The Properties For Biregular Function In Clifford Analysis

In this paper, the regular function with two variables so called the biregular function is studied. In Clifford analysis, the biregular function is the extension of the regular function in high dimensional space. The classical theorems of analytic function of one complex variable such as Morera theorem and Liouville theorem were generalized to the regular function, similarly to biregular function.In chapter 1, the preliminaries of this paper are given and after the definition of the biregular function in Clifford analysis is introduced.In chapter 2, firstly, the lemma explains that speaking of the proper integral in Clifford analysis, the integral can exchange the order. From this, the Cauchy integral theorem of biregular function is obtained. It's integral expression of biregular function which can express interior value with boundary value. The Morera theorem has given a necessary and sufficient condition about determination of the biregular function. The Painleve theorem has developed the biregular function from small region to wide range. These fundamental theorems have all promulgated the characteristic of biregular function, and will play a very big impetus role for following research of biregular function.The progression is an important tool in studying the biregular function. Expressing the biregular function to progression not only has theoretical significance, but also has the practical significance. For example, we may calculate the approximate value of the biregular function by using progression. In chapter 3, firstly, the power series of the biregular function is obtained by means of the uniform convergence of the kernel function in sphere of convergence. And also, when we may discuss the nature of the biregular function from different angle, for example, the uniqueness theorem may be proved to be the Painleve theorem deduction by using the similar method of the regular function. But this time, the Taylor expansion has already been given. Wards off the mentality in addition, the uniqueness theorem is obtained directly from the Taylor expansion and the region connectivity. This kind of method replies to the analytic function of one complex variable and the holomorphic function of multi-complex variable, as well as to the regular function and the biregular function in Clifford analysis. From the uniqueness theorem, it can be seen that its values are totally determined by the partial region value of the biregular function in its domain of definition. But in the previous chapter, the Cauchy integral formula of the biregular function manifests all interior values may be obtained by region boundary values. Therefore, the uniqueness theorem may be regarded as the complement of Cauchy integral formula. They reflect an essential characteristic of the biregular function from different side. Secondly, by researching the kernel function of the biregular function, the kernel estimate is obtained, and then Cauchy inequality is deduced. In fact, it has given a coefficient estimate in the development. Moreover, the promotion Cauchy inequality to compact set reflects this coefficient can be approximately controlled by the function value. From this, where the request is not too precise, a rough estimate of the biregular function can be carried on. Finally, the Weierstrass theorem has given the convergence of the biregular function sequence.In the preceding chapter, it can be seen that it's convenient to express the biregular function in spherical region using Taylor expansion. But some special functions could not express to Taylor expansion in the neighborhood of singular point. For instance, a function takes zero point as its singular point. Therefore, in chapter 4, the Laurent expansion of the biregular function is given in Laurent territory. The properties of the biregular function in the neighborhood of singular point is studied by using it. Firstly, the residues theorem is concluded. It is the continuation of Cauchy integral theorem and has close relation to the calculation integral. One kind of new power series of the biregular function is received in Laurent territory by Cauchy kernel expansion. It is convenient to research properties for the biregular function. In fact, this development and the Laurent expansion are equivalent. The Cauchy estimate is obtained by means of imputation four items in expansion. After that, movable singular point definition and its necessary and sufficient condition are given. It points out that under certain condition, the infinite power series summation to the biregular function may transform into the limited summation in the new development. From this, it is concluded that the integral function is the biregular function in Clifford analysis.The biregular function has very good nature. It is worth studying in many aspects. For instance, Morera theorem is obtained with the closed rectangle to approach in this article. But in practice, not all regions have the closed rectangle to approach. This aspect work is being studied by author. Moreover, its boundary value problems and approximation problems need to be considered further.