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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR**0448814****[2]**Tore HÈ§vie,*Remarks on an expansion for integrals of rapidly oscillating functions*, Nordisk Tidskr. Informationsbehandling (BIT)**13**(1973), 16–29. MR**0323077****[3]**L. C. Hsu,*A refinement of the line integral approximation method and its application*, Sci. Record (N.S.)**2**(1958), 193–196. MR**0101982****[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)

Retrieve articles in *Mathematics of Computation*
with MSC:
41A60

Retrieve articles in all journals with MSC: 41A60

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