An asymptotic formula for a type of singular oscillatory integrals

Authors:
L. C. Hsu and Y. S. Chou

Journal:
Math. Comp. **37** (1981), 503-507

MSC:
Primary 41A60

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628711-X

MathSciNet review:
628711

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Abstract: This paper offers a general expansion formula for oscillatory integrals of the form in which *N* is a large parameter, *Nx* denotes the fractional part of *Nx*, and is a fixed real number in . Our formula is expressed in terms of some ordinary integrals with integrands containing periodic Bernoulli functions and the generalized Riemann zeta function.

**[1]**P. J. Davis & P. Rabinowitz,*Methods of Numerical Integration*, Blaisdell, Waltham, Mass., 1975, p. 120. MR**0448814 (56:7119)****[2]**T. Havie, "Remarks on an expansion for integrals of rapidly oscillating functions,"*BIT*, v. 13, 1973, pp. 16-29. MR**0323077 (48:1435)****[3]**L. C. Hsu, "A refinement of the line integral approximation method and its application,"*Sci. Record*(N. S. Academia Sinica), No. 6, 1958, pp. 193-196. MR**0101982 (21:784)****[4]**L. C. Hsu & Y. S. Chou,*Numerical Integration in Higher Dimensions*, Science Press, Peking, 1980, Chapter 14. (Chinese)**[5]**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

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628711-X

Keywords:
Periodic Bernoulli function,
generalized Riemann zeta function,
Euler summation formula

Article copyright:
© Copyright 1981
American Mathematical Society