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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
up vote 10 down vote accepted

Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:

The result appears here in Page 60. with proof.

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This should be accepted as official answer. – GH from MO Jul 3 '12 at 13:44

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