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1: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
8.19.1 E p ( z ) = z p - 1 Γ ( 1 - p , z ) .
For an extensive treatment of E 1 ( z ) see Chapter 6. …
See accompanying text
Figure 8.19.5: E 2 ( x + i y ) , - 3 x 3 , - 3 y 3 . … Magnify 3D Help
§8.19(x) Integrals
2: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
6.2.1 E 1 ( z ) = z e - t t d t , z 0 ,
6.2.3 Ein ( z ) = 0 z 1 - e - t t d t .
Ein ( z ) is sometimes called the complementary exponential integral. …
3: 6.5 Further Interrelations
§6.5 Further Interrelations
6.5.1 E 1 ( - x ± i 0 ) = - Ei ( x ) i π ,
6.5.2 Ei ( x ) = - 1 2 ( E 1 ( - x + i 0 ) + E 1 ( - x - i 0 ) ) ,
6.5.3 1 2 ( Ei ( x ) + E 1 ( x ) ) = Shi ( x ) = - i Si ( i x ) ,
6.5.6 Ci ( z ) = - 1 2 ( E 1 ( i z ) + E 1 ( - i z ) ) ,
4: 6.3 Graphics
See accompanying text
Figure 6.3.1: The exponential integrals E 1 ( x ) and Ei ( x ) , 0 < x 2 . Magnify
See accompanying text
Figure 6.3.3: | E 1 ( x + i y ) | , - 4 x 4 , - 4 y 4 . …Also, | E 1 ( z ) | logarithmically as z 0 . Magnify 3D Help
5: 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.

  • 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.

  • 6: 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 ) . …
    7: 6.8 Inequalities
    §6.8 Inequalities
    6.8.1 1 2 ln ( 1 + 2 x ) < e x E 1 ( x ) < ln ( 1 + 1 x ) ,
    6.8.2 x x + 1 < x e x E 1 ( x ) < x + 1 x + 2 ,
    6.8.3 x ( x + 3 ) x 2 + 4 x + 2 < x e x E 1 ( x ) < x 2 + 5 x + 2 x 2 + 6 x + 6 .
    8: 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).

  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • 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. 411–414) gives rational approximations for Ein ( z ) .

  • 9: 6.4 Analytic Continuation
    §6.4 Analytic Continuation
    Analytic continuation of the principal value of E 1 ( z ) yields a multi-valued function with branch points at z = 0 and z = . The general value of E 1 ( z ) is given by
    6.4.1 E 1 ( z ) = Ein ( z ) - Ln z - γ ;
    6.4.2 E 1 ( z e 2 m π i ) = E 1 ( z ) - 2 m π i , m ,
    10: 6.9 Continued Fraction
    §6.9 Continued Fraction
    6.9.1 E 1 ( z ) = e - z z + 1 1 + 1 z + 2 1 + 2 z + 3 1 + 3 z + , | ph z | < π .