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1: 28.6 Expansions for Small q
28.6.2 a 1 ( q ) = 1 + q 1 8 q 2 1 64 q 3 1 1536 q 4 + 11 36864 q 5 + 49 5 89824 q 6 + 55 94 37184 q 7 83 353 89440 q 8 + ,
28.6.3 b 1 ( q ) = 1 q 1 8 q 2 + 1 64 q 3 1 1536 q 4 11 36864 q 5 + 49 5 89824 q 6 55 94 37184 q 7 83 353 89440 q 8 + ,
28.6.4 a 2 ( q ) = 4 + 5 12 q 2 763 13824 q 4 + 10 02401 796 26240 q 6 16690 68401 45 86471 42400 q 8 + ,
For m = 3 , 4 , 5 , , … For the corresponding expansions of se m ( z , q ) for m = 3 , 4 , 5 , change cos to sin everywhere in (28.6.26). …
2: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of s ζ ( s + 1 ) and ζ ( s + k ) , k = 2 , 3 , 4 , 5 , 8 , for 0 s 1 (23D).

  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals < x 2 and 2 x < , with s = 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 4 , 10 8 , 10 12 .

  • 3: 4 Elementary Functions
    Chapter 4 Elementary Functions
    4: 24.2 Definitions and Generating Functions
    Table 24.2.1: Bernoulli and Euler numbers.
    n B n E n
    4 1 30 5
    Table 24.2.2: Bernoulli and Euler polynomials.
    n B n ( x ) E n ( x )
    4 x 4 2 x 3 + x 2 1 30 x 4 2 x 3 + x
    Table 24.2.3: Bernoulli numbers B n = N / D .
    n N D B n
    4 1 30 3.33333 3333 ×10⁻²
    Table 24.2.4: Euler numbers E n .
    n E n
    4 5
    Table 24.2.6: Coefficients e n , k of the Euler polynomials E n ( x ) = k = 0 n e n , k x k .
    k
    4 0 1 0 2 1
    5: 22.9 Cyclic Identities
    These identities are cyclic in the sense that each of the indices m , n in the first product of, for example, the form s m , p ( 4 ) s n , p ( 4 ) are simultaneously permuted in the cyclic order: m m + 1 m + 2 p 1 2 m 1 ; n n + 1 n + 2 p 1 2 n 1 . …
    22.9.17 d 1 , 4 ( 2 ) d 2 , 4 ( 2 ) d 3 , 4 ( 2 ) ± d 2 , 4 ( 2 ) d 3 , 4 ( 2 ) d 4 , 4 ( 2 ) + d 3 , 4 ( 2 ) d 4 , 4 ( 2 ) d 1 , 4 ( 2 ) ± d 4 , 4 ( 2 ) d 1 , 4 ( 2 ) d 2 , 4 ( 2 ) = k ( ± d 1 , 4 ( 2 ) + d 2 , 4 ( 2 ) ± d 3 , 4 ( 2 ) + d 4 , 4 ( 2 ) ) ,
    22.9.18 ( d 1 , 4 ( 2 ) ) 2 d 3 , 4 ( 2 ) ± ( d 2 , 4 ( 2 ) ) 2 d 4 , 4 ( 2 ) + ( d 3 , 4 ( 2 ) ) 2 d 1 , 4 ( 2 ) ± ( d 4 , 4 ( 2 ) ) 2 d 2 , 4 ( 2 ) = k ( d 1 , 4 ( 2 ) ± d 2 , 4 ( 2 ) + d 3 , 4 ( 2 ) ± d 4 , 4 ( 2 ) ) ,
    §22.9(iv) Typical Identities of Rank 4
    22.9.23 s 1 , 3 ( 4 ) d 1 , 3 ( 4 ) c 2 , 3 ( 4 ) c 3 , 3 ( 4 ) + s 2 , 3 ( 4 ) d 2 , 3 ( 4 ) c 3 , 3 ( 4 ) c 1 , 3 ( 4 ) + s 3 , 3 ( 4 ) d 3 , 3 ( 4 ) c 1 , 3 ( 4 ) c 2 , 3 ( 4 ) = κ 2 1 κ 2 ( s 1 , 3 ( 4 ) d 1 , 3 ( 4 ) + s 2 , 3 ( 4 ) d 2 , 3 ( 4 ) + s 2 , 3 ( 4 ) d 2 , 3 ( 4 ) ) .
    6: 10.12 Generating Function and Associated Series
    10.12.4 1 = J 0 ( z ) + 2 J 2 ( z ) + 2 J 4 ( z ) + 2 J 6 ( z ) + ,
    cos z = J 0 ( z ) 2 J 2 ( z ) + 2 J 4 ( z ) 2 J 6 ( z ) + ,
    1 2 z sin z = 4 J 2 ( z ) 16 J 4 ( z ) + 36 J 6 ( z ) .
    7: 36.5 Stokes Sets
    36.5.2 y 3 = 27 4 ( 27 5 ) x 2 = 1.32403 x 2 .
    x = B ± | y | 4 / 3 ,
    36.5.4 80 x 5 40 x 4 55 x 3 + 5 x 2 + 20 x 1 = 0 ,
    36.5.10 160 u 6 + 40 u 4 = Y 2 .
    36.5.20 f ( u , X ) = X 2 12 w + 4 w 3 2 w 2 ,
    8: Bibliography P
  • J. B. Parkinson (1969) Optical properties of layer antiferromagnets with K 2 NiF 4 structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.
  • E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.
  • P. C. B. Phillips (1986) The exact distribution of the Wald statistic. Econometrica 54 (4), pp. 881–895.
  • T. Prellberg and A. L. Owczarek (1995) Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.
  • W. H. Press and S. A. Teukolsky (1990) Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.
  • 9: Bibliography K
  • E. G. Kalnins, W. Miller, and P. Winternitz (1976) The group O ( 4 ) , separation of variables and the hydrogen atom. SIAM J. Appl. Math. 30 (4), pp. 630–664.
  • E. Kanzieper (2002) Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).
  • S. Koizumi (1976) Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.
  • K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.
  • T. H. Koornwinder (1989) Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.
  • 10: 6.3 Graphics
    See accompanying text
    Figure 6.3.3: | E 1 ( x + i y ) | , 4 x 4 , 4 y 4 . … Magnify 3D Help