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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

An asymptotic approximation for a type of Fourier integral


Author: Paul W. Schmidt
Journal: Math. Comp. 32 (1978), 1171-1182
MSC: Primary 41A60; Secondary 42A76
MathSciNet review: 0510821
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Abstract | References | Similar Articles | Additional Information

Abstract: A uniform asymptotic approximation which can be used for all $ qh \geqslant 0$ is developed for the Fourier integral

$\displaystyle I(h) = \int_q^z {\frac{{f(\sqrt {{y^2} - {q^2})} }}{{{{({y^2} - {q^2})}^{1/2}}}}\sin yh\,dy} $

under the assumptions that $ hz > > 1$, that the first $ L + 2$ derivatives of $ f(y)$ are continuous for $ 0 \leqslant y \leqslant {({z^2} - {q^2})^{1/2}}$, and that the first $ 2L + 2$ derivatives of $ f(y)$ are continuous at $ y = 0$.

The approximation is especially convenient when $ z > > q$.


References [Enhancements On Off] (What's this?)

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  • [4] Norman Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Comm. Pure Appl. Math. 19 (1966), 353–370. MR 0204943 (34 #4778)
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  • [8] Reference 6, p. 16, Equations (2a) and (2b).

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1978-0510821-9
PII: S 0025-5718(1978)0510821-9
Article copyright: © Copyright 1978 American Mathematical Society