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)
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