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complementary exponential integral

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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: 6.20 Approximations
  • 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. 321–322) covers Ein ( x ) and - Ein ( - x ) for 0 x 8 (the Chebyshev coefficients are given to 20D); E 1 ( x ) for x 5 (20D), and Ei ( x ) for x 8 (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

  • 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.

  • Luke (1969b, pp. 411–414) gives rational approximations for Ein ( z ) .

  • 3: 6.4 Analytic Continuation
    4: 6.7 Integral Representations
    6.7.7 0 1 e - a t sin ( b t ) t d t = Ein ( a + i b ) , a , b ,
    6.7.8 0 1 e - a t ( 1 - cos ( b t ) ) t d t = Ein ( a + i b ) - Ein ( a ) , a , b .
    6.7.10 Ein ( z ) - Cin ( z ) = 0 π / 2 e - z cos t sin ( z sin t ) d t ,
    6.7.11 0 1 ( 1 - e - a t ) cos ( b t ) t d t = Ein ( a + i b ) - Cin ( b ) , a , b .
    5: 6.2 Definitions and Interrelations
    6.2.3 Ein ( z ) = 0 z 1 - e - t t d t .
    Ein ( z ) is sometimes called the complementary exponential integral. …
    6.2.7 Ei ( ± x ) = - Ein ( x ) + ln x + γ .
    6: 6.6 Power Series
    6.6.4 Ein ( z ) = n = 1 ( - 1 ) n - 1 z n n ! n ,
    7: 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.

  • 8: 6.10 Other Series Expansions
    6.10.8 Ein ( z ) = z e - z / 2 ( i 0 ( 1 ) ( 1 2 z ) + n = 1 2 n + 1 n ( n + 1 ) i n ( 1 ) ( 1 2 z ) ) .
    9: 7.18 Repeated Integrals of the Complementary Error Function
    i - 1 erfc ( z ) = 2 π e - z 2 ,
    7.18.4 d n d z n ( e z 2 erfc z ) = ( - 1 ) n 2 n n ! e z 2 i n erfc ( z ) , n = 0 , 1 , 2 , .
    7.18.9 i n erfc ( z ) = e - z 2 ( 1 2 n Γ ( 1 2 n + 1 ) M ( 1 2 n + 1 2 , 1 2 , z 2 ) - z 2 n - 1 Γ ( 1 2 n + 1 2 ) M ( 1 2 n + 1 , 3 2 , z 2 ) ) ,
    10: 19.37 Tables
    Function exp ( - π K ( k ) / K ( k ) ) ( = q ( k ) )