On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I: real variable

Richard B. Paris

Research output: Contribution to journalArticle

  • 16 Citations

Abstract

We review and discuss the application of Hadamard expansions to the hyperasymptotic evaluation of Laplace integrals where, for simplicity, in this paper x is restricted to be a positive real variable. The integration path C can be taken over both finite and semi-infinite intervals in the complex plane. In general, these expansions take the form of compound expansions, each associated with a different exponential level, and involve absolutely convergent series containing the incomplete gamma function as a smoothing factor. The early terms in each convergent expansion possess a rapid asymptotic-like decay (when the variable x is large) with late terms that can be transformed into a rapid decay comparable with that of the early terms. The Hadamard expansion of the above integral when the phase function p(t) is linear is shown to depend significantly on the singularity structure of the amplitude function f(t). The application of the theory to Laplace-type integrals with quadratic, cubic and nonpolynomial phase functions is considered; in addition to the amplitude function, the location of the saddle points satisfying p′(t)=0 also plays a role in the detailed structure of the different exponential levels in the resulting Hadamard expansion. Numerical examples are given to illustrate the accuracy that can be achieved with this new procedure.
Original languageEnglish
Pages (from-to)293-319
Number of pages27
JournalJournal of Computational and Applied Mathematics
Volume167
Issue number2
DOIs
StatePublished - Jun 2004

Fingerprint

Term
Laplace
Decay
Evaluation
Real variables
Laplace integral
Incomplete gamma function
Path integration
Infinite interval
Saddlepoint
Argand diagram
Smoothing
Simplicity
Singularity
Numerical examples
Series

Cite this

Paris, Richard B. / On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I : real variable.

In: Journal of Computational and Applied Mathematics, Vol. 167, No. 2, 06.2004, p. 293-319.

Research output: Contribution to journalArticle

@article{0492cbd9e44a466b8979426836b2c717,
title = "On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I: real variable",
abstract = "We review and discuss the application of Hadamard expansions to the hyperasymptotic evaluation of Laplace integrals where, for simplicity, in this paper x is restricted to be a positive real variable. The integration path C can be taken over both finite and semi-infinite intervals in the complex plane. In general, these expansions take the form of compound expansions, each associated with a different exponential level, and involve absolutely convergent series containing the incomplete gamma function as a smoothing factor. The early terms in each convergent expansion possess a rapid asymptotic-like decay (when the variable x is large) with late terms that can be transformed into a rapid decay comparable with that of the early terms. The Hadamard expansion of the above integral when the phase function p(t) is linear is shown to depend significantly on the singularity structure of the amplitude function f(t). The application of the theory to Laplace-type integrals with quadratic, cubic and nonpolynomial phase functions is considered; in addition to the amplitude function, the location of the saddle points satisfying p′(t)=0 also plays a role in the detailed structure of the different exponential levels in the resulting Hadamard expansion. Numerical examples are given to illustrate the accuracy that can be achieved with this new procedure.",
author = "Paris, {Richard B.}",
year = "2004",
month = "6",
doi = "10.1016/j.cam.2003.10.005",
volume = "167",
pages = "293--319",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",
number = "2",

}

On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I : real variable. / Paris, Richard B.

In: Journal of Computational and Applied Mathematics, Vol. 167, No. 2, 06.2004, p. 293-319.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I

T2 - Journal of Computational and Applied Mathematics

AU - Paris,Richard B.

PY - 2004/6

Y1 - 2004/6

N2 - We review and discuss the application of Hadamard expansions to the hyperasymptotic evaluation of Laplace integrals where, for simplicity, in this paper x is restricted to be a positive real variable. The integration path C can be taken over both finite and semi-infinite intervals in the complex plane. In general, these expansions take the form of compound expansions, each associated with a different exponential level, and involve absolutely convergent series containing the incomplete gamma function as a smoothing factor. The early terms in each convergent expansion possess a rapid asymptotic-like decay (when the variable x is large) with late terms that can be transformed into a rapid decay comparable with that of the early terms. The Hadamard expansion of the above integral when the phase function p(t) is linear is shown to depend significantly on the singularity structure of the amplitude function f(t). The application of the theory to Laplace-type integrals with quadratic, cubic and nonpolynomial phase functions is considered; in addition to the amplitude function, the location of the saddle points satisfying p′(t)=0 also plays a role in the detailed structure of the different exponential levels in the resulting Hadamard expansion. Numerical examples are given to illustrate the accuracy that can be achieved with this new procedure.

AB - We review and discuss the application of Hadamard expansions to the hyperasymptotic evaluation of Laplace integrals where, for simplicity, in this paper x is restricted to be a positive real variable. The integration path C can be taken over both finite and semi-infinite intervals in the complex plane. In general, these expansions take the form of compound expansions, each associated with a different exponential level, and involve absolutely convergent series containing the incomplete gamma function as a smoothing factor. The early terms in each convergent expansion possess a rapid asymptotic-like decay (when the variable x is large) with late terms that can be transformed into a rapid decay comparable with that of the early terms. The Hadamard expansion of the above integral when the phase function p(t) is linear is shown to depend significantly on the singularity structure of the amplitude function f(t). The application of the theory to Laplace-type integrals with quadratic, cubic and nonpolynomial phase functions is considered; in addition to the amplitude function, the location of the saddle points satisfying p′(t)=0 also plays a role in the detailed structure of the different exponential levels in the resulting Hadamard expansion. Numerical examples are given to illustrate the accuracy that can be achieved with this new procedure.

U2 - 10.1016/j.cam.2003.10.005

DO - 10.1016/j.cam.2003.10.005

M3 - Article

VL - 167

SP - 293

EP - 319

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 2

ER -