on finite point sets
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11: 36.15 Methods of Computation
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►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7.
Close to the bifurcation set but far from , the uniform asymptotic approximations of §36.12 can be used.
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►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real -axis containing all real critical points of and is deformed outside this range so as to reach infinity along the asymptotic valleys of .
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§36.15(iv) Integration along Finite Contour
►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …12: 1.5 Calculus of Two or More Variables
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►A function is continuous on a point set
if it is continuous at all points of .
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Finite Integrals
… ►Moreover, if are finite or infinite constants and is piecewise continuous on the set , then … ►Again the mapping is one-to-one except perhaps for a set of points of volume zero. …13: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►A (finite or countably infinite, generalizing the definition of (1.2.40)) set
is an orthonormal set if the are normalized and pairwise orthogonal.
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►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where is continuous, with convergence to if is an isolated point of discontinuity.
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►Assume has no point spectrum, i.
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►More generally, continuous spectra may occur in sets of disjoint finite intervals , often called bands, when is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7).
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►We assume a continuous spectrum , and a finite or countably infinite point spectrum with elements .
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14: 18.39 Applications in the Physical Sciences
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►where is a spatial coordinate, the mass of the particle with potential energy , is the reduced Planck’s constant, and a finite or infinite interval.
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►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being and forming a complete set.
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►The finite system of functions is orthonormal in , see (18.34.7_3).
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►These, taken together with the infinite sets of bound states for each , form complete sets.
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►This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional functions of (18.39.53), provided that such information involves potentials, or projections onto functions, exactly expressed, or well approximated, in the finite basis of (18.39.44).
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15: 2.3 Integrals of a Real Variable
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►is finite and bounded for , then the th error term (that is, the difference between the integral and th partial sum in (2.3.2)) is bounded in absolute value by when exceeds both and .
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►assume and are finite, and is infinitely differentiable on .
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►Since need not be continuous (as long as the integral converges), the case of a finite integration range is included.
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►If is finite, then both endpoints contribute:
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►Assume also that and are continuous in and , and for each the minimum value of in is at , at which point
vanishes, but both and are nonzero.
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16: 3.1 Arithmetics and Error Measures
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►A nonzero normalized binary floating-point machine number
is represented as
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IEEE Standard
… ►Rounding
…17: Mathematical Introduction
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►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3).
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complex plane (excluding infinity). | |
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is finite, or converges. | |
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or | half-closed intervals. |
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least limit point. | |
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set subtraction. | |
set of all integers. | |
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18: 2.8 Differential Equations with a Parameter
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►in which is a real or complex parameter, and asymptotic solutions are needed for large that are uniform with respect to in a point set
in or .
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§2.8(ii) Case I: No Transition Points
… ►§2.8(iii) Case II: Simple Turning Point
… ►These results are valid when and are finite. … ►§2.8(v) Multiple and Fractional Turning Points
…19: 26.12 Plane Partitions
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►A plane partition, , of a positive integer , is a partition of in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns.
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►An equivalent definition is that a plane partition is a finite subset of with the property that if and , then must be an element of .
…It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point
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26.12.20
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26.12.26
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20: 1.8 Fourier Series
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►(1.8.10) continues to apply if either or or both are infinite and/or has finitely many singularities in , provided that the integral converges uniformly (§1.5(iv)) at , and the singularities for all sufficiently large .
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►Let be an absolutely integrable function of period , and continuous except at a finite number of points in any bounded interval.
…at every point at which has both a left-hand derivative (that is, (1.4.4) applies when ) and a right-hand derivative (that is, (1.4.4) applies when ).
The convergence is non-uniform, however, at points where ; see §6.16(i).
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►If a function is periodic, with period , then the series obtained by differentiating the Fourier series for term by term converges at every point to .
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