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11: 19.30 Lengths of Plane Curves
As λ increases, the eccentricity k decreases and the rate of change of arclength for a fixed value of ϕ is given by …
12: 23.5 Special Lattices
As functions of ω 3 , e 1 and e 2 are decreasing and e 3 is increasing. …
13: Bibliography G
  • B. Gabutti and B. Minetti (1981) A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.
  • 14: 9.8 Modulus and Phase
    As x increases from to 0 each of the functions M ( x ) , M ( x ) , | x | 1 / 4 N ( x ) , M ( x ) N ( x ) , θ ( x ) , ϕ ( x ) is increasing, and each of the functions | x | 1 / 4 M ( x ) , θ ( x ) , ϕ ( x ) is decreasing. …
    15: 2.4 Contour Integrals
    Paths on which ( z p ( t ) ) is constant are also the ones on which | exp ( z p ( t ) ) | decreases most rapidly. …
    16: 19.22 Quadratic Transformations
    Ascent and descent correspond respectively to increase and decrease of k in Legendre’s notation. …
    17: 18.2 General Orthogonal Polynomials
    If polynomials p n ( x ) are generated by recurrence relation (18.2.8) under assumption of inequality (18.2.9_5) (or similarly for the other three forms) then the p n ( x ) are orthogonal by Favard’s theorem, see §18.2(viii), in that the existence of a bounded non-decreasing function α ( x ) on ( a , b ) yielding the orthogonality realtion (18.2.4_5) is guaranteed. … For OP’s p n on [ a , b ] with weight function w ( x ) and orthogonality relation (18.2.5_5) assume that b < and w ( x ) is non-decreasing in the interval [ a , b ] . …
    18: 2.11 Remainder Terms; Stokes Phenomenon
    When a rigorous bound or reliable estimate for the remainder term is unavailable, it is unsafe to judge the accuracy of an asymptotic expansion merely from the numerical rate of decrease of the terms at the point of truncation. …
    19: 18.14 Inequalities
    The successive maxima of | H n ( x ) | form a decreasing sequence for x 0 , and an increasing sequence for x 0 . …
    20: 10.21 Zeros
    j ν , m / ν and j ν , m / ν are decreasing functions of ν when ν > 0 for m = 1 , 2 , 3 , . …