The development of the elliptic functions according to Ramanujan : a thesis presented in partial fulfillment of the requirements for the degree of Master of Information Sciences in Mathematics at Massey University
Srinivasa Ramanujan (1887-1920) was one of the world's greatest mathematical geniuses. He made substantial contributions to elliptic functions, continued fractions, infinite series, and the theory of numbers. For many years people have studied Ramanujan's work and tried to obtain a better understanding of his work. The main purpose of my thesis will be to consider some important classical results on elliptic functions and give proofs of these results using the methods which could have been used by Ramanujan. This will give an insight into how Ramanujan may have proved many of his results since his own proofs are often unknown. This thesis contains five chapters. Chapter 1 is the introduction and this is related to Chapter 2 up to Chapter 4. The goal for Chapter 2 is to write the transformation of S2n+1(q), Φr,s(q), U2n(q), and V2n(q) in terms of P(p), Q (p), and R(p). Chapter 3 discusses Ramanujan's congruence for partitions and we give a proof for Ramanujan's modulus 5 partition congruence. In Chapter 4, we investigate a method of determining the number of representations of an integer n as the sum of two, four, six, and eight squares and triangular numbers. Then we present two computer programs which are for the sums of squares and triangles. Finally, some interesting relations between the sums of squares and the sums of triangles are shown.