2026-09-22T03:26:32Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/86824 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Berndt, Bruce C. Yeap, Boon Pin 2015-09-28T15:19:43Z 2015-09-28T15:19:43Z 10000-01-01 2003 2003 "This thesis explores two different topics in number theory. The first portion of this thesis concentrates on trigonometric sums, giving also a survey on the history of evaluations and reciprocity theorems on such sums. We establish some general theorems on explicit evaluations and reciprocity theorems using contour integration as our main method. The second portion of this thesis focuses on Ramanujan's mock theta functions. In the second chapter, we extend the work of G. N. Watson on the fifth order mock theta functions, and show that in comparison to the third order functions, the fifth order mock theta functions possess a more complex behavior on radial approach to rational points on the unit circle. We prove a special case of a general transformation formula for the fifth order mock theta functions, extending the work of G. E. Andrews on the third order functions. Finally, the last chapter is devoted to proving four mock theta identities found in Ramanujan's ""lost"" notebook. We devise a method based on Ramanujan's y11 summation formula which effectively allows the computation of the residues of all the simple poles of certain infinite products. We explain how the method may be applied to obtain new identities of a similar type. The main ingredient in our proofs is a lemma of A. O. L. Atkin and P. Swinnerton-Dyer." Made available in DSpace on 2015-09-28T15:19:43Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3102001.pdf: 5187429 bytes, checksum: 77ce380f2e34aa719f584342d9af3926 (MD5) Previous issue date: 2003 Embargo set by: Seth Robbins for item 88105 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 158 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003. http://hdl.handle.net/2142/86824 (MiAaPQ)AAI3102001 eng Mathematics Contributions to Trigonometric Sums and Fifth -Order Mock Theta Functions text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.