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1: 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.

  • 2: William P. Reinhardt
    Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …
    3: 24.2 Definitions and Generating Functions
    Table 24.2.3: Bernoulli numbers B n = N / D .
    n N D B n
    30 861 58412 76005 14322 6.01580 8739 ×10⁸
    Table 24.2.4: Euler numbers E n .
    n E n
    30 44 15438 93249 02310 45536 82821
    Table 24.2.5: Coefficients b n , k of the Bernoulli polynomials B n ( x ) = k = 0 n b n , k x k .
    k
    n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
    11 0 5 6 0 11 2 0 11 0 11 0 55 6 11 2 1
    Table 24.2.6: Coefficients e n , k of the Euler polynomials E n ( x ) = k = 0 n e n , k x k .
    k
    12 0 2073 0 3410 0 1683 0 396 0 55 0 6 1
    4: Mathematical Introduction
    The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … As a consequence, in addition to providing more information about the special functions that were covered in AMS 55, the NIST Handbook includes several special functions that have appeared in the interim in applied mathematics, the physical sciences, and engineering, as well as in other areas. … Two other ways in which this Handbook differs from AMS 55, and other handbooks, are as follows. … For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …
    5: 4.14 Definitions and Periodicity
    4.14.3 cos z ± i sin z = e ± i z ,
    6: 8.21 Generalized Sine and Cosine Integrals
    8.21.4 si ( a , z ) = z t a 1 sin t d t , a < 1 ,
    8.21.5 ci ( a , z ) = z t a 1 cos t d t , a < 1 ,
    8.21.18 f ( a , z ) = si ( a , z ) cos z ci ( a , z ) sin z ,
    8.21.22 f ( a , z ) = 0 sin t ( t + z ) 1 a d t ,
    8.21.23 g ( a , z ) = 0 cos t ( t + z ) 1 a d t .
    7: 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 + ,
    28.6.10 a 5 ( q ) = 25 + 1 48 q 2 + 11 7 74144 q 4 + 1 1 47456 q 5 + 37 8918 13888 q 6 + ,
    28.6.11 b 5 ( q ) = 25 + 1 48 q 2 + 11 7 74144 q 4 1 1 47456 q 5 + 37 8918 13888 q 6 + ,
    28.6.21 2 1 / 2 ce 0 ( z , q ) = 1 1 2 q cos 2 z + 1 32 q 2 ( cos 4 z 2 ) 1 128 q 3 ( 1 9 cos 6 z 11 cos 2 z ) + ,
    8: 6.14 Integrals
    For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).
    9: 10.12 Generating Function and Associated Series
    For z and t { 0 } , …
    10: 22.15 Inverse Functions
    Comprehensive treatments are given by Carlson (2005), Lawden (1989, pp. 52–55), Bowman (1953, Chapter IX), and Erdélyi et al. (1953b, pp. 296–301). …