§6.20 Approximations

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§6.20(i) Approximations in Terms of Elementary Functions

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Keywords:: approximations, auxiliary functions for sine and cosine integrals, cosine integrals, exponential integrals, sine integrals
Permalink:: http://dlmf.nist.gov/6.20.i
See also:: Annotations for §6.20 and Ch.6

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Hastings (1955) gives several minimax polynomial and rational approximations for $E_{1}\left(x\right)+\ln x$ , $xe^{x}E_{1}\left(x\right)$ , and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ . These are included in Abramowitz and Stegun (1964, Ch. 5).
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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.
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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.
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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

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Keywords:: Chebyshev-series expansions, auxiliary functions for sine and cosine integrals, cosine integrals, exponential integrals, sine integrals
Permalink:: http://dlmf.nist.gov/6.20.ii
See also:: Annotations for §6.20 and Ch.6

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Clenshaw (1962) gives Chebyshev coefficients for $-E_{1}\left(x\right)-\ln|x|$ for $-4\leq x\leq 4$ and $e^{x}E_{1}\left(x\right)$ for $x\geq 4$ (20D).
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Luke and Wimp (1963) covers $\operatorname{Ei}\left(x\right)$ for $x\leq-4$ (20D), and $\operatorname{Si}\left(x\right)$ and $\operatorname{Ci}\left(x\right)$ for $x\geq 4$ (20D).
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Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\operatorname{Ein}\left(ax\right)$ , $\operatorname{Si}\left(ax\right)$ , and $\operatorname{Cin}\left(ax\right)$ for $-1\leq x\leq 1$ , $a\in\mathbb{C}$ . The coefficients are given in terms of series of Bessel functions.
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Luke (1969b, pp. 321–322) covers $\operatorname{Ein}\left(x\right)$ and $-\operatorname{Ein}\left(-x\right)$ for $0\leq x\leq 8$ (the Chebyshev coefficients are given to 20D); $E_{1}\left(x\right)$ for $x\geq 5$ (20D), and $\operatorname{Ei}\left(x\right)$ for $x\geq 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.
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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 $E_{1}\left(z\right)$ , $\mathrm{f}\left(z\right)$ , and $\mathrm{g}\left(z\right)$ 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 $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.

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Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for $\operatorname{Ein}\left(z\right)$ , $\operatorname{Si}\left(z\right)$ , $\operatorname{Cin}\left(z\right)$ (valid near the origin), and $E_{1}\left(z\right)$ (valid for large $|z|$ ); approximate errors are given for a selection of $z$ -values.
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Luke (1969b, pp. 411–414) gives rational approximations for $\operatorname{Ein}\left(z\right)$ .