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Numerical and asymptotic aspects of parabolic cylinder functions. (English) Zbl 0966.65023

The so-called Weber parabolic cylinder functions are the solutions of the differential equation

${y}^{\text{'}\text{'}}-\left(a+\frac{1}{4}{z}^{2}\right)y=0\phantom{\rule{2.em}{0ex}}\left(*\right)$

and stand for entire functions of $z$ for all values of $a$. The aim of this paper is to get uniform asymptotic expansions for the two standard independent solutions of $\left(*\right)$ usually denoted by $U\left(a,z\right)$ and $V\left(a,z\right)$. By only considering real values of the parameters, some of the asymptotic expansions derived from $\left(*\right)$ for these functions by F. W. J. Olver [J. Res. Natl. Bur. Stand., Sect. B 63, 131-169 (1959; Zbl 0090.04602)] valid as $|a|$ is large, are suitably modified by the author to obtain other new expansions that hold for computing $U\left(a,z\right)$ and $V\left(a,z\right)$ if at least one of the two parameters $a$, $z$ is large.

From a numerical point of view, several asymptotic properties of these modified expansions improve the corresponding ones of the results given by Olver in terms of elementary functions and Airy functions. The advantages for using the modified expansions in numerical algorithms are showed in a number of interesting remarks. Some of the expansions are also obtained from well-known integral representations of $U\left(a,z\right)$ and $V\left(a,z\right)$.

Numerical tests for some expansions are finally given.

##### MSC:
 65D20 Computation of special functions, construction of tables 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)