zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 ν (z) 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.

33E20Functions defined by series and integrals
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
34E05Asymptotic expansions (ODE)