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T. Amdeberhan, O. Espinosa, V. H. Moll and A. Straub
The American Mathematical Monthly
Vol. 117, No. 7 (August 2010), pp. 618-632
DOI: 10.4169/000298910x496741
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Page Count: 15
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Abstract One of the earliest examples of analytic representations for π is given by an infinite product provided by Wallis in 1655. The modern literature often presents this evaluation based on the integral formula \frac{2}{\pi }\int_0^\infty {\frac{{dx}}{{(x^2 + 1)^{n + 1} }} = \frac{1}{{2^{2n} }}\left({\begin{array}{c} {2n} \\ n \end{array} } \right).} In trying to understand the behavior of this integral when the integrand is replaced by the inverse of a product of distinct quadratic factors, the authors encounter relations to some formulas of Ramanujan, expressions involving Schur functions, and Matsubara sums that have appeared in the context of Feynman diagrams.

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