Parabolic cyclinder functions : examples of error bounds for asymptotic expansions

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.