Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On a uniform asymptotic expansion of a Fourier-type integral

Author: R. Wong
Journal: Quart. Appl. Math. 38 (1980), 225-234
MSC: Primary 41A60
DOI: https://doi.org/10.1090/qam/580880
MathSciNet review: 580880
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Abstract: An alternative derivation is given for a uniform asymptotic expansion of the integral

$\displaystyle I\left( h \right) = \int_q^z {g\left( {\sqrt {{y^2} - {q^2}} } \right)\sin yh dy \qquad \left( {h \to + \infty } \right)} $

, which has been obtained recently by Schmidt. Here $ q$ varies in the interval $ \left[ {0,{q_0}} \right]$, $ {q_0}$ is some fixed number less than $ z$, and $ z$ is finite. A similar result is obtained for integrals having an infinite range of integration. Realistic bounds are also provided for the error terms associated with the expansions. Our approach is based on a summability method introduced by Olver.

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DOI: https://doi.org/10.1090/qam/580880
Article copyright: © Copyright 1980 American Mathematical Society

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