Modular Properties Of Mock Theta Functions

Mock theta functions are an analogue and generalization of the classic concept of modular forms.The concept was first introduced by Ramanujan in the year 1920,and has undergone extensive research ever since.With the rapid development of modern mathematics,mock theta functions are becoming increasingly important,with connec-tions to problems in various subjects.This paper summarises S.Zwegers' work on the theory of mock theta functions,focusing on the derivation of their modular properties and the relationship between mock theta functions and classic modular forms.Modular properties are of fundamental importance in the theory of mock theta functions.After a brief background introduction,Chapter 1 gives an example of the modular properties of third order mock theta functions.This example illustrates certain features of the modular properties of mock theta functions,and provides some motivations and directions for the following discussion.Chapter 2 lays out the theoretical preparation.The main tool introduced is the so-called indefinite theta functions.Chapter 3 is devoted to deriving the modular properties of seventh order and fifth order mock theta functions.The key observation is through building a connection between the examples considered and the indefinite theta functions discussed in the previous chapter.