منابع مشابه
Surprising Sinc Sums and Integrals
We show that a variety of trigonometric sums have unexpected closed forms by relating them to cognate integrals. 1 Motivation Recall that sinc(x) := sin(x)/x when x 6= 0 and sinc(0) := 1. In [4] and [8], it was shown that, for N = 0, 1, 2, 3, 4, 5, and 6, ∫ ∞
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Using Fourier transform techniques, we establish inequalities for integrals of the form ∫ ∞ 0 n ∏ k=0 sin(ak x) ak x dx . We then give quite striking closed form evaluations of such integrals and finish by discussing various extensions and applications.
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This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that are encountered in Feynman diagram calculations. Together with results for the values of the higher harmonic series at infinity the presented algorithms can...
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In this paper closed-form sums are given for various slowly-convergent infinite series which arise essentially from the differentiation of Dirichlet L-series. Some associated integrations are also considered. A small number of the results appear in standard tables, but most seem to be new.
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We consider sums involving the product of reciprocal binomial coefficient and polynomial terms and develop some double integral identities. In particular cases it is possible to express the sums in closed form, give some general results, recover some known results in Coffey and produce new identities.
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ژورنال
عنوان ژورنال: The American Mathematical Monthly
سال: 2008
ISSN: 0002-9890,1930-0972
DOI: 10.1080/00029890.2008.11920606