【作者】 王海燕；

【导师】 乔玉英；

【作者基本信息】 河北师范大学 ， 基础数学， 2008， 硕士

【摘要】 本文主要研究含有两个变量的正则函数,所谓的双正则函数。Clifford分析中,正则函数是单复分析中全纯函数在高维空间的推广,全纯函数的经典函数理论如Morera定理,刘维尔定理等都可推广到正则函数,同样也可推广到双正则函数。第一章,给出了本文的预备知识和双正则函数的定义。第二章,首先,给出一引理说明对Clifford分析中正常积分而言,积分是可以交换顺序的。由此得到双正则函数的Cauchy积分定理,其导出了一个用边界值表示双正则函数内部值的积分公式,它是双正则函数的积分表达式。Morera定理给出了双正则函数判定的一个充要条件,而开拓定理将小区域的双正则函数开拓到更大区域。它们都揭示了双正则函数的特性,是双正则函数的基本定理,这些结果对下面研究双正则函数的性质起了很大的推动作用。级数是研究双正则函数的一个重要工具,把双正则函数表示为级数不但有理论上的意义,而且也有实用意义,例如利用级数可以计算函数的近似值。第三章,首先由双正则的核函数在收敛球内的一致收敛性,给出双正则函数的幂级数展开形式即Taylor展式。另外,在探讨双正则函数的性质时可从不同角度入手,如双正则函数的唯一性定理可以仿照正则函数的方法作为开拓定理的推论给出。但此时,Taylor展式已经给出,我们可另僻思路,直接从Taylor展式和Ω的连通性得出唯一性定理。这种方法对单复变中全纯函数,多复变中全纯函数,以及实Clifford分析中正则函数,双正则函数都适用。从得到的唯一性定理还可以看出,双正则函数在其定义域中,部分区域的取值情况完全决定着它在其他部分的值。而上章的Cauchy积分公式中,体现出双正则函数的区域边界上的值可以推得它在区域内部的一切值,因此唯一性定理可以看成Cauchy积分公式的补充,它们从不同侧面反映出双正则函数的本质特征。接下来,对双正则的核函数进行研究,得到了核的估计,进而得到了Cauchy不等式,它实际上给出了Taylor展式中系数的一个估算。另外Cauchy不等式在紧集上的推广,反映出此系数可以由函数值进行大致控制。由此在要求不太精确的情况下,可对双正则函数进行估值,最后Weierstrass定理给出了双正则函数列的收敛性。在前一章已经看出,用Taylor展式来表示球形域内的双正则函数是很方便的,但是对于一些特殊的函数,比如以原点为奇点的函数,就不能在奇点邻域内表示成Taylor展式。为此,第四章,建立了Laurent域内双正则函数的级数表示即Laurent展式,并以它为工具研究双正则函数在奇点邻域内的性质。首先给出了留数定理。它是Cauchy积分定理的继续。与积分的计算问题有密切联系。为了更方便的研究双正则函数的性质,由Cauchy核的展开,还给出了Laurent域内双正则函数一种新的幂级数展开,实际上,此展式与Laurent展式是等价的。然后对展式中四项进行估算,得到了Cauchy估计,而后给出了可去奇点的定义及其充要条件,并且指出双正则函数在一定条件限制下,新的展式中无穷项幂级数求和可转换为有限项求和,并由此得出结论,在Clifford分析中,有界整函数必为双正则函数。双正则函数是一类性质很好的函数,它有很多地方值得我们去研究,比如在研究Morera定理时,是用有界闭矩形进行逼近的,但实际问题中,不是所有区域都可用有界闭矩形逼近,关于这方面的工作本文作者正在继续研究。另外,其边值问题及逼近问题有待进一步研究。更多还原

【Abstract】 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.更多还原