On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type

D. Kaminski, Richard B. Paris

Research output: Contribution to journal › Article

3 Citations (Scopus)
11 Downloads (Pure)

Abstract

We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability.
Original languageEnglish
Pages (from-to)59-178
Number of pages20
JournalProceedings of the Royal Societ of Edinburgh: Section A Mathematics
Volume134
Issue number1
DOIs
Publication statusPublished - 2004

Fingerprint

Differential equation
Linear differential equation
Evaluation
Special Functions
Remainder
Asymptotic Expansion
Series

Cite this

Kaminski, D. ; Paris, Richard B. / On the use of Hadamard expansions in hyperasymptotic evaluation : differential equations of hypergeometric type. In: Proceedings of the Royal Societ of Edinburgh: Section A Mathematics. 2004 ; Vol. 134, No. 1. pp. 59-178.
@article{dbe99daaa5d24829b788ffffff976d53,
title = "On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type",
abstract = "We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability.",
author = "D. Kaminski and Paris, {Richard B.}",
year = "2004",
doi = "10.1017/S0308210500003139",
language = "English",
volume = "134",
pages = "59--178",
journal = "Proceedings of the Royal Society of Edinburgh Section A: Mathematics",
issn = "0308-2105",
publisher = "Cambridge University Press",
number = "1",

}

Kaminski, D & Paris, RB 2004, 'On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type', Proceedings of the Royal Societ of Edinburgh: Section A Mathematics, vol. 134, no. 1, pp. 59-178. https://doi.org/10.1017/S0308210500003139

On the use of Hadamard expansions in hyperasymptotic evaluation : differential equations of hypergeometric type. / Kaminski, D.; Paris, Richard B.

In: Proceedings of the Royal Societ of Edinburgh: Section A Mathematics, Vol. 134, No. 1, 2004, p. 59-178.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the use of Hadamard expansions in hyperasymptotic evaluation

T2 - differential equations of hypergeometric type

AU - Kaminski, D.

AU - Paris, Richard B.

PY - 2004

Y1 - 2004

N2 - We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability.

AB - We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability.

U2 - 10.1017/S0308210500003139

DO - 10.1017/S0308210500003139

M3 - Article

VL - 134

SP - 59

EP - 178

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 1

ER -

Kaminski D, Paris RB. On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type. Proceedings of the Royal Societ of Edinburgh: Section A Mathematics. 2004;134(1):59-178. https://doi.org/10.1017/S0308210500003139

Access to Document