zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
On the use of Hadamard expansions in hyperasymptotic evaluation. I: Real variables. (English) Zbl 1039.33500
It is known that the level of accuracy afforded by optimal truncation of an asymptotic expansion (superasymptotic level) results in a remainder term that is exponentially small in the asymptotic variable. In order to achieve higher levels of accuracy a hyperasymptotic expansion method was introduced by M. V. Berry and C. J. Howls [Proc. R. Soc. Lond., Ser. A 434, No. 1892, 657–675 (1991; Zbl 0764.30031); ibid. 430, No. 1880, 653–668 (1990; Zbl 0745.34052)] by re-expanding the superasymptotic remainder in a new asymptotic series whose remainder is exponentially small compared with the first remainder. This process is repeated by expanding the remainder in a new expansion up to any desired level of accuracy. In this paper the author proposes a new method of hyperasymptotic evaluation using (absolutely convergent) Hadamard expansions. The essence of the method is previously illustrated by evaluating a slowly convergent Hadamard expansion of a well-known integral representation of the modified Bessel function I ν (x) (for x>0) written in terms of the incomplete gamma function [see G. N. Watson, A treatise on the theory of Bessel functions, 2nd. ed., Cambridge Math. Library (ed. 1995; Zbl 0849.33001)]. This expansion is here transformed by a simple rearrangement of the tail to produce a rapidly convergent series which is suitable for hyperasymptotic levels of precision. Then, starting from an integral representation for the confluent hypergeometric function 1 F 1 (a;b;x) (when x>0) [see M. Abramowitz and I. Stegun (eds.), Handbook of mathematical functions, 10 th. printing, New-York: Wiley, p. 505 (1972; Zbl 0543.33001)] (function that includes I ν (x) as a particular typical case), the author derives a modified Hadamard expansion that consists of a single sum with coefficients involving the normalized incomplete gamma function P(b-a+k,x) Re b>Rea>0). When a=ν+1 2, b=2ν+1 the results agree with those obtained for I ν (x). By an extension of the previous analysis, the Hadamard expansion for the second type of confluent hypergeometric function U(a;b;x) (x>0), represented by a Laplace integral over an infinite interval [Abramowitz and Stegun, ibid, p. 505] it requires a decomposition of the path of integration to yield an infinite number of Hadamard expansions, associated with a decreasing sequence of subdominant exponentials. Numerical examples involving I ν (x) and K ν (x) are also given to illustrate the accuracy of the asymptotic expansions.

MSC:
33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1
33F05Numerical approximation and evaluation of special functions
65D20Computation of special functions, construction of tables