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1: 4.17 Special Values and Limits
Table 4.17.1: Trigonometric functions: values at multiples of 1 12 π .
θ sin θ cos θ tan θ csc θ sec θ cot θ
π / 4 1 2 2 1 2 2 1 2 2 1
2 π / 3 1 2 3 1 2 3 2 3 3 2 1 3 3
3 π / 4 1 2 2 1 2 2 1 2 2 1
11 π / 12 1 4 2 ( 3 1 ) 1 4 2 ( 3 + 1 ) ( 2 3 ) 2 ( 3 + 1 ) 2 ( 3 1 ) ( 2 + 3 )
4.17.3 lim z 0 1 cos z z 2 = 1 2 .
2: 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 + ,
The coefficients of the power series of a 2 n ( q ) , b 2 n ( q ) and also a 2 n + 1 ( q ) , b 2 n + 1 ( q ) are the same until the terms in q 2 n 2 and q 2 n , respectively. … Here j = 1 for a 2 n ( q ) , j = 2 for b 2 n + 2 ( q ) , and j = 3 for a 2 n + 1 ( q ) and b 2 n + 1 ( q ) . … where k is the unique root of the equation 2 E ( k ) = K ( k ) in the interval ( 0 , 1 ) , and k = 1 k 2 . …
3: 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).

  • Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover ζ ( s ) for 0 s 1 (15D), ζ ( s + 1 ) for 0 s 1 (20D), and ln ξ ( 1 2 + i x ) 25.4) for 1 x 1 (20D). For errata see Piessens and Branders (1972).

  • Morris (1979) gives rational approximations for Li 2 ( x ) 25.12(i)) for 0.5 x 1 . Precision is varied with a maximum of 24S.

  • 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 .

  • 4: 24.2 Definitions and Generating Functions
    B 2 n + 1 = 0 ,
    ( 1 ) n + 1 B 2 n > 0 , n = 1 , 2 , .
    E 2 n + 1 = 0 ,
    ( 1 ) n E 2 n > 0 .
    24.2.9 E n = 2 n E n ( 1 2 ) = integer ,
    5: 28.16 Asymptotic Expansions for Large q
    Let s = 2 m + 1 , m = 0 , 1 , 2 , , and ν be fixed with m < ν < m + 1 . …
    28.16.1 λ ν ( h 2 ) 2 h 2 + 2 s h 1 8 ( s 2 + 1 ) 1 2 7 h ( s 3 + 3 s ) 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
    6: 28.15 Expansions for Small q
    28.15.1 λ ν ( q ) = ν 2 + 1 2 ( ν 2 1 ) q 2 + 5 ν 2 + 7 32 ( ν 2 1 ) 3 ( ν 2 4 ) q 4 + 9 ν 4 + 58 ν 2 + 29 64 ( ν 2 1 ) 5 ( ν 2 4 ) ( ν 2 9 ) q 6 + .
    28.15.2 a ν 2 q 2 a ( ν + 2 ) 2 q 2 a ( ν + 4 ) 2 = q 2 a ( ν 2 ) 2 q 2 a ( ν 4 ) 2 .
    28.15.3 me ν ( z , q ) = e i ν z q 4 ( 1 ν + 1 e i ( ν + 2 ) z 1 ν 1 e i ( ν 2 ) z ) + q 2 32 ( 1 ( ν + 1 ) ( ν + 2 ) e i ( ν + 4 ) z + 1 ( ν 1 ) ( ν 2 ) e i ( ν 4 ) z 2 ( ν 2 + 1 ) ( ν 2 1 ) 2 e i ν z ) + ;
    7: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
    For k = 2 M 2 is the number of permutations of { 1 , 2 , , n } with a 1 cycles of length 1, a 2 cycles of length 2, , and a n cycles of length n : … M 3 is the number of set partitions of { 1 , 2 , , n } with a 1 subsets of size 1, a 2 subsets of size 2, , and a n subsets of size n : …For each n all possible values of a 1 , a 2 , , a n are covered. … where the summation is over all nonnegative integers n 1 , n 2 , , n k such that n 1 + n 2 + + n k = n . …
    8: 10.12 Generating Function and Associated Series
    cos ( z sin θ ) = J 0 ( z ) + 2 k = 1 J 2 k ( z ) cos ( 2 k θ ) ,
    sin ( z sin θ ) = 2 k = 0 J 2 k + 1 ( z ) sin ( ( 2 k + 1 ) θ ) ,
    cos ( z cos θ ) = J 0 ( z ) + 2 k = 1 ( 1 ) k J 2 k ( z ) cos ( 2 k θ ) ,
    cos z = J 0 ( z ) 2 J 2 ( z ) + 2 J 4 ( z ) 2 J 6 ( z ) + ,
    sin z = 2 J 1 ( z ) 2 J 3 ( z ) + 2 J 5 ( z ) ,
    9: 24.19 Methods of Computation
    If N ~ 2 n denotes the right-hand side of (24.19.1) but with the second product taken only for p ( π e ) 1 2 n + 1 , then N 2 n = N ~ 2 n for n 2 . … For other information see Chellali (1988) and Zhang and Jin (1996, pp. 1–11). … For number-theoretic applications it is important to compute B 2 n ( mod p ) for 2 n p 3 ; in particular to find the irregular pairs ( 2 n , p ) for which B 2 n 0 ( mod p ) . …
  • Buhler et al. (1992) uses the expansion

    24.19.3 t 2 cosh t 1 = 2 n = 0 ( 2 n 1 ) B 2 n t 2 n ( 2 n ) ! ,

    and computes inverses modulo p of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

  • A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs ( 2 n , p ) . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

  • 10: 4.19 Maclaurin Series and Laurent Series
    4.19.3 tan z = z + z 3 3 + 2 15 z 5 + 17 315 z 7 + + ( 1 ) n 1 2 2 n ( 2 2 n 1 ) B 2 n ( 2 n ) ! z 2 n 1 + , | z | < 1 2 π ,
    4.19.4 csc z = 1 z + z 6 + 7 360 z 3 + 31 15120 z 5 + + ( 1 ) n 1 2 ( 2 2 n 1 1 ) B 2 n ( 2 n ) ! z 2 n 1 + , 0 < | z | < π ,
    4.19.5 sec z = 1 + z 2 2 + 5 24 z 4 + 61 720 z 6 + + ( 1 ) n E 2 n ( 2 n ) ! z 2 n + , | z | < 1 2 π ,
    4.19.8 ln ( cos z ) = n = 1 ( 1 ) n 2 2 n 1 ( 2 2 n 1 ) B 2 n n ( 2 n ) ! z 2 n , | z | < 1 2 π ,
    4.19.9 ln ( tan z z ) = n = 1 ( 1 ) n 1 2 2 n ( 2 2 n 1 1 ) B 2 n n ( 2 n ) ! z 2 n , | z | < 1 2 π .