留数及应用--毕业论文.docVIP

PAGE - 2012届毕业生 毕业论文 题 目: 留数及其应用 2012年5月25日 摘 要 留数是复变函数论中重要概念的其中之一，它和解析函数在孤立奇点上的洛朗展开式、柯西复合闭路定理等有着紧密的联系.留数理论是复积分与复级数理论相结合的成果，正确运用留数定理可以使沿闭路的积分转变为计算孤立点处的留数.另外，我们运用留数定理可以把要求的积分转化成为复变函数沿闭曲线的积分，从而把等待求解积分转化为留数的计算.本文首先介绍留数的定义和留数定理及其相关定理，随后针对具体的不同的积分的类型通过举例子来说明这几类特殊函数的定积分的计算问题. 关键词：留数 留数理论 实积分 应用 Title Residues and its Application Abstract The residue is an important concept in the theory of complex functions which closely contacts with the analytic functions in Laurent expansions which are on the isolated singularity and Cauchy composite closed-circuit theorem. Residue theory is the combination of the results of the theory of complex integration with the complex series. The correct use of the residue theorem to the residue can transform isolated point at the residue which is along with the closed-loop integral into the calculation of the isolated point. In addition, we use the residue theorem to the required integral. It can be transformed into a complex function of the integral which is along with the closed curve, and change the integral which is waiting to be solved into the calculation of the residue. In this paper, the definition of the residue and the residue theorem including its related theorems are introduced firstly. And then, focusing in different types of integrals, I illustrate the calculation of the definite integrals of these types of specific functions by using examples. Keywords: residue theorem the definite integral application 目录 TOC \o 1-3 \h \z \u 1引言 5 2留数的起源及其概念 5 2.1 留数的起源及其相关现状 5 2.2留数的铺垫知识--孤立奇点 6 2.2.1孤立奇点的分类 6 2.2.2函数的零点与极点的关系 7 2.2.3函数在无穷远点的性态 8 2.3 留数的定义及留数定理 9 3留数的求法 10 4函数在无穷远点的留数 14 5用留数定理计算实积分 19 5.1 计算 型积分 20 5.2 形如的积分 22 5.3 计算型积分 23 5.4　计算型积分 26 5.5 计算积分路径上有奇点的积分 30 6辐角原理及其应用 32 6.1 对数留数 32 6.2 辐角原理 34 6.3 儒歇定理 35 结论 39 致谢 40 参考文献 41 1引言 留数在复变函数中是一个非常重要的基本概念，是数学中的一个非常重