Hastings (1955) gives several minimax polynomial and rational approximations for $E_1\left(x\right)+\ln x$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, 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).

Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

Cody and Thacher (1969) provides minimax rational approximations for $\mathrm{Ei}\left(x\right)$, with accuracies up to 20S.

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.

Clenshaw (1962) gives Chebyshev coefficients for $-E_1\left(x\right)-\ln |x|$ for $-4\le x\le 4$ and ${\mathrm{e}}^x{E}_1\left(x\right)$ for $x\ge 4$ (20D).

Luke and Wimp (1963) covers $\mathrm{Ei}\left(x\right)$ for $x\le -4$ (20D), and $\mathrm{Si}\left(x\right)$ and $\mathrm{Ci}\left(x\right)$ for $x\ge 4$ (20D).

Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\mathrm{Ein}\left(ax\right)$, $\mathrm{Si}\left(ax\right)$, and $\mathrm{Cin}\left(ax\right)$ for $-1\le x\le 1$, $a\in ℂ$. The coefficients are given in terms of series of Bessel functions.

Luke (1969b, pp. 321–322) covers $\mathrm{Ein}\left(x\right)$ and $-\mathrm{Ein}\left(-x\right)$ for $0\le x\le 8$ (the Chebyshev coefficients are given to 20D); $E_1\left(x\right)$ for $x\ge 5$ (20D), and $\mathrm{Ei}\left(x\right)$ for $x\ge 8$ (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 $E_1\left(z\right)$, $\mathrm{f}\left(z\right)$, 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 the scheme can be used in backward direction.

Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for $\mathrm{Ein}\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Cin}\left(z\right)$ (valid near the origin), and $E_1\left(z\right)$ (valid for large $|z|$); approximate errors are given for a selection of $z$-values.

Luke (1969b, pp. 411–414) gives rational approximations for $\mathrm{Ein}\left(z\right)$.