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

Introduction

Generalizations of C(n,r)

Ramanujan’s Theorems

Identities

Arithmetic Functions

Introduction

Even Functions (Mod r)

A Linear Congruence

Ramanujan’s Significant Formula

Introduction

Relationships between H1(s) through H12(s)

Evaluation of H1(s) through H12(s) for Certain Values of s

The Case s = 2

The Case s = 3 and Ramanujan’s G(1)

Ramanujan’s Formula for ζ(2n+1)

A Proof of (1.3.6.4)

Proof of (1.3.6.1) for n≥1

Proof of (1.3.6.3)

Another Proof of (1.3.6.1) for n≤-1

Acknowledgement

Appendix A

A1 Partial Fraction Decomposition of Cotangent

A2 Ramanujan’s Formulae

A3 Bernoulli Numbers

A4 A Formula for ζ(1-2n)

A5 Halphen’s Formula

A6 Eisenstein Series

2 Ramanujan’s Theorem on p(5m+4)

Combinatorial Properties

Euler’s Identity

Product of Infinite Number of Power Series

Convergence Considerations

Jacobi’s Formula

Power Series with Integer Coefficients

Ramanujan’s Theorem

**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

B1

B2

Reference