§9.19 Approximations

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

Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.19.i

Martín et al. (1992) provides two simple formulas for approximating $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ 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 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)$ . 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 $-\infty<x\leq-3^{{2/3}}$ . The precision in the coefficients is 21S.

§9.19(ii) Expansions in Chebyshev Series

Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.19.ii

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

§9.19(iii) Approximations in the Complex Plane

Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.19.iii

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

§9.19(iv) Scorer Functions

Keywords:: Scorer functions
Permalink:: http://dlmf.nist.gov/9.19.iv

MacLeod (1994) supplies Chebyshev-series expansions to cover $\mathop{\mathrm{Gi}\/}\nolimits\!\left(x\right)$ for $0\leq x<\infty$ and $\mathop{\mathrm{Hi}\/}\nolimits\!\left(x\right)$ 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