5
$\begingroup$

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?

Improve this question
$\endgroup$
5
  • 6
    $\begingroup$ Surely Bruce Berndt has documented these? The only question would be where. $\endgroup$
    – Charles Matthews
    Commented Jul 2, 2012 at 17:09
  • 6
    $\begingroup$ "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 $2i$. $\endgroup$
    – Noam D. Elkies
    Commented Jul 2, 2012 at 19:07
  • $\begingroup$ @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. $\endgroup$
    – C.S.
    Commented Jul 3, 2012 at 2:21
  • $\begingroup$ @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". $\endgroup$
    – C.S.
    Commented Jul 3, 2012 at 2:24
  • $\begingroup$ 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. $\endgroup$
    – David Roberts ♦
    Commented Jul 3, 2012 at 2:29

1 Answer 1

Reset to default
10
$\begingroup$

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.

Improve this answer
$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .