\mathrm{\infty })$, with $\zeta$, $\varphi (\zeta )$, $A_s(\zeta )$, 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 $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. … ►For $Z_1{Z}_2=-1$ and $m=m_{\mathrm{e}}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_0=\mathrm{\hslash }/(m_{\mathrm{e}}c\alpha )$, and to a multiple of the Rydberg constant, … ►Customary variables are $(ϵ,r)$ in atomic physics and $(\eta ,\rho )$ in atomic and nuclear physics. Both variable sets may be used for attractive and repulsive potentials: the $(ϵ,r)$ 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. …