  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.
