Digital Library of Mathematical Functions
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6 Exponential, Logarithmic, Sine, and Cosine IntegralsComputation

§6.20 Approximations


§6.20(i) Approximations in Terms of Elementary Functions

  • Hastings (1955) gives several minimax polynomial and rational approximations for E1(x)+lnx, xxE1(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 E1(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.

§6.20(ii) Expansions in Chebyshev Series

  • Clenshaw (1962) gives Chebyshev coefficients for -E1(x)-ln|x| for -4x4 and xE1(x) for x4 (20D).

  • Luke and Wimp (1963) covers Ei(x) for x-4 (20D), and Si(x) and Ci(x) for x4 (20D).

  • Luke (1969b, pp. 41–42) gives Chebyshev expansions of Ein(ax), Si(ax), and Cin(ax) for -1x1, a. The coefficients are given in terms of series of Bessel functions.

  • Luke (1969b, pp. 321–322) covers Ein(x) and -Ein(-x) for 0x8 (the Chebyshev coefficients are given to 20D); E1(x) for x5 (20D), and Ei(x) for x8 (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

  • Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric U-function (§13.2(i)) from which Chebyshev expansions near infinity for E1(z), f(z), and g(z) follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the U functions. If |phz|<π the scheme can be used in backward direction.

§6.20(iii) Padé-Type and Rational Expansions

  • 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 E1(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).