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11: 36.15 Methods of Computation
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 𝐱 = 𝟎 , the uniform asymptotic approximations of §36.12 can be used. … Direct numerical evaluation can be carried out along a contour that runs along the segment of the real t -axis containing all real critical points of Φ and is deformed outside this range so as to reach infinity along the asymptotic valleys of exp ( i Φ ) . …
§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 Φ , 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
A function is continuous on a point set D if it is continuous at all points of D . … …
Finite Integrals
Moreover, if a , b , c , d are finite or infinite constants and f ( x , y ) is piecewise continuous on the set ( a , b ) × ( c , d ) , 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
A (finite or countably infinite, generalizing the definition of (1.2.40)) set { v n } is an orthonormal set if the v n are normalized and pairwise orthogonal. … Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where f ( x ) is continuous, with convergence to ( f ( x 0 ) + f ( x 0 + ) ) / 2 if x 0 is an isolated point of discontinuity. … Assume T has no point spectrum, i. … More generally, continuous spectra may occur in sets of disjoint finite intervals [ λ a , λ b ] ( 0 , ) , often called bands, when q ( x ) is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … We assume a continuous spectrum λ 𝝈 c = [ 0 , ) , and a finite or countably infinite point spectrum 𝝈 p with elements λ n . …
14: 18.39 Applications in the Physical Sciences
where x is a spatial coordinate, m the mass of the particle with potential energy V ( x ) , = h / ( 2 π ) is the reduced Planck’s constant, and ( a , b ) a finite or infinite interval. … Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being L 2 and forming a complete set. … The finite system of functions ψ n is orthonormal in L 2 ( , d x ) , see (18.34.7_3). … These, taken together with the infinite sets of bound states for each l , form complete sets. … This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional L 2 functions of (18.39.53), provided that such information involves potentials, or projections onto L 2 functions, exactly expressed, or well approximated, in the finite basis of (18.39.44). …
15: 2.3 Integrals of a Real Variable
is finite and bounded for n = 0 , 1 , 2 , , then the n th error term (that is, the difference between the integral and n th partial sum in (2.3.2)) is bounded in absolute value by | q ( n ) ( 0 ) / ( x n ( x σ n ) ) | when x exceeds both 0 and σ n . … assume a and b are finite, and q ( t ) is infinitely differentiable on [ a , b ] . … Since q ( t ) need not be continuous (as long as the integral converges), the case of a finite integration range is included. … If p ( b ) is finite, then both endpoints contribute: … Assume also that 2 p ( α , t ) / t 2 and q ( α , t ) are continuous in α and t , and for each α the minimum value of p ( α , t ) in [ 0 , k ) is at t = α , at which point p ( α , t ) / t vanishes, but both 2 p ( α , t ) / t 2 and q ( α , t ) are nonzero. …
16: 3.1 Arithmetics and Error Measures
A nonzero normalized binary floating-point machine number x is represented as … …
IEEE Standard
Rounding
17: Mathematical Introduction
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). …
complex plane (excluding infinity).
< is finite, or converges.
( a , b ] or [ a , b ) half-closed intervals.
lim inf least limit point.
set subtraction.
set of all integers.
18: 2.8 Differential Equations with a Parameter
in which u is a real or complex parameter, and asymptotic solutions are needed for large | u | that are uniform with respect to z in a point set 𝐃 in or . …
§2.8(ii) Case I: No Transition Points
§2.8(iii) Case II: Simple Turning Point
These results are valid when 𝒱 0 , α 2 ( ξ 1 / 2 B 0 ) and 𝒱 0 , α 2 ( ξ 1 / 2 B n 1 ) are finite. …
§2.8(v) Multiple and Fractional Turning Points
19: 26.12 Plane Partitions
A plane partition, π , of a positive integer n , is a partition of n in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … An equivalent definition is that a plane partition is a finite subset of × × with the property that if ( r , s , t ) π and ( 1 , 1 , 1 ) ( h , j , k ) ( r , s , t ) , then ( h , j , k ) must be an element of π . …It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point ( h , j , k ) π . …
26.12.20 π × × q | π | = k = 1 1 ( 1 q k ) k ,
26.12.26 pp ( n ) ( ζ ( 3 ) ) 7 / 36 2 11 / 36 ( 3 π ) 1 / 2 n 25 / 36 exp ( 3 ( ζ ( 3 ) ) 1 / 3 ( 1 2 n ) 2 / 3 + ζ ( 1 ) ) ,
20: 1.8 Fourier Series
(1.8.10) continues to apply if either a or b or both are infinite and/or f ( x ) has finitely many singularities in ( a , b ) , provided that the integral converges uniformly (§1.5(iv)) at a , b , and the singularities for all sufficiently large λ . … Let f ( x ) be an absolutely integrable function of period 2 π , and continuous except at a finite number of points in any bounded interval. …at every point at which f ( x ) has both a left-hand derivative (that is, (1.4.4) applies when h 0 ) and a right-hand derivative (that is, (1.4.4) applies when h 0 + ). The convergence is non-uniform, however, at points where f ( x ) f ( x + ) ; see §6.16(i). … If a function f ( x ) C 2 [ 0 , 2 π ] is periodic, with period 2 π , then the series obtained by differentiating the Fourier series for f ( x ) term by term converges at every point to f ( x ) . …