About the Project
NIST

outside the interval [0,1]

AdvancedHelp

(0.003 seconds)

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
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 , ) , and ( 1 - z t ) a = 1 at t = 0 . …
4: 30.15 Signal Analysis
Let τ ( > 0 ) and σ ( > 0 ) be given. … Equations (30.15.4) and (30.15.6) show that the functions ϕ n are σ -bandlimited, that is, their Fourier transform vanishes outside the interval [ - σ , σ ] . … The sequence ϕ n , n = 0 , 1 , 2 , forms an orthonormal basis in the space of σ -bandlimited functions, and, after normalization, an orthonormal basis in L 2 ( - τ , τ ) . … for (fixed) Λ 0 < α 1 , is given by …If 0 < α Λ 0 , then B = 1 . …
5: 1.14 Integral Transforms
If f ( t ) and f ( t ) are piecewise continuous on [ 0 , ) with discontinuities at ( 0 = ) t 0 < t 1 < < t n , then … Note: If f ( x ) is continuous and α and β are real numbers such that f ( x ) = O ( x α ) as x 0 + and f ( x ) = O ( x β ) as x , then x σ - 1 f ( x ) is integrable on ( 0 , ) for all σ ( - α , - β ) . … Suppose x - σ f ( x ) and x σ - 1 g ( x ) are absolutely integrable on ( 0 , ) and either g ( σ + i t ) or f ( 1 - σ - i t ) is absolutely integrable on ( - , ) . … If x σ - 1 f ( x ) and x σ - 1 g ( x ) are absolutely integrable on ( 0 , ) , then for s = σ + i t , … Suppose f ( t ) is continuously differentiable on ( - , ) and vanishes outside a bounded interval. …
6: 14.28 Sums
When z 1 > 0 , z 2 > 0 , | ph ( z 1 - 1 ) | < π , and | ph ( z 2 - 1 ) | < π , …where the branches of the square roots have their principal values when z 1 , z 2 ( 1 , ) and are continuous when z 1 , z 2 ( 0 , 1 ] . … where 1 and 2 are ellipses with foci at ± 1 , 2 being properly interior to 1 . The series converges uniformly for z 1 outside or on 1 , and z 2 within or on 2 . … 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …
7: 36.7 Zeros
The zeros in Table 36.7.1 are points in the x = ( x , y ) plane, where ph Ψ 2 ( x ) 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 , , 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. …
8: 12.14 The Function W ( a , x )
For real μ and t oscillations occur outside the t -interval [ - 1 , 1 ] . … uniformly for t [ 1 + δ , ) . … uniformly for t [ - 1 + δ , 1 - δ ] , with η given by (12.10.23) and 𝒜 ~ s ( t ) given by (12.10.24). … uniformly for t [ - 1 + δ , ) , with ζ , ϕ ( ζ ) , A s ( ζ ) , and B s ( ζ ) as in §12.10(vii). … In the oscillatory intervals we write …
9: 18.39 Physical Applications
For (18.39.2) to have a nontrivial bounded solution in the interval - < x < , the constant E (the total energy of the particle) must satisfy
18.39.4 E = E n = ( n + 1 2 ) ω , n = 0 , 1 , 2 , .
where b = ( / m ω ) 1 / 2 , and H n is the Hermite polynomial. … 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 ( α = β = 0 ) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974). …
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 π ε 0 s ) = Z 1 Z 2 α c / s , where Z j are atomic numbers, ε 0 is the electric constant, α is the fine structure constant, and 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 = / ( m e c α ) , and to a multiple of the Rydberg constant, … Customary variables are ( ϵ , r ) in atomic physics and ( η , ρ ) 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 ( η , ρ ) set cannot be used for zero energy E because this would imply ρ = 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. …