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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?

edited Jul 3 at 14:53

asked Jul 2 at 17:01

Chandrasekhar
2,010●1●11●27

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Surely Bruce Berndt has documented these? The only question would be where. – Charles Matthews Jul 2 at 17:09

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"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 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. – Chandrasekhar Jul 3 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". – Chandrasekhar Jul 3 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 at 2:29

Chandrasekhar · Accepted Answer · 2012-07-04 10:55:49Z UTC

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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, Mess. Math. 44 (1915), 75-85.

The result appears here in Page 60. with proof.

edited Jul 4 at 10:55

answered Jul 3 at 6:53

Chandrasekhar
2,010●1●11●27

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

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

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