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auxiliary functions for sine and cosine integrals

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1: 6.1 Special Notation
2: 6.4 Analytic Continuation
6.4.6 f ( z e ± π i ) = π e i z - f ( z ) ,
6.4.7 g ( z e ± π i ) = π i e i z + g ( z ) .
3: 6.2 Definitions and Interrelations
§6.2(iii) Auxiliary Functions
6.2.17 f ( z ) = Ci ( z ) sin z - si ( z ) cos z ,
6.2.18 g ( z ) = - Ci ( z ) cos z - si ( z ) sin z .
4: 6.11 Relations to Other Functions
5: 6.20 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for E 1 ( x ) + ln x , x e x E 1 ( x ) , and the auxiliary functions f ( x ) and g ( x ) . These are included in Abramowitz and Stegun (1964, Ch. 5).

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

  • Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric U -function13.2(i)) from which Chebyshev expansions near infinity for E 1 ( z ) , f ( z ) , and g ( z ) follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the U functions. If | ph z | < π the scheme can be used in backward direction.

  • 6: 6.5 Further Interrelations
    7: 6.19 Tables
    §6.19(ii) Real Variables
    8: 6.7 Integral Representations
    §6.7(iii) Auxiliary Functions
    6.7.13 f ( z ) = 0 sin t t + z d t = 0 e - z t t 2 + 1 d t ,
    6.7.15 f ( z ) = 2 0 K 0 ( 2 z t ) cos t d t ,
    6.7.16 g ( z ) = 2 0 K 0 ( 2 z t ) sin t d t .
    9: 6.12 Asymptotic Expansions
    6.12.3 f ( z ) 1 z ( 1 - 2 ! z 2 + 4 ! z 4 - 6 ! z 6 + ) ,
    6.12.4 g ( z ) 1 z 2 ( 1 - 3 ! z 2 + 5 ! z 4 - 7 ! z 6 + ) ,
    6.12.5 f ( z ) = 1 z m = 0 n - 1 ( - 1 ) m ( 2 m ) ! z 2 m + R n ( f ) ( z ) ,
    10: 8.21 Generalized Sine and Cosine Integrals
    §8.21(vii) Auxiliary Functions
    8.21.18 f ( a , z ) = si ( a , z ) cos z - ci ( a , z ) sin z ,
    8.21.19 g ( a , z ) = si ( a , z ) sin z + ci ( a , z ) cos z .
    8.21.21 ci ( a , z ) = - f ( a , z ) sin z + g ( a , z ) cos z .
    §8.21(viii) Asymptotic Expansions