## An asymptotic formula for a type of singular oscillatory integrals

HTML articles powered by AMS MathViewer

- 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