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9 Airy and Related FunctionsComputation

§9.19 Approximations

Contents

§9.19(i) Approximations in Terms of Elementary Functions

  • Martín et al. (1992) provides two simple formulas for approximating Ai(x) to graphical accuracy, one for -<x0, the other for 0x<.

  • Moshier (1989, §6.14) provides minimax rational approximations for calculating Ai(x), Ai(x), Bi(x), Bi(x). They are in terms of the variable ζ, where ζ=23x3/2 when x is positive, ζ=23(-x)3/2 when x is negative, and ζ=0 when x=0. The approximations apply when 2ζ<, that is, when 32/3x< or -<x-32/3. 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 -<xa, ax0, 0xb, bx<. The constants a and b are chosen numerically, with a view to equalizing the effort required for summing the series.

  • Prince (1975) covers Ai(x), Ai(x), Bi(x), Bi(x). The Chebyshev coefficients are given to 10-11D. Fortran programs are included. See also Razaz and Schonfelder (1981).

  • Németh (1992, Chapter 8) covers Ai(x), Ai(x), Bi(x), Bi(x), and integrals 0xAi(t)dt, 0xBi(t)dt, 0x0vAi(t)dtdv, 0x0vBi(t)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 Ai(x), Ai(x), Bi(x), Bi(x), again to 15D.

  • Razaz and Schonfelder (1980) covers Ai(x), Ai(x), Bi(x), Bi(x). 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 into a triangular mesh, with values of Ai(z), Ai(z) stored at the nodes. Ai(z) and Ai(z) 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 Ai(z), Ai(z) at the node. Similarly for Bi(z), Bi(z).

§9.19(iv) Scorer Functions

  • MacLeod (1994) supplies Chebyshev-series expansions to cover Gi(x) for 0x< and Hi(x) for -<x0. The Chebyshev coefficients are given to 20D.