Abstract: ABSTRACT: Methods for the numerical evaluation of the Weber parabolic cylinder functions W (a, x), which are independent solutions of the in- verted harmonic oscillator y?? + (x2/4 a)y = 0, are described. The functions appear in the solution of many physical problems, and no- tably in quantum mechanics. It is shown that the combined use of Maclaurin series, Chebyshev series, uniform asymptotic expansions for large a and/or x and the integration of the differential equation by local Taylor series are enough for computing accurately the functions in a wide rage of parameters. Differently from previous methods, the com- putational scheme is stable in the sense that high accuracy is retained: r 3 digits may be lost in double precision computations.

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