Some Topics On General Cauchy-Riemann Equations

The system of Cauchy-Riemann partial differential equations in complex analysis provides sufficient and necessary conditions for complex differentiable functions to sat-isfy holomorphiG functions in open sets.Holomorphic functions are one of the core of complex theory research.They are differentiable functions from complex manifolds to complex fields.Analytic function is the main object of the study of complex variable function theory,that is,the differentiable complex function everywhere in the region.It is a kind of differentiable analytic function with certain characteristics,which is the main object of the study of complex variable function theory.It is a kind of dif-ferentiable function with certain characteristics.The main condition for judging the differentiability and analysis of complex function i5 Cauchy-Riemann strip.CR condi-tion is the main condition for judging that a complex variable function is differentiable at a point or analytic in a region.A single complex variable function is holomorphic if and only if it is real differentiable and satisfies Caudiy-Riemann equation.Ana-lytic function is the main object of the study of complex variable function theory.Cauchy-Riemann equation is the main condition for judging the differentiability and analysis of complex variable function.Its important role and position in complex vari-able function theory is self-evident.However,with the development and deepening of complex analysis,scholars have found that the existing linear Cauchy-Riemann equa-tion.Some nonlinear complex function problems can not be well described,that is,the Cauchy-Riemann equation has limitations.Euler,Riemann,Cauchy,d,Alembert and others are pioneers in exploring the Cauchy-Riemann equation.Therefore,many scholars have discussed and studied the Cauchy-Riemann equation for a long time.Analytic function is the main object of the study of complex variable function theory,that is,the Differentiable Complex Function everywhere in the region.It is a kind of differentiable analytic function with certain characteristics,which is the main object of the study of complex variable function theory.It is a kind of differentiable func-tion with certain characteristics.The main condition for judging the differentiability and analysis of complex function is Cauchy-Riemann strip.CR condition is the main condition for judging that a complex variable function is differentiable at a point or analytic in a region.I.N.Vekua,L.Bers and T.Carleman et al first developed the Cauchy-Riemann equation in the following generalized form called Carleman-Bers-Vekua equation.Its corresponding solution is called generalized analytic function.Z.D.Usmanov,M.Reis-sig,A.Timofeev,Giorgadze G,Jikia V,G.T.Makatsaria et al are famous scholars on Generalized Cauchy-Riemann Systems,they have studied Carleman-Bers-Vekua equation in detail from different angles and obtained abundant results.Therefore,this paper first converts the differentiable logic relation of complex function into algebraic form by K-structure transformation,and then carries on the thorough researcli.It is a common method to study mathematical objects by means of transformation.The most general conclusion and theoretical significance are obtained by studying the changing law of objective function before and after transformation.In this paper,the generalized Cauchy-Riemann equation is studied by means of K-structure transformation,which has general advantages,because of the arbitrary value of K-function.The main contents and innovations of this paper are as follows:The first chapter describes the background of this study.First,it introduces the development of linear Cauchy-Riemann equation and the related knowledge of analytic functions,including Cauchy integral theorem and integration,Liouville theorem,maxi-mum modulus principle and Schwarz lemma,and the algebraic expression of non-lineax Cauchy-Riemann equation.In Chapter 2,we study K-transformation and K-structure holomorphism.Ana-lytical or holomorphism is the core problem of complex function or complex analysis.It can explain and solve some phenomena in the field of complex analysis,such as the usability of constant theorem,the application scope of Liouville theorem and related problems.We use K-structure holomorphism.Conditions,the relevant issues were analyzed,and their scope of application and special forms were reconsidered.More further,we study K-structure holomorphic conditions for functions with multiple com-plex variables.Firstly,we extend the complex field of single complex variable C to the case of multiple complex variable Cn,and obtain some sufficient and necessary conditions for determining whether any given complex function is K-structurally holo-morphic.Then,we give Cn.Wirtinger derivative operators with generalized structure are studied.In Chapter 3,we study the second-order nonlinear K-structural Laplace equation.By using the K-structural holomorphic condition of functions with multiple complex variables,we obtain the generalized K-structural extra differential operator and the D operator,which extend the known(?)operator.In the fourth chapter,under the condition of K-structure holomorphism,the gen-eralized Cauchy integral theorem and the generalized Cauchy integral formula are studied.