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Parabolic cylinder functions of large order. (English) Zbl 1087.33010
Summary: The asymptotic behaviour of parabolic cylinder functions of large real order is considered. Various expansions in terms of elementary functions are derived. They hold uniformly for the variable in appropriate parts of the complex plane. Some of the expansions are doubly asymptotic with respect to the order and the complex variable which is an advantage for computational purposes. Error bounds are determined for the truncated versions of the asymptotic series.
33E30Functions coming from differential, difference and integral equations
Full Text: DOI
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[2] Olver, F. W. J.: Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter. Philos. trans. Roy. soc. A 250, 479-517 (1958) · Zbl 0083.05701
[3] F.W.J. Olver, Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders, J. Res. Nat. Bur. Stand. 63B (Math. Math., Phys.) (1959) 131 -- 169. · Zbl 0090.04602
[4] Olver, F. W. J.: Asymptotics and special functions. (1974) · Zbl 0303.41035
[5] Temme, N. M.: Numerical and asymptotic aspects of parabolic cylinder functions. J. comput. Appl. math. 121, 221-246 (2000) · Zbl 0966.65023