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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
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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?)

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  • [2] Tore Hȧvie, Remarks on an expansion for integrals of rapidly oscillating functions, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 16–29. MR 0323077
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Keywords: Periodic Bernoulli function, generalized Riemann zeta function, Euler summation formula
Article copyright: © Copyright 1981 American Mathematical Society