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Is there any explanation based on algebraic number theory that the integral $$ \int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}\tag{1} $$ has a closed form? Analytic proof of this integral is given in this MSE post, however this proof does not explain why a similar looking integral $$ \int_{-\infty}^\infty\frac{e^{ix\sqrt{3}}\ dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^3}=\frac{\sqrt{3}}{8\pi}\int_{0}^\infty\frac{dx}{\left(1+\frac{x^3}{1^3}\right)\left(1+\frac{x^3}{2^3}\right)\left(1+\frac{x^3}{3^3}\right)\ldots} $$ probably does not have a closed form. Is it possible that $(1)$ is related to Eisenstein integers?

Alternative formulation of the integral $(1)$ is $$ \int_0^\infty\frac{dt}{(1+t+t^{\,\alpha})^2}=\frac23, \quad \alpha=\frac{1+i\sqrt3}{2}. \tag{1a} $$

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  • $\begingroup$ Why do you think there is an explanation based on algebraic number theory? You could equally ask for an explanation based on algebraic topology or probability or combinatorics or what not. $\endgroup$
    – GH from MO
    Commented Mar 2, 2017 at 13:06
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    $\begingroup$ @GHfromMO , it seems that there might be connection with Eisenstein integers, but I wouldn't be too surprised if this was wrong. I also couldn't find a parametric generalization of (1), although there are other integrals of this kind with $(e^x+e^{-x}+e^{ix\sqrt{3}})^{a+3n}$, where $a=2,1/2$ and $n\ge 0$ an integer, in the denominator. There is some possibility that an isolated integral evaluation has relation to algebraic number theory. Only one proof of (1) is known so far, so any other proofs are welcomed, because they might show how to generalize (1) or explain why it is isolated result. $\endgroup$
    – Nemo
    Commented Mar 2, 2017 at 14:13
  • $\begingroup$ For example the Herglotz integral is from the category of integrals related to number theory math.stackexchange.com/questions/522913/… $\endgroup$
    – Nemo
    Commented Mar 2, 2017 at 14:16

1 Answer 1

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The following formula gives a parametric extension of $(1)$ for $|a|$ sufficiently small \begin{align} \small\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{a+ix\sqrt{3}}\right)^2}+e^a\int_{-\infty}^\infty\frac{dx}{\left(e^{a+x}+e^{-x}+e^{ix\sqrt{3}}\right)^2}+e^a\int_{-\infty}^\infty\frac{dx}{\left(e^{a+x}+e^{-x}+e^{-ix\sqrt{3}}\right)^2}=1 \end{align} This means that $(1)$ is not an isolated result. In view of this one might be very sceptical that any number theoretic interpretation of the integral exists.

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