Abstract.
In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and \(\overline{\overline{G}}(p)\), the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, \(\overline{\overline{G}}(p)\) is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the Euler-Maclaurin expansion for the quadrature error functional.
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Received June 11, 1997 / Revised version received December 15, 1997
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Monegato, G., Lyness, J. The Euler-Maclaurin expansion and finite-part integrals. Numer. Math. 81, 273–291 (1998). https://doi.org/10.1007/s002110050392
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DOI: https://doi.org/10.1007/s002110050392