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Berndt, Bruce C. Ahlgren, Scott Berndt, Bruce C. Stolarsky, Kenneth B. Zaharescu, Alexandru Xu, Ping 2013-08-22T16:37:57Z 2013-08-22T16:37:57Z 2013-08 2013-08-22T16:37:57Z 2013-08 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. Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-07-09T16:07:48Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Xu_Ping.pdf: 488152 bytes, checksum: 7316f588bedf939190e2b3e500be8c68 (MD5) Made available in DSpace on 2013-08-22T16:37:57Z (GMT). No. of bitstreams: 2 Ping_Xu.pdf: 488152 bytes, checksum: 7316f588bedf939190e2b3e500be8c68 (MD5) license.txt: 4056 bytes, checksum: 585245f3c9eb952638524554e4742239 (MD5) http://hdl.handle.net/2142/45365 en Copyright 2013 Ping Xu Circular summation formula elementary proof two-dimensional lattice sums Poisson equation theta functions analogue of theta functions analogue of Gauss sums Identities involving theta functions and analogues of theta functions text Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD