This book provides an introduction to the large expanse of Ramanujan’s work in Number Theory, Analysis, and several other areas of Mathematics. The first chapter discusses about trigonometric sums and Ramanujan’s significant formulae. Ramanujan’s own proof of “P(5m + 4) is divisible by 5” is explained in chapter second. The Ramanujan differential equations in third chapter lead on to the Weierstrass invariants, and from there the final chapter, which provides a full account of elliptic functions as viewed by Ramanujan.

Needless to say, we only covered only a very small fraction of Ramanujan’s work. However, after developing a few facts about his work, we equipped to prove many interesting theorems. Our intent here is not to give a rigorous course in analysis but to emphasize the most important ideas about his work and how they interplay with other areas of Mathematics.

Dr. M.L. Thivagar, Professor and Chairperson of School of Mathematics, Madurai Kamaraj University, Madurai, India

1. Ramanujan’s Contribution

• On Ramanujan’s Trigonometrical Sums
• Arithmetic Functions
• Ramanujan’s Significant Formula
• Acknowledgement
• Appendix A

2. Ramanujan’s Theorem on p(5m+4)

• Combinatorial Properties
• Euler’s Identity
• Jacobi’s Formula

3. Ramanujan’s Differential Equations

• The Basic Identity
• The Differential Equations of P, Q, R
• The Jordan – Kronecker Functions

4. Ramanujan’s Sense of Weierstrassian Invariants

• Generalized Ramanujan’s Identity
• Hyper-geometric Equations
• Lambert Series Expansions

5. Ramanujan Sense of Development of Elliptic Functions

• Elliptic Functions
• Picard’s Theorem
• Elliptic Integrals
• Acknowledgement
• Appendix B

References