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1: 5.4 Special Values and Extrema
§5.4 Special Values and Extrema
§5.4(iii) Extrema
5.4.20 x n = - n + 1 π arctan ( π ln n ) + O ( 1 n ( ln n ) 2 ) .
2: 18.4 Graphics
See accompanying text
Figure 18.4.2: Jacobi polynomials P n ( 1.25 , 0.75 ) ( x ) , n = 7 , 8 . This illustrates inequalities for extrema of a Jacobi polynomial; see (18.14.16). … Magnify
3: 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.
  • R. Wong and J.-M. Zhang (1994b) On the relative extrema of the Jacobi polynomials P n ( 0 , - 1 ) ( x ) . SIAM J. Math. Anal. 25 (2), pp. 776–811.
  • 4: Bibliography S
  • O. Szász (1950) On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.
  • O. Szász (1951) On the relative extrema of the Hermite orthogonal functions. J. Indian Math. Soc. (N.S.) 15, pp. 129–134.
  • 5: 3.5 Quadrature
    For the latter a = - 1 , b = 1 , and the nodes x k are the extrema of the Chebyshev polynomial T n ( x ) 3.11(ii) and §18.3). …
    6: Errata
  • Table 5.4.1


    The table of extrema for the Euler gamma function Γ had several entries in the x n column that were wrong in the last 2 or 3 digits. These have been corrected and 10 extra decimal places have been included.

    n x n Γ ( x n )
    0 1.46163 21449 68362 34126 0.88560 31944 10888 70028
    1 - 0.50408 30082 64455 40926 - 3.54464 36111 55005 08912
    2 - 1.57349 84731 62390 45878 2.30240 72583 39680 13582
    3 - 2.61072 08684 44144 65000 - 0.88813 63584 01241 92010
    4 - 3.63529 33664 36901 09784 0.24512 75398 34366 25044
    5 - 4.65323 77617 43142 44171 - 0.05277 96395 87319 40076
    6 - 5.66716 24415 56885 53585 0.00932 45944 82614 85052
    7 - 6.67841 82130 73426 74283 - 0.00139 73966 08949 76730
    8 - 7.68778 83250 31626 03744 0.00018 18784 44909 40419
    9 - 8.69576 41638 16401 26649 - 0.00002 09252 90446 52667
    10 - 9.70267 25400 01863 73608 0.00000 21574 16104 52285

    Reported 2018-02-17 by David Smith.