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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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An asymptotic approximation for a type of Fourier integral
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by Paul W. Schmidt PDF
Math. Comp. 32 (1978), 1171-1182 Request permission

Abstract:

A uniform asymptotic approximation which can be used for all $qh \geqslant 0$ is developed for the Fourier integral \[ I(h) = \int _q^z {\frac {{f(\sqrt {{y^2} - {q^2})} }}{{{{({y^2} - {q^2})}^{1/2}}}}\sin yh dy} \] 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
    H. WU & P. W. SCHMIDT, research to be published. P. W. SCHMIDT, J. Math. Phys., v. 7, 1966, p. 1295.
  • A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956. MR 0078494
  • Norman Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Comm. Pure Appl. Math. 19 (1966), 353–370. MR 204943, DOI 10.1002/cpa.3160190403
  • A. Erdélyi, Asymptotic evaluation of integrals involving a fractional derivative, SIAM J. Math. Anal. 5 (1974), 159–171. MR 348360, DOI 10.1137/0505018
  • W. MAGNUS & F. OBERHETTINGER, Formulas and Theorems for the Functions of Mathematical Physics, Chelsea, New York, 1954, p. 27.
  • F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
  • Reference 6, p. 16, Equations (2a) and (2b).
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Math. Comp. 32 (1978), 1171-1182
  • MSC: Primary 41A60; Secondary 42A76
  • DOI: https://doi.org/10.1090/S0025-5718-1978-0510821-9
  • MathSciNet review: 0510821