[Srinivasa Ramanujan](http://en.wikipedia.org/wiki/Srinivasa_Ramanujan) in his [first letter](http://books.google.co.in/books?id=Of5G0r6DQiEC&lpg=PP1&pg=PA21#v=onepage&q&f=false) to [G.H. Hardy](http://en.wikipedia.org/wiki/G.H._Hardy) stated many results for which he didn't give proofs. Among them the result taken from [this link](http://books.google.co.in/books?id=Of5G0r6DQiEC&lpg=PP1&pg=PA24#v=onepage&q&f=false) seems interesting :


* If $$\int\limits\_{0}^{\infty} \frac{\cos{nx}}{e^{2\pi\sqrt{x}}-1} \ dx = \phi(n)$$ then $\displaystyle\int\limits\_{0}^{\infty} \frac{\sin{nx}}{e^{2\pi\sqrt{x}}-1} \ dx = \phi(n)-\frac{1}{2n} + \phi\biggl(\frac{\pi^2}{n}\biggr)\sqrt{\frac{2\pi^3}{n^3}}$.


The link also mentions that $\phi(n)$ is a complicated function. The following are certain special values and shows some values.


Questions which I would like to ask here are:


* Where can I find the proof of the above result?
* "*The following are certain special values*": Whats so special about the values?



Request for the proof of a result from Ramanujan's letter to Hardy.
Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:


* *[Some definite integrals connected with Gauss's sums](http://books.google.co.in/books?id=oSioAM4wORMC&lpg=PA357&dq=Ramanujan%20collected%20papers&pg=PA59#v=onepage&q=Ramanujan%20collected%20papers&f=false)*, Mess. Math. 44 (1915), 75-85.


The result appears here in [Page 60.](http://books.google.co.in/books?id=oSioAM4wORMC&lpg=PA357&dq=Ramanujan%2520collected%2520papers&pg=PA60#v=onepage&q=Ramanujan%2520collected%2520papers&f=false) with proof.


