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The Euler Maclaurin expansion for the Cauchy Principal Value integral

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Summary

The classical Euler Maclaurin Summation Formula expresses the difference between a definite integral over [0, 1] and its approximation using the trapezoidal rule with step lengthh=1/m as an asymptotic expansion in powers ofh together with a remainder term. Many variants of this exist some of which form the basis of extrapolation methods such as Romberg Integration. in this paper a variant in which the integral is a Cauchy Principal Value integral is derived. The corresponding variant of the Fourier Coefficient Asymptotic Expansion is also derived. The possible role of the former in numerical quadrature is discussed.

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, 60439, Argonne, IL, USA

    J. N. Lyness

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  1. J. N. Lyness
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Additional information

This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38

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Lyness, J.N. The Euler Maclaurin expansion for the Cauchy Principal Value integral. Numer. Math. 46, 611–622 (1985). https://doi.org/10.1007/BF01389662

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