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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 Poincaré-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:
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