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11—20 of 20 matching pages
11: 19.30 Lengths of Plane Curves
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►As increases, the eccentricity
decreases and the rate of change of arclength for a fixed value of is given by
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12: 23.5 Special Lattices
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►As functions of , and are decreasing and is increasing.
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13: Bibliography G
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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.
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14: 9.8 Modulus and Phase
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►As increases from to each of the functions , , , , , is increasing, and each of the functions , , is decreasing.
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15: 2.4 Contour Integrals
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►Paths on which is constant are also the ones on which
decreases most rapidly.
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16: 19.22 Quadratic Transformations
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►Ascent and descent correspond respectively to increase and decrease of in Legendre’s notation.
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17: 18.2 General Orthogonal Polynomials
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►If polynomials 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 are orthogonal by Favard’s theorem, see §18.2(viii), in that the existence of a bounded non-decreasing function on yielding the orthogonality realtion (18.2.4_5) is guaranteed.
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►For OP’s on with weight function and orthogonality relation (18.2.5_5) assume that and is non-decreasing in the interval .
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18: 2.11 Remainder Terms; Stokes Phenomenon
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►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.
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19: 18.14 Inequalities
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►The successive maxima of form a decreasing sequence for , and an increasing sequence for .
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20: 10.21 Zeros
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and are decreasing functions of when for .
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