On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. II: Complex variable
type: Item
instance of: scholarly article
title: On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. II: Complex variable
zbMATH Open document ID: 1052.41017
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-10T00:00:00Z
review text: This paper considers the application of the Hadamard expansions to the hyperasymptotic evaluation of Laplace integrals of the form \[ I(x)\equiv\int_C e^{zp(t)}f(t)\,dt \] for large \(z\), where \(z\) is a complex variable and the integration path \(C\) is a finite or semi-infinite interval in the complex plane. In Part I of this work [J. Comput. Appl. Math. 167, No. 2, 293-319 (2004; Zbl 1052.41016)] the author improves the convergence properties of the standard Hadamard expansion for real \(z\). In this paper (part II) the author extends his analysis to the case of complex \(z\). The author shows how the resulting Hadamard expansions can be employed in the neighbourhood of a Stokes line. Applying this modification of the Hadamard method, the author obtains a fast convergent expansion of the the Airy function. Several numerical examples illustrate the accuracy and the computational improvement achieved with this new method.
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 2085544
zbMATH Keywords: asymptotics
zbMATH Keywords: hyperasymptotics
zbMATH Keywords: Hadamard expansions
zbMATH Keywords: Laplace-type integrals
author: Richard B. Paris
reviewed by: José L. López
MaRDI profile type: Publication
full work available at URL: j.cam.2003.10.021
OpenAlex ID: W4230049051
cites work: Uniform asymptotic smoothing of Stokes’s discontinuities
cites work: Hyperasymptotics for integrals with finite endpoints
cites work: On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type
cites work: On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables
cites work: On the use of Hadamard expansions in hyperasymptotic evaluation. II Complex variables
cites work: On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I: Real variable
cites work: Asymptotics and Mellin-Barnes integrals
cites work: Q4856171
cites work: Q3844537
DOI: 10.1016/J.CAM.2003.10.021
Recommended article: On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I: Real variable
Recommended article: On the use of Hadamard expansions in hyperasymptotic evaluation. II Complex variables
Recommended article: On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. IV: Poles
Recommended article: On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type
Recommended article: On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables
Recommended article: On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. III: Clusters of saddle points
Recommended article: Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions
Recommended article: Asymptotic expansions for two-dimensional hypersingular integrals
Recommended article: Asymptotic expansions of Laplace-type integrals. III
Recommended article: Q5270403
description: scientific article; zbMATH DE number 2085544