Several asymptotic expansions of parabolic cylinder functions are discussedand error bounds for remainders in the expansions are presented. Inparticular Poincaré-type expansions for large values of the argument$z$ and uniform expansions for large values of the parameter areconsidered. It is shown how expansions can be derived by using thedifferential equation, and, for a special case, how an integralrepresentation can be used. The expansions are based on those given inOlver (1959) and on modifications of these expansions given in Temme (2000).Computer algebra techniques are used for obtaining representations of thebounds and for numerical computations.

Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$ (msc 33C15), Numerical approximation and evaluation (msc 33F05), Computation of special functions, construction of tables (msc 65D20)
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Vidunas, R, & Temme, N.M. (2002). Parabolic cyclinder functions : examples of error bounds for asymptotic expansions. Modelling, Analysis and Simulation [MAS]. CWI.