# §9.19 Approximations

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

• Martín et al. (1992) provides two simple formulas for approximating $\operatorname{Ai}\left(x\right)$ to graphical accuracy, one for $-\infty, the other for $0\leq x<\infty$.

• Moshier (1989, §6.14) provides minimax rational approximations for calculating $\operatorname{Ai}\left(x\right)$, $\operatorname{Ai}'\left(x\right)$, $\operatorname{Bi}\left(x\right)$, $\operatorname{Bi}'\left(x\right)$. They are in terms of the variable $\zeta$, where $\zeta=\tfrac{2}{3}x^{3/2}$ when $x$ is positive, $\zeta=\tfrac{2}{3}(-x)^{3/2}$ when $x$ is negative, and $\zeta=0$ when $x=0$. The approximations apply when $2\leq\zeta<\infty$, that is, when $3^{2/3}\leq x<\infty$ or $-\infty. The precision in the coefficients is 21S.

## §9.19(ii) Expansions in Chebyshev Series

These expansions are for real arguments $x$ and are supplied in sets of four for each function, corresponding to intervals $-\infty, $a\leq x\leq 0$, $0\leq x\leq b$, $b\leq x<\infty$. The constants $a$ and $b$ are chosen numerically, with a view to equalizing the effort required for summing the series.

• Prince (1975) covers $\operatorname{Ai}\left(x\right)$, $\operatorname{Ai}'\left(x\right)$, $\operatorname{Bi}\left(x\right)$, $\operatorname{Bi}'\left(x\right)$. The Chebyshev coefficients are given to 10-11D. Fortran programs are included. See also Razaz and Schonfelder (1981).

• Németh (1992, Chapter 8) covers $\operatorname{Ai}\left(x\right)$, $\operatorname{Ai}'\left(x\right)$, $\operatorname{Bi}\left(x\right)$, $\operatorname{Bi}'\left(x\right)$, and integrals $\int_{0}^{x}\operatorname{Ai}\left(t\right)\,\mathrm{d}t$, $\int_{0}^{x}\operatorname{Bi}\left(t\right)\,\mathrm{d}t$, $\int_{0}^{x}\int_{0}^{v}\operatorname{Ai}\left(t\right)\,\mathrm{d}t\,\mathrm{% d}v$, $\int_{0}^{x}\int_{0}^{v}\operatorname{Bi}\left(t\right)\,\mathrm{d}t\,\mathrm{% d}v$ (see also (9.10.20) and (9.10.21)). The Chebyshev coefficients are given to 15D. Chebyshev coefficients are also given for expansions of the second and higher (real) zeros of $\operatorname{Ai}\left(x\right)$, $\operatorname{Ai}'\left(x\right)$, $\operatorname{Bi}\left(x\right)$, $\operatorname{Bi}'\left(x\right)$, again to 15D.

• Razaz and Schonfelder (1980) covers $\operatorname{Ai}\left(x\right)$, $\operatorname{Ai}'\left(x\right)$, $\operatorname{Bi}\left(x\right)$, $\operatorname{Bi}'\left(x\right)$. The Chebyshev coefficients are given to 30D.

## §9.19(iii) Approximations in the Complex Plane

• Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\operatorname{Ai}\left(z\right)$, $\operatorname{Ai}'\left(z\right)$ stored at the nodes. $\operatorname{Ai}\left(z\right)$ and $\operatorname{Ai}'\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\operatorname{Ai}\left(z\right)$, $\operatorname{Ai}'\left(z\right)$ at the node. Similarly for $\operatorname{Bi}\left(z\right)$, $\operatorname{Bi}'\left(z\right)$.

## §9.19(iv) Scorer Functions

• MacLeod (1994) supplies Chebyshev-series expansions to cover $\operatorname{Gi}\left(x\right)$ for $0\leq x<\infty$ and $\operatorname{Hi}\left(x\right)$ for $-\infty. The Chebyshev coefficients are given to 20D.