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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)

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.)
Full Text: DOI
[1] Abramowitz M., National Bureau of Standards Applied Mathematics Series, in: Handbook of Mathematical Functions (1964) · Zbl 0171.38503
[2] DOI: 10.1007/978-3-642-88396-5 · doi:10.1007/978-3-642-88396-5
[3] Miller J. C. P., Tables of Weber Parabolic Cylinder Functions – Giving Solutions of the Differential Equation d2y/dx2 + (\(\tfrac14\)x2– a) y = 0 (1955)
[4] DOI: 10.1137/S0036141090187685 · Zbl 0799.41028 · doi:10.1137/S0036141090187685
[5] Olver F. W. J., J. Research NBS 63 pp 131–
[6] DOI: 10.1137/0702017 · Zbl 0173.33901 · doi:10.1137/0702017
[7] DOI: 10.1016/S0377-0427(00)00347-2 · Zbl 0966.65023 · doi:10.1016/S0377-0427(00)00347-2
[8] DOI: 10.1007/BF01447384 · JFM 02.0217.01 · doi:10.1007/BF01447384
[9] Whittaker E. T., A Course in Modern Analysis (1952)
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