Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we don't need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.

Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Singular perturbations, turning point theory, WKB methods (msc 34E20), Bessel and Airy functions, cylinder functions, ${}_0F_1$ (msc 33C10), Computation of special functions, construction of tables (msc 65D20)
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Temme, N.M. (1997). Numerical algorithms for uniform Airy-type asymptotic expansions. Modelling, Analysis and Simulation [MAS]. CWI.