# outside the interval [0,1]

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## 10 matching pages

##### 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

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►There is no guaranteed convergence: the first approximation ${x}_{2}$ may be outside
$[{x}_{0},{x}_{1}]$.
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##### 3: 15.6 Integral Representations

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►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 $t=0$.
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##### 4: 30.15 Signal Analysis

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►Let $\tau $
$(>0)$ and $\sigma $
$(>0)$ be given.
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►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: 1.14 Integral Transforms

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►If $f(t)$ and ${f}^{\prime}(t)$ are piecewise continuous on $[0,\mathrm{\infty})$ with discontinuities at ($0=$) $$, then
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►Note: If $f(x)$ is continuous and $\alpha $ and $\beta $ are real numbers such that $f(x)=O\left({x}^{\alpha}\right)$ as $x\to 0+$ and $f(x)=O\left({x}^{\beta}\right)$ as $x\to \mathrm{\infty}$, then ${x}^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty})$ for all $\sigma \in (-\alpha ,-\beta )$.
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►Suppose ${x}^{-\sigma}f(x)$ and ${x}^{\sigma -1}g(x)$ are absolutely integrable on $(0,\mathrm{\infty})$ and either $\mathcal{M}g\left(\sigma +\mathrm{i}t\right)$ or $\mathcal{M}f\left(1-\sigma -\mathrm{i}t\right)$ is absolutely integrable on $(-\mathrm{\infty},\mathrm{\infty})$.
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►If ${x}^{\sigma -1}f(x)$ and ${x}^{\sigma -1}g(x)$ are absolutely integrable on $(0,\mathrm{\infty})$, then for $s=\sigma +\mathrm{i}t$,
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►Suppose $f(t)$ is continuously differentiable on $(-\mathrm{\infty},\mathrm{\infty})$ and vanishes outside a bounded interval.
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##### 6: 14.28 Sums

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►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 ${z}_{1},{z}_{2}\in \u2102\setminus (0,1]$.
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►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 ${\mathcal{E}}_{2}$.
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►1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively.
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##### 7: 36.7 Zeros

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►The zeros in Table 36.7.1 are points in the $\mathbf{x}=(x,y)$ plane, where $\mathrm{ph}{\mathrm{\Psi}}_{2}\left(\mathbf{x}\right)$ is undetermined.
All zeros have $$, and fall into two classes.
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►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
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►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.
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##### 8: 12.14 The Function $W(a,x)$

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►For real $\mu $ and $t$ oscillations occur outside the $t$-interval
$[-1,1]$.
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►uniformly for $t\in [1+\delta ,\mathrm{\infty})$.
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►uniformly for $t\in [-1+\delta ,1-\delta ]$, with $\eta $ given by (12.10.23) and ${\stackrel{~}{\mathcal{A}}}_{s}(t)$ given by (12.10.24).
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►uniformly for $t\in [-1+\delta ,\mathrm{\infty})$, with $\zeta $, $\varphi (\zeta )$, ${A}_{s}(\zeta )$, and ${B}_{s}(\zeta )$ as in §12.10(vii).
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►In the oscillatory intervals we write
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##### 9: 18.39 Physical Applications

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►For (18.39.2) to have a nontrivial bounded solution in the interval
$$, the constant $E$ (the total energy of the particle) must satisfy
►

18.39.4
$$E={E}_{n}=\left(n+\frac{1}{2}\right)\mathrm{\hslash}\omega ,$$
$n=0,1,2,\mathrm{\dots}$.

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►where $b={(\mathrm{\hslash}/m\omega )}^{1/2}$, and ${H}_{n}$ is the Hermite polynomial.
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►For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983).
►For applications and an extension of the Szegő–Szász inequality (18.14.20) for Legendre polynomials ($\alpha =\beta =0$) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974).
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##### 10: 33.22 Particle Scattering and Atomic and Molecular Spectra

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►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.
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►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,
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►Customary variables are $(\u03f5,r)$ in atomic physics and $(\eta ,\rho )$ in atomic and nuclear physics.
Both variable sets may be used for attractive and repulsive potentials: the $(\u03f5,r)$ 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.
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►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.
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