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1: 6.20 Approximations
  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • 2: 6.19 Tables
    §6.19(ii) Real Variables
  • Abramowitz and Stegun (1964, Chapter 5) includes x 1 Si ( x ) , x 2 Cin ( x ) , x 1 Ein ( x ) , x 1 Ein ( x ) , x = 0 ( .01 ) 0.5 ; Si ( x ) , Ci ( x ) , Ei ( x ) , E 1 ( x ) , x = 0.5 ( .01 ) 2 ; Si ( x ) , Ci ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x = 2 ( .1 ) 10 ; x f ( x ) , x 2 g ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x 1 = 0 ( .005 ) 0.1 ; Si ( π x ) , Cin ( π x ) , x = 0 ( .1 ) 10 . Accuracy varies but is within the range 8S–11S.

  • Zhang and Jin (1996, pp. 652, 689) includes Si ( x ) , Ci ( x ) , x = 0 ( .5 ) 20 ( 2 ) 30 , 8D; Ei ( x ) , E 1 ( x ) , x = [ 0 , 100 ] , 8S.

  • Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of z e z E 1 ( z ) , x = 19 ( 1 ) 20 , y = 0 ( 1 ) 20 , 6D; e z E 1 ( z ) , x = 4 ( .5 ) 2 , y = 0 ( .2 ) 1 , 6D; E 1 ( z ) + ln z , x = 2 ( .5 ) 2.5 , y = 0 ( .2 ) 1 , 6D.

  • Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of E 1 ( z ) , ± x = 0.5 , 1 , 3 , 5 , 10 , 15 , 20 , 50 , 100 , y = 0 ( .5 ) 1 ( 1 ) 5 ( 5 ) 30 , 50 , 100 , 8S.

  • 3: 7.24 Approximations
    §7.24(i) Approximations in Terms of Elementary Functions
  • Hastings (1955) gives several minimax polynomial and rational approximations for erf x , erfc x and the auxiliary functions f ( x ) and g ( x ) .

  • 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).

  • Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for F ( z ) , erf z , erfc z , C ( z ) , and S ( z ) ; approximate errors are given for a selection of z -values.

  • 4: Bibliography F
  • FDLIBM (free C library)
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
  • H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
  • H. E. Fettis (1976) Complex roots of sin z = a z , cos z = a z , and cosh z = a z . Math. Comp. 30 (135), pp. 541–545.