§ 9.19. Approximations

Referenced by:: §9.18(i)
Permalink:: http://dlmf.nist.gov/9.19

§9.19(i)Approximations in Terms of Elementary Functions
§9.19(ii)Expansions in Chebyshev Series
§9.19(iii)Approximations in the Complex Plane
§9.19(iv)Scorer Functions

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

Permalink:: http://dlmf.nist.gov/9.19.SS1

Martín et al. (1992) provides two simple formulas for approximating $\mathrm{Ai}\!\left(x\right)$ to graphical accuracy, one for $-\infty<x\leq 0$ , the other for $0\leq x<\infty$ .
Moshier (1989, §6.14) provides minimax rational approximations for calculating $\mathrm{Ai}\!\left(x\right)$ , ${{\mathrm{Ai}}^{{\prime}}}\!\left(x\right)$ , $\mathrm{Bi}\!\left(x\right)$ , ${{\mathrm{Bi}}^{{\prime}}}\!\left(x\right)$ . They are in terms of the variable $\zeta$ , where $\zeta=\tfrac{2}{3}x^{{3/2}}$ when $x$ is positive, or $\zeta=\tfrac{2}{3}(-x)^{{3/2}}$ when $x$ is negative. The approximations apply when $2\leq\zeta<\infty$ , that is, when $3^{{2/3}}\leq x<\infty$ or $-\infty<x\leq-3^{{2/3}}$ . The precision in the coefficients is 21S.

§ 9.19(ii). Expansions in Chebyshev Series

Permalink:: http://dlmf.nist.gov/9.19.SS2

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

Prince (1975) covers $\mathrm{Ai}\!\left(x\right)$ , ${{\mathrm{Ai}}^{{\prime}}}\!\left(x\right)$ , $\mathrm{Bi}\!\left(x\right)$ , ${{\mathrm{Bi}}^{{\prime}}}\!\left(x\right)$ . The Chebyshev coefficients are given to 10-11D. Fortran programs are included. See also Razaz and Schonfelder (1981).
Németh (1992, Chapter 8) covers $\mathrm{Ai}\!\left(x\right)$ , ${{\mathrm{Ai}}^{{\prime}}}\!\left(x\right)$ , $\mathrm{Bi}\!\left(x\right)$ , ${{\mathrm{Bi}}^{{\prime}}}\!\left(x\right)$ , and integrals $\int _{0}^{x}\!\!\mathrm{Ai}\!\left(t\right)dt$ , $\int _{0}^{x}\!\!\mathrm{Bi}\!\left(t\right)dt$ , $\int _{0}^{x}\!\!\int _{0}^{v}\mathrm{Ai}\!\left(t\right)dtdv$ , $\int _{0}^{x}\!\!\int _{0}^{v}\mathrm{Bi}\!\left(t\right)dtdv$ (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 $\mathrm{Ai}\!\left(x\right)$ , ${{\mathrm{Ai}}^{{\prime}}}\!\left(x\right)$ , $\mathrm{Bi}\!\left(x\right)$ , ${{\mathrm{Bi}}^{{\prime}}}\!\left(x\right)$ , again to 15D.
Razaz and Schonfelder (1980) covers $\mathrm{Ai}\!\left(x\right)$ , ${{\mathrm{Ai}}^{{\prime}}}\!\left(x\right)$ , $\mathrm{Bi}\!\left(x\right)$ , ${{\mathrm{Bi}}^{{\prime}}}\!\left(x\right)$ . The Chebyshev coefficients are given to 30D.

§ 9.19(iii). Approximations in the Complex Plane

Permalink:: http://dlmf.nist.gov/9.19.SS3

Corless et al. (1992) describe a method of approximation based on subdividing $\Complex$ into a triangular mesh, with values of $\mathrm{Ai}\!\left(z\right)$ , ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)$ stored at the nodes. $\mathrm{Ai}\!\left(z\right)$ and ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)$ 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 $\mathrm{Ai}\!\left(z\right)$ , ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\!\left(z\right)$ , ${{\mathrm{Bi}}^{{\prime}}}\!\left(z\right)$ .

§ 9.19(iv). Scorer Functions

Permalink:: http://dlmf.nist.gov/9.19.SS4

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

§ 9.19. Approximations

Contents

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

§ 9.19(ii). Expansions in Chebyshev Series

§ 9.19(iii). Approximations in the Complex Plane

§ 9.19(iv). Scorer Functions