# §6.20 Approximations

## §6.20(i) Approximations in Terms of Elementary Functions

• Hastings (1955) gives several minimax polynomial and rational approximations for $\mathop{E_{1}\/}\nolimits\!\left(x\right)+\mathop{\ln\/}\nolimits x$, $xe^{x}\mathop{E_{1}\/}\nolimits\!\left(x\right)$, and the auxiliary functions $\mathop{\mathrm{f}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{g}\/}\nolimits\!\left(x\right)$. These are included in Abramowitz and Stegun (1964, Ch. 5).

• Cody and Thacher (1968) provides minimax rational approximations for $\mathop{E_{1}\/}\nolimits\!\left(x\right)$, with accuracies up to 20S.

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

• MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathop{\mathrm{f}\/}\nolimits$ and $\mathop{\mathrm{g}\/}\nolimits$, with accuracies up to 20S.

## §6.20(ii) Expansions in Chebyshev Series

• Clenshaw (1962) gives Chebyshev coefficients for $-\mathop{E_{1}\/}\nolimits\!\left(x\right)-\mathop{\ln\/}\nolimits|x|$ for $-4\leq x\leq 4$ and $e^{x}\mathop{E_{1}\/}\nolimits\!\left(x\right)$ for $x\geq 4$ (20D).

• Luke and Wimp (1963) covers $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ for $x\leq-4$ (20D), and $\mathop{\mathrm{Si}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{Ci}\/}\nolimits\!\left(x\right)$ for $x\geq 4$ (20D).

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

• Luke (1969b, pp. 321–322) covers $\mathop{\mathrm{Ein}\/}\nolimits\!\left(x\right)$ and $-\mathop{\mathrm{Ein}\/}\nolimits\!\left(-x\right)$ for $0\leq x\leq 8$ (the Chebyshev coefficients are given to 20D); $\mathop{E_{1}\/}\nolimits\!\left(x\right)$ for $x\geq 5$ (20D), and $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ for $x\geq 8$ (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 $\mathop{U\/}\nolimits$-function (§13.2(i)) from which Chebyshev expansions near infinity for $\mathop{E_{1}\/}\nolimits\!\left(z\right)$, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$, and $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ 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 $\mathop{U\/}\nolimits$ functions. If $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ 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 $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$, $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ (valid near the origin), and $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ (valid for large $|z|$); approximate errors are given for a selection of $z$-values.

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