# Request for the proof of a result from Ramanujan's letter to Hardy.

Srinivasa Ramanujan in his first letter to G.H. Hardy stated many results for which he didn't give proofs. Among them the result taken from this link 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?

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Surely Bruce Berndt has documented these? The only question would be where. – Charles Matthews Jul 2 '12 at 17:09
"Special value of $\phi$" simply means that there might not be a closed form for $\phi(n)$ for general $n$ but there are special cases where $\phi(n)$ can be evaluated. For example, $\sin x$ has the special value $(\sqrt{5}-1)/4$ at $x = \pi/10$, and the $j$ function has the special value $66^3$ at $2i$. – Noam D. Elkies Jul 2 '12 at 19:07
@Noam: Thanks for the explanation. A small doubt: Here you say $\phi$ doesn't have a closed form, but you give an example of a function $\sin{x}$ which does seem to have a closed form. – S.C. Jul 3 '12 at 2:21
@Charles: Yes even I thought Bruce Berndt's webpage should contain this. But I couldn't find it in his webpage, though what I did was the following: go to his publication page, search for papers which contain the title "Integral". – S.C. Jul 3 '12 at 2:24
Usually 'closed form' refers to 'being able to write down a function in terms of elementary functions and no integrals'. One can write $Li(x)$ as an integral, but it is generally held to have no closed form. – David Roberts Jul 3 '12 at 2:29