Hyperbolic Functions On Hurwitz Series

In this thesis,we mainly study the Hurwitz series.We define hyperbolic functions on Hurwitz series,and obtain some equalities and properties,which are similar to ones in analysis.We first review the definition of Hurwitz series and a natural topology on it.Then we review some operations of the Hurwitz series,such as divided power,composition,Exp and Log.This thesis consists of three chapters.The first chapter introduces the background and its recent development,and analyzes the motivations of this thesis,and gives some basic notions and related notations.The second chapter reviews a natural topology and some operations on Hurwitz series,such as divided power,composition,Exp and Log.We introduce the concept of odd and even of Hurwitz series,and give some specific examples.We also compare the related concepts in analysis.The third chapter mainly studies some properties of hyperbolic functions on Hurwitz series.Using the relevant concepts and properties defined in the above two chapters,we can get some similar equalities on hyperbolic function of Hurwitz series,which are similar to the ones in analysis.