Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0743.41025
Olver, F.W.J.
Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. (English)
[J] SIAM J. Math. Anal. 22, No.5, 1460-1474 (1991). ISSN 0036-1410; ISSN 1095-7154/e

Author's abstract: ``By allowing the number of terms in an asymptotic expansion to depend on the asymptotic variable, it is possible to obtain an error term that is exponentially small as the asymptotic variable tends to its limit. This procedure is called `` exponential improvement''. It is shown how to improve exponentially the well-known Poincaré expansions for the generalized exponential integral (or incomplete Gamma function) of large argument. New uniform expansions are derived in terms of elementary functions, and also in terms of the error function. Inter alia, the results supply a rigorous foundation for some of the recent work of M. V. Berry on a smooth interpretation of the Stokes phenomenon.'' Note: In the words of the author the purpose of this investigation is to show how to construct expansions for a wide class of functions. The new theory goes beyond that one presented in chapter 14 of the author's celebrated book, Introduction to asymptotics and special functions (1974) in three ways: (i) the number of functions that can be treated successfully is greatly increased, (ii) Non elementary functions may be used in constructing expansions, (iii) as a consequence of (ii), regions of validity are extended considerably. He further states that {\it D. S. Jones} [Lect. Notes Pure Appl. Math. 124, 241-264 (1990; Zbl 0693.41034)] has independently investigated the generalized exponential integrals or incomplete Gamma function substantially overlaping the work of the author but his work differs significantly in the method of proof and results achieved.
[C.M.Joshi (Udaipur)]

MSC 2000:: *41A60 Asymptotic problems in approximation
33E99 Special functions

Keywords: exponential improvement; Poincaré expansions; Stokes phenomenon

Citations: Zbl 0693.41034

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]