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11: 10.21 Zeros
§10.21(iv) Monotonicity Properties
For further monotonicity properties see Elbert (2001), Lorch (1990, 1993, 1995), Lorch and Muldoon (2008), Lorch and Szegő (1990, 1995), and Muldoon (1981). For inequalities for zeros arising from monotonicity properties see Laforgia and Muldoon (1983). …
12: Bibliography W
  • R. Wong and J.-M. Zhang (1994a) Asymptotic monotonicity of the relative extrema of Jacobi polynomials. Canad. J. Math. 46 (6), pp. 1318–1337.
  • 13: Bibliography
  • H. Alzer and S. Qiu (2004) Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.
  • 14: Bibliography G
  • P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.
  • 15: Bibliography K
  • S. Koumandos and M. Lamprecht (2010) Some completely monotonic functions of positive order. Math. Comp. 79 (271), pp. 1697–1707.
  • 16: 22.19 Physical Applications
    See accompanying text
    Figure 22.19.1: Jacobi’s amplitude function am ( x , k ) for 0 x 10 π and k = 0.5 , 0.9999 , 1.0001 , 2 . When k < 1 , am ( x , k ) increases monotonically indicating that the motion of the pendulum is unbounded in θ , corresponding to free rotation about the fulcrum; compare Figure 22.16.1. … Magnify
    17: 10.17 Asymptotic Expansions for Large Argument
    where 𝒱 denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that | t | changes monotonically. …
    18: 10.40 Asymptotic Expansions for Large Argument
    where 𝒱 denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that | t | changes monotonically. …
    19: Bibliography B
  • R. W. Barnard, K. Pearce, and K. C. Richards (2000) A monotonicity property involving F 2 3 and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.
  • 20: Bibliography M
  • M. E. Muldoon (1977) Higher monotonicity properties of certain Sturm-Liouville functions. V. Proc. Roy. Soc. Edinburgh Sect. A 77 (1-2), pp. 23–37.