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1: 6.1 Special Notation
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 ) . …
2: 8.21 Generalized Sine and Cosine Integrals
From §§8.2(i) and 8.2(ii) it follows that each of the four functions si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) is a multivalued function of z with branch point at z = 0 . Furthermore, si ( a , z ) and ci ( a , z ) are entire functions of a , and Si ( a , z ) and Ci ( a , z ) are meromorphic functions of a with simple poles at a = 1 , 3 , 5 , and a = 0 , 2 , 4 , , respectively. … When ph z = 0 (and when a 1 , 3 , 5 , , in the case of Si ( a , z ) , or a 0 , 2 , 4 , , in the case of Ci ( a , z ) ) the principal values of si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … 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. … For the corresponding expansions for si ( a , z ) and ci ( a , z ) apply (8.21.20) and (8.21.21). …
3: 6.15 Sums
6.15.2 n = 1 si ( π n ) n = 1 2 π ( ln π 1 ) ,
6.15.4 n = 1 ( 1 ) n si ( 2 π n ) n = π ( 3 2 ln 2 1 ) .
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: 6.14 Integrals
6.14.3 0 e a t si ( t ) d t = 1 a arctan a , a > 0 .
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 .
6: 6.19 Tables
  • 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.

  • 7: 6.3 Graphics
    See accompanying text
    Figure 6.3.2: The sine and cosine integrals Si ( x ) , Ci ( x ) , 0 x 15 . Magnify
    8: 6.13 Zeros
    Ci ( x ) and si ( x ) each have an infinite number of positive real zeros, which are denoted by c k , s k , respectively, arranged in ascending order of absolute value for k = 0 , 1 , 2 , . …
    6.13.2 c k , s k α + 1 α 16 3 1 α 3 + 1673 15 1 α 5 5 07746 105 1 α 7 + ,
    9: 6.2 Definitions and Interrelations
    6.2.9 Si ( z ) = 0 z sin t t d t .
    Si ( z ) is an odd entire function.
    6.2.10 si ( z ) = z sin t t d t = Si ( z ) 1 2 π .
    lim x Si ( x ) = 1 2 π ,
    6.2.17 f ( z ) = Ci ( z ) sin z si ( z ) cos z ,
    10: 8.1 Special Notation
    The functions treated in this chapter are the incomplete gamma functions γ ( a , z ) , Γ ( a , z ) , γ ( a , z ) , P ( a , z ) , and Q ( a , z ) ; the incomplete beta functions B x ( a , b ) and I x ( a , b ) ; the generalized exponential integral E p ( z ) ; the generalized sine and cosine integrals si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) . Alternative notations include: Prym’s functions P z ( a ) = γ ( a , z ) , Q z ( a ) = Γ ( a , z ) , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); ( a , z ) ! = γ ( a + 1 , z ) , [ a , z ] ! = Γ ( a + 1 , z ) , Dingle (1973); B ( a , b , x ) = B x ( a , b ) , I ( a , b , x ) = I x ( a , b ) , Magnus et al. (1966); Si ( a , x ) Si ( 1 a , x ) , Ci ( a , x ) Ci ( 1 a , x ) , Luke (1975).