Integrals containing the infinite product ${\prod }_{n=0}^{\infty }[1+(\frac{x}{b+n}{)}^3]$

Abstract

We study several integrals that contain the infinite product $\prod _{n=0}^{\infty }\left[1+{\left(\frac{x}{b+n}\right)}^3\right]$ in the denominator of their integrand. These considerations lead to closed form evaluation ${\int }_{-\infty }^{\infty }\frac{dx}{{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)}^2}=\frac{1}{3}$ and to some other formulas.

8 pages