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1: 20 Theta Functions
Chapter 20 Theta Functions
2: 7.24 Approximations
§7.24(i) Approximations in Terms of Elementary Functions
  • Cody (1969) provides minimax rational approximations for erf x and erfc x . The maximum relative precision is about 20S.

  • Cody et al. (1970) gives minimax rational approximations to Dawson’s integral F ( x ) (maximum relative precision 20S–22S).

  • 3: 8 Incomplete Gamma and Related
    Functions
    4: 28 Mathieu Functions and Hill’s Equation
    5: 6.20 Approximations
    §6.20(i) Approximations in Terms of Elementary Functions
  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • 6: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 7: 23 Weierstrass Elliptic and Modular
    Functions
    8: 19.36 Methods of Computation
    When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …
    19.36.1 1 1 10 E 2 + 1 14 E 3 + 1 24 E 2 2 3 44 E 2 E 3 5 208 E 2 3 + 3 104 E 3 2 + 1 16 E 2 2 E 3 ,
    where the elementary symmetric functions E s are defined by (19.19.4). …
    19.36.2 1 3 14 E 2 + 1 6 E 3 + 9 88 E 2 2 3 22 E 4 9 52 E 2 E 3 + 3 26 E 5 1 16 E 2 3 + 3 40 E 3 2 + 3 20 E 2 E 4 + 45 272 E 2 2 E 3 9 68 ( E 3 E 4 + E 2 E 5 ) .
    For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …
    9: 10.75 Tables
  • Achenbach (1986) tabulates J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) , x = 0 ( .1 ) 8 , 20D or 18–20S.

  • Bickley et al. (1952) tabulates x n I n ( x ) or e x I n ( x ) , x n K n ( x ) or e x K n ( x ) , n = 2 ( 1 ) 20 , x = 0 (.01 or .1) 10(.1) 20, 8S; I n ( x ) , K n ( x ) , n = 0 ( 1 ) 20 , x = 0 or 0.1 ( .1 ) 20 , 10S.

  • Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of K n ( z ) and K n ( z ) , for n = 2 ( 1 ) 20 , 9S.

  • Zhang and Jin (1996, p. 322) tabulates ber x , ber x , bei x , bei x , ker x , ker x , kei x , kei x , x = 0 ( 1 ) 20 , 7S.

  • Zhang and Jin (1996, p. 323) tabulates the first 20 real zeros of ber x , ber x , bei x , bei x , ker x , ker x , kei x , kei x , 8D.

  • 10: 36 Integrals with Coalescing Saddles