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If you try to prove Ramanujan-like identities involving integrals or series, there is a wide variety of techniques to approach those problems (such as contour integrals, Fourier transformation, Dirichlet series and much more).

I wonder if there is a nice book that subsumes these techniques and offers a systematic approach to learn to solve hard integral identities. What would you recommend regarding this subject?

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This is not exactly what you're after, but you might find this book interesting: springerlink.com/content/978-81-322-0769-6/… –  Andrew Dec 18 '12 at 21:21

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up vote 3 down vote accepted

Look up "Ramanujan's Notebooks" in both Google and Amazon. In Amazon, you will get a chance to spend a lot to learn a lot. In Google, you will get links like this: http://www.plouffe.fr/simon/math/Ramanujan's%20Notebooks%20I.pdf.

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As an alternative, you could possibly do what Ramanujan himself is reputed to have done, which is to take the book by Carr: "Formulas and theorems in pure mathematics" (it is probably available on Google books), and work your way through the book trying to prove all the results that Carr quotes without proof, and do this in parallel with reading "The Mathematical Legacy of Srinivasa Ramanujan" by Murty and Murty.

Unfortunately, Ramanujan seems to have been rather poor writing out formal proofs of the amazing results he obtained, so I am not sure that my idea would be a very productive way to proceed, but it might possibly provide some insight into his way of making mathematical discoveries.

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Dear Old John, I know someone who did this actually (when they were a teenager, more-or-less) and is now a very strong number theorist. In fact, they were posting for a short while on this site just recently, until they were suspended! Cheers, –  Matt E Dec 19 '12 at 1:17
    
Fascinating! Back in the late 60s, I was very tempted to try this approach, and got as far as getting a copy of Carr's book ... and then I went to university, and they insisted I produced rigorous proofs of everything, finally I ended up working in analysis rather than number theory. –  Old John Dec 19 '12 at 1:23
    
Dear Old John, I think my friend was given a copy of Carr by his older brother (also now a gifted mathematician, though not in number theory). I had the impression that you are now interested in number theory, even if it's not what you worked in previously. Best wishes, –  Matt E Dec 19 '12 at 6:01

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