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Fast Integration for Cauchy Principal Value Integrals of Oscillatory Kind

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Abstract

In this paper we give a simple, but high order and rapid convergence method for computing the Cauchy principal value integrals of the form \(\int_{-1}^{1}e^{i\omega x}\frac{f(x)}{x-\tau}dx\) and its error bounds, where f(x) is a given smooth function, ωR + may be large and −1<τ<1. The proposed method is constructed by approximating \((\frac{f(x)-f(\tau)}{x-\tau})^{(s)}\) by using the special Hermite interpolation polynomial, which is a Taylor series. The validity of the method has been demonstrated by the results of several numerical experiments and the comparisons with other methods.

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Acknowledgements

The author is deeply grateful to the associated editor and referees for their valuable comments and suggestions for great improvement of this paper. Remarks are cited from the referee’s comments.

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Correspondence to Ruyun Chen.

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This work was supported partially Doctor Funds of Guangdong Ocean University (No. E09230).

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Chen, R. Fast Integration for Cauchy Principal Value Integrals of Oscillatory Kind. Acta Appl Math 123, 21–30 (2013). https://doi.org/10.1007/s10440-012-9709-z

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  • DOI: https://doi.org/10.1007/s10440-012-9709-z

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