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Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. (English) Zbl 0772.33015

If a function has a compound asymptotic expansion consisting of two series, one of which dominates the other in a certain sector of $argz$, it is seen that the multipliers of subdominant terms, called Stokes multipliers, in the expansion in the vicinity of certain rays, called Stokes lines, show a discontinuous behaviour which is called Stokes phenomenon and this was discovered by G. G. Stokes in 1857. M. V. Berry [Proc. R. Soc. Lond., Ser. A 422, No. 1862, 7-21 (1989; Zbl 0683.33004)] has recently put forward a theory of the smoothing of Stokes phenomenon based upon optimal truncation of the asymptotic expansion and Borel summation of resulting exponentially small remainder terms. For certain classes of functions defined by Laplace and Stieltjes transforms, D. S. Jones and F. W. J. Olver [Asymptotic and computational analysis. Conf. in honor of Frank W. J. Olver’s 65th birthday, Proc. Int. Symp., Winnipeg/Can. 1989, Lect. Notes Pure Appl. Mat. 124, 241-264 (1990; Zbl 0693.41034); ibid., 329-355 (1990; Zbl 0704.33001)] and W. G. C. Boyd [Proc. R. Soc. Lond., Ser. A 429, No. 1876, 227-246 (1990; Zbl 0704.33002)] obtained rigorous uniform exponentially-improved asymptotic expansions which describe the smooth transition of a Stokes multiplier across a Stokes line. In this paper an alternative theory has been presented for functions defined by Mellin-Barnes’ integrals by which uniform exponentially-improved expansions can be established in a direct manner.

The theory has been illustrated by considering the parabolic cylinder function, which has a compound asymptotic expansion in a certain sector, using Mellin-Barnes integrals. The author first studies the most important properties of basic terminants ${T}_{\nu }\left(z\right)$ introduced by D. B. Dingle [Asymptotic expansions: their derivatives and interpretation (1973)] and then gives a rigorous discussion of the estimates of ${R}_{n}$, the remainder terms in the asymptotic series of parabolic cylinder function, in the form of Mellin-Barnes integrals and expressible in terms of basic terminants, which were derived by F. W. J. Olver [SIAM J. Math. Anal. 22, No. 5, 1475-1489 (1991; Zbl 0738.41030)] by a different method.

##### MSC:
 33E20 Functions defined by series and integrals 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 34E05 Asymptotic expansions (ODE)