Algorithm 498: Airy Functions Using Chebyshev Series Approximations
P.J. Prince · ACM Transactions on Mathematical Software · 1975
PurposeThis subroutine evaluates the Airy functions Ai (z) and Bi (z) and their respective derivatives Ai' and Bi', for all real values of z within computer capability, by means of Chebyshev series approximations. Method(a) For z < --7 the following asymptotic expansions are used (see [-1, eqs.10.4.60, 10.4.64, 10.4.62, and 10.4.67-]) :where s = sin(~" ~-7r/4), c = cos(~ + ~/4), ~ = 2/3 z 3/~, and f, g, p, and q are approximated by the following Chebyshev expansions: m I m2 m8 m~ f ~--~,' arT,* (t), g ~---~-1 ~__, brT,* (t), p ~-~ ~_,' crTr* (t), q ~-~-~ ~-'~' drT,* (t) r~O r~O r~O r~O where t = -(7/z) s, the Tr* are the shifted Chebyshev polynomials, and the ar, b,, cr, and dr are the prescribed Chebyshev coefficients.