Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



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
MathSciNet review: 628711
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [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)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 41A60

Retrieve articles in all journals with MSC: 41A60

Additional Information

Keywords: Periodic Bernoulli function, generalized Riemann zeta function, Euler summation formula
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society