Foreword; Preface; Contents; 1 Introduction ; 1.1 Introduction; 1.2 An Identity of Euler; 1.3 Our First Proof of Jacobi's Triple Product Identity; 1.4 Our Second Proof of Jacobi's Triple Product Identity; 1.5 Some Important Special Cases of Jacobi's Triple Product Identity; 1.6 Euler's Pentagonal Numbers Theorem; 1.7 Jacobi's Formula for the Cube of Euler's Product; 1.8 Polynomial Versions of Earlier Results; 1.9 m-Dissection; 1.10 Linear Transformations; 2 Jacobi's Two-Squares and Four-Squares Theorems; 2.1 Introduction; 2.2 Our First Proof of Jacobi's Two-Squares Theorem

2.3 Our Second Proof of Jacobi's Two-Squares Theorem2.4 A Proof of Jacobi's Four-Squares Theorem; 3 Ramanujan's Partition Congruences; 3.1 Introduction; 3.2 p(5n+4)equiv08mu(mod6mu5); 3.3 p(5n+4)equiv08mu(mod6mu5) Again; 3.4 p(7n+5)equiv08mu(mod6mu7); 3.5 p(11n+6)equiv08mu(mod6mu11); 3.6 The Atkin
Swinnerton-Dyer Congruences for the Modulus 5; 3.7 The Atkin
Swinnerton-Dyer Congruences for the Modulus 7; 3.8 The Atkin
Swinnerton-Dyer Congruences for the Modulus 11; 4 Ramanujan's Partition Congruences
A Uniform Proof; 4.1 Introduction; 4.2 p(5n+4)equiv08mu(mod6mu5)

7 Ramanujan's Partition Congruences for Powers of 77.1 Introduction; 7.2 The 7-Dissection of Euler's Product; 7.3 The 7-Analogue of Ramanujan's Most Beautiful Identity; 7.4 The Modular Equation of Degree 7; 7.5 Ramanujan's Partition Congruences for Powers of 7; 7.6 Two More Formulas; 8 Ramanujan's 5-Dissection of Euler's Product; 8.1 Introduction; 8.2 Our First Proof of Ramanujan's 5-Dissection of Euler's Product; 8.3 Our Second Proof of Ramanujan's 5-Dissection of Euler's Product; 8.4 Review of Earlier Results; 8.5 A Factorisation of Identity (8.4.3)

8.6 Ramanujan's Versions of (8.3.1) and (8.3.2), (8.5.4) and (8.5.5)9 A "Difficult and Deep'' Identity of Ramanujan; 9.1 Introduction; 9.2 Proof Inspired by Chadwick Gugg; 9.3 Proof Inspired by G.N. Watson; 10 The Quintuple Product Identity; 10.1 Introduction; 10.2 Proof of the Quintuple Product Identity; 10.3 A Generalisation of the Quintuple Product Identity; 10.4 Ramanujan's 5-Dissection of Euler's Product Again; 10.5 The 7-Dissection of Euler's Product; 10.6 The 11-Dissection of Euler's Product; 10.7 Two Identities of Ramanujan; 10.8 Continuation; 10.9 Some Interesting Results