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Title:  Contributions to Trigonometric Sums and Fifth Order Mock Theta Functions 
Author(s):  Yeap, Boon Pin 
Doctoral Committee Chair(s):  Berndt, Bruce C. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  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. SwinnertonDyer. 
Issue Date:  2003 
Type:  Text 
Language:  English 
Description:  158 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2003. 
URI:  http://hdl.handle.net/2142/86824 
Other Identifier(s):  (MiAaPQ)AAI3102001 
Date Available in IDEALS:  20150928 
Date Deposited:  2003 
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Dissertations and Theses  Mathematics

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