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9 Airy and Related FunctionsComputation

§9.19 Approximations

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Permalink:
http://dlmf.nist.gov/9.19
See also:
Annotations for Ch.9
Contents
  1. §9.19(i) Approximations in Terms of Elementary Functions
  2. §9.19(ii) Expansions in Chebyshev Series
  3. §9.19(iii) Approximations in the Complex Plane
  4. §9.19(iv) Scorer Functions

§9.19(i) Approximations in Terms of Elementary Functions

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

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    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<x\leq-3^{2/3}. 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<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.

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    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).

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    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.

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    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

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Keywords:
Airy functions, approximations, in the complex plane, integrals
Permalink:
http://dlmf.nist.gov/9.19.iii
See also:
Annotations for §9.19 and Ch.9
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    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

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    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<x\leq 0. The Chebyshev coefficients are given to 20D.

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