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1: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
From here on it is assumed that unless indicated otherwise the functions si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) have their principal values. …
§8.21(v) Special Values
For the corresponding expansions for si ( a , z ) and ci ( a , z ) apply (8.21.20) and (8.21.21).
2: 6.2 Definitions and Interrelations
§6.2(ii) Sine and Cosine Integrals
Hyperbolic Analogs of the Sine and Cosine Integrals
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 .
3: 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 ) .
Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions E 1 ( z ) , Ci ( z ) , Chi ( z ) , f ( z ) , and g ( z ) assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.
4: 6.21 Software
§6.21(ii) E 1 ( x ) , Ei ( x ) , Si ( x ) , Ci ( x ) , Shi ( x ) , Chi ( x ) , x
§6.21(iii) E 1 ( z ) , Si ( z ) , Ci ( z ) , Shi ( z ) , Chi ( z ) , z
5: 6.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the exponential integrals Ei ( x ) , E 1 ( z ) , and Ein ( z ) ; the logarithmic integral li ( x ) ; the sine integrals Si ( z ) and si ( z ) ; the cosine integrals Ci ( z ) and Cin ( z ) .
6: 6.5 Further Interrelations
§6.5 Further Interrelations
7: 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 and Wimp (1963) covers Ei ( x ) for x - 4 (20D), and Si ( x ) and Ci ( x ) for x 4 (20D).

  • Luke (1969b, pp. 41–42) gives Chebyshev expansions of Ein ( a x ) , Si ( a x ) , and Cin ( a x ) for - 1 x 1 , a . The coefficients are given in terms of series of Bessel functions.

  • Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for Ein ( z ) , Si ( z ) , Cin ( z ) (valid near the origin), and E 1 ( z ) (valid for large | z | ); approximate errors are given for a selection of z -values.

  • 8: 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.

  • 9: 6.11 Relations to Other Functions
    10: 6.14 Integrals
    §6.14(i) Laplace Transforms
    §6.14(ii) Other Integrals
    6.14.5 0 cos t Ci ( t ) d t = 0 sin t si ( t ) d t = - 1 4 π ,
    6.14.6 0 Ci 2 ( t ) d t = 0 si 2 ( t ) d t = 1 2 π ,
    6.14.7 0 Ci ( t ) si ( t ) d t = ln 2 .