1 - 10 of 44 Chapters
[Jacobi’s triple product identity, the fundamental tool for all that follows, is established. A technique that is relied on heavily is illustrated.]
[The power of q-series is demonstrated by proofs of Jacobi’s 2- and 4-squares theorems.]
[The topic of partitions is introduced and Ramanujan’s simplest partition congruences are presented and proved. The proofs depend only on the expansion of Euler’s product and Jacobi’s cube of Euler’s product.]
[A uniform proof, based on an idea of Garvan and Stanton, is presented of Ramanujan’s simplest partition congruences.]
[A simple proof is given of what is often described as “Ramanujan’s most beautiful identity”.]
[A proof is given of Ramanujan’s partition congruences for powers of 5.]
[Garvan’s proof of Ramanujan’s partition congruences for powers of 7 is presented.]
[Earlier results are re-presented from Ramanujan’s point of view. Two proofs of Ramanujan’s celebrated 5-dissection of Euler’s product are given, including one by factorisation.]
[An identity, sent by Ramanujan in his first letter to Hardy, one of a handful that convinced Hardy that Ramanujan was a “mathematician of the highest class”, is presented. The proof is begun here, and finished after the Rogers–Ramanujan continued fraction is given in Chapter 15.]
[The quintuple product identity is stated and proved, and used to obtain the 5-, 7- and 11-dissections of Euler’s product and to obtain further identities of Ramanujan. The quintuple product identity is used at various points in the book.]
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