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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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An asymptotic formula for a type of singular oscillatory integrals
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by L. C. Hsu and Y. S. Chou PDF
Math. Comp. 37 (1981), 503-507 Request permission

Abstract:

This paper offers a general expansion formula for oscillatory integrals of the form $\smallint _0^1{x^{ - \alpha }}f(x,\{ Nx\} ) dx$ in which N is a large parameter, Nx denotes the fractional part of Nx, and $\alpha$ is a fixed real number in $0 < \alpha < 1$. Our formula is expressed in terms of some ordinary integrals with integrands containing periodic Bernoulli functions and the generalized Riemann zeta function.
References
  • Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0448814
  • Tore Hȧvie, Remarks on an expansion for integrals of rapidly oscillating functions, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 16–29. MR 323077, DOI 10.1007/bf01933520
  • L. C. Hsu, A refinement of the line integral approximation method and its application, Sci. Record (N.S.) 2 (1958), 193–196. MR 101982
  • L. C. Hsu & Y. S. Chou, Numerical Integration in Higher Dimensions, Science Press, Peking, 1980, Chapter 14. (Chinese) E. Riekstenš, "On asymptotic expansions of some integrals involving a large parameter," Učen. Zap. Leningrad. Gos. Univ., v. 41, 1961, pp. 5-23. (Russian)
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Math. Comp. 37 (1981), 503-507
  • MSC: Primary 41A60
  • DOI: https://doi.org/10.1090/S0025-5718-1981-0628711-X
  • MathSciNet review: 628711