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These are two famous identities of Ramanujan. Where can I find the proofs of them:

  1. $ \displaystyle \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum\limits_{k=0}^{\infty} \frac{(4k!)(1103 + 26390k)}{(k!)^{2} (396)^{4k}}$

  2. $\displaystyle \int\limits_{0}^{\infty} \frac{1 + x^{2}/(b+1)^{2}}{1+x^{2}/a^{2}} \times \frac{1+ x^{2}/(b+2)^{2}}{1 + x^{2}/(a+1)^{2}} \times \cdots dx= \frac{\sqrt{\pi}}{2} \times \frac{\Gamma(a+1) \Gamma(b+\frac{1}{2}) \Gamma(b-a+\frac{1}{2}}{\Gamma(a)\Gamma(b+\frac{1}{2} \Gamma(b-a+1)}$ for $0 < a < b+\frac{1}{2}$.

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Consider adding some specifics to your title, maybe "Where can I find the proofs of the following identities of Ramanujan?" Long titles show up very well on the main page. – Tom Stephens Aug 9 '10 at 19:45… may be of use (I don't understand what machinery they use but it claims to prove the series) it also says the full details are in 'Pi and the AGM: a study in the analytic number theory and computational complexity'. – anon Aug 9 '10 at 22:19
Hurrah! Motivation! – Charles Stewart Aug 10 '10 at 12:45
up vote 4 down vote accepted

Since you seem to be in Chennai, walk into any mathematics library and pick up the collected works of Ramanujan. That is the best. Or look into the notes edited by Bruce Berndt.

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