Research On Some Theorems And Related Problem Of The Bi-analytic Functions And Analytic Functions

Complex variable function is an important branch in mathematics.It also has been applied widely in masses of subjects.Analytic functions are the centre object of complex variable function.The connection of complex variable function and other branches in mathematics is more and more intimate.In recent years,many scholars have studied the bi-analytic function that derived from analytic functions and obtained rich results.However,the theory of bi-analytic function is still not perfect and needs further study.This dissections is mainly to discuss the problem of estimating in the derivation of bounded analytic function and some theorem and nature about bi-analytic function.Firstly,background and status of the derivative of bounded analytic functions null functions,non-zero and bi-analytic function are introduced,and some result is obtained in this paper.Secondly,the problem of estimating in the derivation of bounded analytic null functions and non-zero function are discussed.By using the principle of the inductive model and some lemmas,more accurate estimation of the n th derivation of the functions are obtained.Hence there are some better result in the problem of estimating the derivation of bounded regular functions.Finally,the concept and some theorem are introduced for analytic function and bi-analytic function.And get some theorem of bi-analytic function.Such as:Liouville theorem of bi-analytic function,Cauchy inequality theorem of bi-analytic function.According to the relationship between analytic function and harmonic function,the expression of two integral forms of bi-analytic function is obtained,(?).(A,B,C are constant)u(x,y),s(x,y)are real part;and(?).(A,B,Care constant)v(x,y),t(x,y)are imaginary part.This enrichs the theory of bi-analytic function.Otherwise,another kind of bi-analytic function is studied in this paper.