Holomorphic Function On The Set Of The Roots Of Unity

In analytic function theory,the general analytic function theory is relatively mature.In order to extend the analytic function theory,this paper studies a new analytic function-analytic function on the set of roots of unit.In [1],the author gives the concept of an analytic function on the set of roots of unit.They are function series with one variable x by definition.This series are value only when x is a root of unit.Analogy general analytic function,the author constructs the derivative function of analytic function and the Taylor expansion on the set of roots of unit through a special method.This paper mainly discusses some computations and generalizations of this analytic function,which enriches the research results of analytic function theory and extends analytic function theory very well.This paper always based on the analytical function theory,to study Some computations and generalizations of holomorphic functions on the set of the roots of unity.Specific arrangements are as follows:In the first chapter,we first introduce some of the historical background of this article and research status,and then briefly about the work done in this article.In the second chapter,we mainly discuss the related definition theorems of group and ring,give the concept of polynomial rings,give some special cases and proof process to understand,provide some preliminary knowledge for this article,and increase the understanding of the article.In the third chapter,we write about the related definition theorem of derivative and Taylor expansion,and the related theorem proving.In Chapter four,according to Kazuo Habiro in article [1] gives the concept of unit root set analytic function.Conduct a new definition of "derivative" and Taylor expansion,and explore some of the conclusions of analytic functions under the new definition by calculating.we introduce a conception of “sets system”generalizing “the set of roots of unit”,and present a conception of “analytic functions on roots systems”generalizing the conception of analytic functions on the set of roots of unit.In the fifth chapter,we made a summary and outlook.