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1: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
§8.19(ii) Graphics
§8.19(iv) Series Expansions
§8.19(ix) Inequalities
§8.19(x) Integrals
2: 4.2 Definitions
log e x = ln x is also called the Napierian or hyperbolic logarithm. …
§4.2(iii) The Exponential Function
The function exp is an entire function of z , with no real or complex zeros. …
4.2.32 e z = exp z ,
but the general value of e z is …
3: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
The principal value of the exponential integral E 1 ( z ) is defined by … Ein ( z ) is sometimes called the complementary exponential integral. … The logarithmic integral is defined by …
4: 6.20 Approximations
  • 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.

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

  • Clenshaw (1962) gives Chebyshev coefficients for E 1 ( x ) ln | x | for 4 x 4 and e x E 1 ( x ) for x 4 (20D).

  • Luke and Wimp (1963) covers Ei ( x ) for x 4 (20D), and Si ( x ) and Ci ( x ) for x 4 (20D).

  • 5: 2.11 Remainder Terms; Stokes Phenomenon
    If the results agree within S significant figures, then it is likely—but not certain—that the truncated asymptotic series will yield at least S correct significant figures for larger values of x . … Truncation after 5 terms yields 0. … The process just used is equivalent to re-expanding the remainder term of the original asymptotic series (2.11.24) in powers of 1 / ( x + 5 ) and truncating the new series optimally. … Optimum truncation occurs just prior to the numerically smallest term, that is, at s 4 . … For example, using double precision d 20 is found to agree with (2.11.31) to 13D. …
    6: 20 Theta Functions
    Chapter 20 Theta Functions
    7: 22.3 Graphics
    See accompanying text
    Figure 22.3.13: sn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
    See accompanying text
    Figure 22.3.14: cn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
    See accompanying text
    Figure 22.3.15: dn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
    See accompanying text
    Figure 22.3.26: Density plot of | sn ( 5 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . Grayscale, running from 0 (black) to 10 (white), with | ( sn ( 5 , k ) ) | > 10 truncated to 10. … Magnify
    See accompanying text
    Figure 22.3.28: Density plot of | sn ( 20 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . Grayscale, running from 0 (black) to 10 (white), with | sn ( 20 , k ) | > 10 truncated to 10. … Magnify
    8: Bibliography W
  • P. L. Walker (1991) Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • E. J. Weniger (2007) Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.
  • A. D. Wheelon (1968) Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.
  • R. Wong and Y. Zhao (2002a) Exponential asymptotics of the Mittag-Leffler function. Constr. Approx. 18 (3), pp. 355–385.
  • 9: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • §8.26(iv) Generalized Exponential Integral
  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 10: 11.6 Asymptotic Expansions
    If the series on the right-hand side of (11.6.1) is truncated after m ( 0 ) terms, then the remainder term R m ( z ) is O ( z ν 2 m 1 ) . … More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). …
    c 3 ( λ ) = 20 λ 6 4 λ 4 ,