| My dissertation is mainly about various identities involving theta functions and analogues of theta functions. In Chapter 1, we give a completely elementary proof of Ramanujan's circular summation formula of theta functions and its generalizations given by S. H. Chan and Z.-G. Liu, and J. M. Zhu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given. In Chapter 2, we analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many nice explicit examples are given. In Chapter 3, we study one page in Ramanujan's lost notebook that is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this chapter, we prove all the claims made by Ramanujan about this integral. |
