# §7.24 Approximations

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

• Hastings (1955) gives several minimax polynomial and rational approximations for $\mathop{\mathrm{erf}\/}\nolimits x$, $\mathop{\mathrm{erfc}\/}\nolimits x$ and the auxiliary functions $\mathop{\mathrm{f}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{g}\/}\nolimits\!\left(x\right)$.

• Cody (1969) provides minimax rational approximations for $\mathop{\mathrm{erf}\/}\nolimits x$ and $\mathop{\mathrm{erfc}\/}\nolimits x$. The maximum relative precision is about 20S.

• Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

• Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $\mathop{F\/}\nolimits\!\left(x\right)$ (maximum relative precision 20S–22S).

## §7.24(ii) Expansions in Chebyshev Series

• Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi}\mathop{\mathrm{erf}\/}\nolimits x$ and $e^{x^{2}}\mathop{F\/}\nolimits\!\left(x\right)$ for $-3\leq x\leq 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi}xe^{x^{2}}\mathop{\mathrm{erfc}\/}\nolimits x$ and $2x\mathop{F\/}\nolimits\!\left(x\right)$ for $x\geq 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

• Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathop{\mathrm{f}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{g}\/}\nolimits\!\left(x\right)$ for $x\geq 3$ (15D).

• Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\mathop{\mathrm{erf}\/}\nolimits x$ on $0\leq x\leq 2$, for $xe^{x^{2}}\mathop{\mathrm{erfc}\/}\nolimits x$ on $[2,\infty)$, and for $e^{x^{2}}\mathop{\mathrm{erfc}\/}\nolimits x$ on $[0,\infty)$ (30D).

• Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x)e^{x^{2}}\mathop{\mathrm{erfc}\/}\nolimits x$ on $(0,\infty)$ (22D).

• Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $\mathop{F\/}\nolimits\!\left(z\right)$, $\mathop{\mathrm{erf}\/}\nolimits z$, $\mathop{\mathrm{erfc}\/}\nolimits z$, $\mathop{C\/}\nolimits\!\left(z\right)$, and $\mathop{S\/}\nolimits\!\left(z\right)$; approximate errors are given for a selection of $z$-values.