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Title:  Identities involving theta functions and analogues of theta functions 
Author(s):  Xu, Ping 
Director of Research:  Berndt, Bruce C. 
Doctoral Committee Chair(s):  Ahlgren, Scott 
Doctoral Committee Member(s):  Berndt, Bruce C.; Stolarsky, Kenneth B.; Zaharescu, Alexandru 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Circular summation formula
elementary proof twodimensional lattice sums Poisson equation theta functions analogue of theta functions analogue of Gauss sums 
Abstract:  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 twodimensional 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. 
Issue Date:  20130822 
URI:  http://hdl.handle.net/2142/45365 
Rights Information:  Copyright 2013 Ping Xu 
Date Available in IDEALS:  20130822 
Date Deposited:  201308 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois