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3 classical analysis tag added

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?

2 added 18 characters in body

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

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 are:

• Where can I find the proof of the above result?

• "The following are certain special values": Whats so special about the values?

1

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

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

Questions I would like to ask are:

• Where can I find the proof of the above result?

• "The following are certain special values": Whats so special about the values?