Cauchy Integral Formula For K-Monogenic Function With ?-weight In Superspace

Clifford analysis,as a function theory,also known as hyper complex function theory,is a new mathematical branch inspired by algebraic systems in geometry.The main research is regular function(solution of Dirac equation).The domain of object is fixed8)dimension orthogonal vector space and range is Clifford algebra.Different from the theory of simple complex variable,the theory is an extension to high-dimensional space.In addition,the main object of the theory of simple complex variable function is analytic function,and Clifford analysis mainly studies regular function.Superspace,founded in the middle of the last century,is an algebraic space derived from quantum mechanics.The original intention of this theory is that physicists need to use mathematical models to study the motion properties of some particles in the microscopic field.In recent years,people have begun to study hyperspace from the perspective of function theory.Then by using Clifford analysis theory,the two algebraic spaces are integrated to create a new algebraic system composed of two inverse commutative variable functions.Based on the above research,this paper uses the existing Clifford analysis theory to study the properties of6)-regular with?-weight functions in hyperspace.Preliminary preparation,construct the kernel function of Cauchy-Pompeiu formula.Through Clifford's analysis,we pay attention to the definition of6)-regular function,define6)-regular function with?-weight in hyperspace,and then use some lemmas to prove several main results in the previous work,including that the function|x|^(k-j)?f(x)is6)-regular with?-weight,the change of the form after the function H _k(x)is acted on by the super-Dirac operator,etc.Then we use these results to modify the kernel function of the Cauchy integral of the6)-regular with?-weight function in Clifford analysis to get the form we need.In the later work,the kernel function in the given binary domain is obtained by constantly using the Stokes formula in hyperspace,and we finally get the Cauchy-Pompeiu formula in hyperspace.In particular,when the function is6)-regular with?-weight,continue to use the Stokes formula to prove the Cauchy integral formula for6)-regular with?-weight function in hyperspace.