  One of the earliest examples of analytic representations for $\pi$ is given
by an infinite product provided by Wallis in 1655. The modern literature often
presents this evaluation based on the integral formula $$ \frac{2}{\pi}
\int_0^\infty \frac{dx}{(x^2+1)^{n+1}} = \frac{1}{2^{2n}} \binom{2n}{n}. $$ 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.
