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Uniformly asymptotic expansions for an integral with a large and a small parameter

Published online by Cambridge University Press:  24 October 2008

F. Ursell
F. Ursell
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL
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Abstract

The integral

involves a large real parameter N and a small real parameter ∈. Its asymptotic behaviour is non-uniform when N → ∞ and ∈ → 0. Thus, when ∈ > 0 is kept fixed and N → ∞, the integral decays exponentially at a rate depending on ∈; when ∈ → 0 the integral tends to

which decays algebraically when N → ∞. It is shown that several distinct uniformly asymptotic expansions can be obtained which each involve an infinite set of functions of the combination Certain related integrals are also treated.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 2 , March 1987 , pp. 349 - 362
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

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