| Clifford analysis mainly studies the properties and related theories of functions defined on R~n and valued in Clifford algebra.Clifford analysis is a generalization of simple complex variable function theory in higher dimensional space,and it has been widely used in many branches of mathematics.The universal Clifford analysis is a more general extension of Clifford analysis.In this article,we first study Cauchy type integral and Cauchy type principal value integral of LR biregular functions taking values in the universal Clifford algebra.The L regular function,R regular function,LR regular function and LR biregular function are defined by the introduced Dirac operator.Then we define the Cauchy type integral of LR biregular function in universal Clifford analysis and construct Cauchy kernel function satisfying LR regularization.By using some lemmas,the problem of integral transformation of bivariable in universal Clifford analysis is solved.Then we obtain the Cauchy integral formula in the universal Clifford algebra by using repeated integral.Finally,we define Cauchy type principal value integrals and prove the existence of principal value integrals by using interpolation method.Then we define the T_ioperator of the universal Clifford analysis and study the commutative property of the T_i operator and Dirac operator.Then we use the Dirac operator and T_i operator to construct the L_k operator and R_k operator,define the k-vector regular function in the universal Clifford algebra,and study the relationship between k-vector regular function and other regular function classes and generalized harmonic functions.Finally,we give the generalized Stokes formula related to L_k and R_k operators in universal Clifford algebraic space,which is the key to find out the integral representation of k-vector regular functions. |
