# Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

The following is a re-post from MSE because I did not get any answer even after offering a bounty.

Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern Analysis") paper "The Final Problem: An Account of the Mock Theta Functions" the following formula of Ramanujan is mentioned: \begin{align}&\int_{0}^{\infty}e^{-3\pi x^{2}}\frac{\sinh \pi x}{\sinh 3\pi x}\,dx\notag\\ &\,\,\,\,\,\,\,\,= \frac{1}{e^{2\pi/3}\sqrt{3}}\sum_{n = 0}^{\infty}\frac{e^{-2n(n + 1)\pi}}{(1 + e^{-\pi})^{2}(1 + e^{-3\pi})^{2}\dots(1 + e^{-(2n + 1)\pi})^{2}}\tag{1} \end{align} where the term corresponding to $n = 0$ in the sum on the right is $1$.

Is there way to establish this exotic integral formula? Or a reference to any existing proof of $(1)$ would be of great help.

This relation seems to be part of the story of Ramanujan's mock theta function. The identity you discuss is a specialization of transformation properties of a mock theta function evaluated at $\tau =i$. See this paper of Zwegers for proofs of the general transformation identity (and he gives other references to earlier work of Watson): in particular, look at section 3 and Lemma 3.2 of Zwegers's paper. Other surveys that you may find of interest are Ono and Duke.
• Thanks. I now get the full picture. This integral is related to $\omega(-q)$ of Watson's paper. I had not read those transformations formulas (just read proofs of algebraic identities between mock theta functions) otherwise the answer to my question would have been obvious. Commented Aug 1, 2015 at 7:43