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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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

  • [1] N. Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Comm. Pure Appl. Math. 19, 353-370 (1966) MR 0204943
  • [2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher transcendental functions. Vol. 2, McGraw-Hill, New York, 1953
  • [3] A. Erdélyi, Asymptotic expansions, Dover, New York, 1956 MR 0078494
  • [4] A. Erdélyi, Asymptotic evaluation of integrals involving a fractional derivative, SIAM J. Math. Anal. 5, 159-171 (1974) MR 0348360
  • [5] G. H. Hardy, Divergent series, Oxford University Press (Clarendon), London, 1949 MR 0030620
  • [6] F. W. J. Olver, Asymptotics and special functions, Academic Press, New York (1974) MR 0435697
  • [7] F. W. J. Olver, Error bounds for stationary phase approximations, SIAM J. Math. Anal. 5, 19-29 (1974) MR 0333545
  • [8] P. W. Schmidt, Small-angle x-ray scattering from rods and platelets, J. Math. Phys. 7, 1295-1300 (1966)
  • [9] P. W. Schmidt, An asymptotic approximation for a type of Fourier integral, Math. Comp. 32, 1171-1182 (1978) MR 0510821
  • [10] R. Wong, Error bounds for asymptotic expansions of Hankel transforms, SIAM J. Math. Anal. 7, 799-808 (1976) MR 0415224
  • [11] H. Wu and P. W. Schmidt, to be published

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 41A60

Retrieve articles in all journals with MSC: 41A60

Additional Information

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

American Mathematical Society