Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.

Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$ (msc 33C15), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Computation of special functions, construction of tables (msc 65D20)
CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Temme, N.M. (2000). Numerical and asymptotic aspects of parabolic cylinder functions. Modelling, Analysis and Simulation [MAS]. CWI.