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Parabolic cylinder functions: Examples of error bounds for asymptotic expansions. (English) Zbl 1049.33003
The parabolic cylinder functions \(U(a,z)\) and \(V(a,z)\) are solutions of the differential equation \[ \frac{d^2 y}{d z^2}-\Big( \frac{z^2}{4} + a \Big) y = 0. \] The authors derive PoincarĂ©-type expansions of \(U\) and \(V\) for large \(z\) and uniform expansions for large \(a\). They extend F. W. J. Olver’s [J. Soc. Ind. Appl. Math., Ser. B, Numer. Anal. 2, 225–243 (1965; Zbl 0173.33901)] error bounds and discuss error bounds when integral representations of \(U\) are used. Numerical aspects of computing those bounds and of evaluating the parabolic cylinder functions are also described.
Reviewer: Axel Riese (Linz)

MSC:
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
33F05 Numerical approximation and evaluation of special functions
65D20 Computation of special functions and constants, construction of tables
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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References:
[1] Abramowitz M., National Bureau of Standards Applied Mathematics Series, in: Handbook of Mathematical Functions (1964) · Zbl 0171.38503
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