山西师范大学本科毕业论文

留数在积分计算中的应用

姓 名 学 院 专 业 班 级 学 号 指导教师 答辩日期 成 绩

杨瑛

数学与计算机科学学院

数学与应用数学

12级双学位

1154050131

籍慧洁

留数在积分计算中的应用

内容摘要

积分计算不但是高等数学中的一大主要内容，还是其他学科在处理实际生活问题时需要解决的一大问题。有的被积函数往往很难求出原函数，这时我们需要用到新的计算积分的方法——留数。留数是积分计算的又一重要工具，一般的积分计算我们可以采用牛顿—莱布尼茨公式、柯西积分定理、高阶求导公式、换元法等方法，而相对复杂的积分计算则需要采用新的运算方法，而留数及留数理论就起到至关重要的作用。本文首先，系统的归纳总结了留数在有限奇点及无穷远点处的定义，留数定理及相关理论以及留数的计算方法；其次具体的介绍了留数定理在定积分计算中的应用，主要包括三角函数有理式积分计算、有理函数积分计算、有理函数乘三角函数积分计算、两类特殊的广义积分计算以及利用泊松积分 ?e?xdx?0??2?2作辅助函数计算弗莱聂耳积分?cosx2dx及

0?????0sinx2dx；最后对本文进行了小结.

本文对留数理论的应用进行了分析总结，旨在为解决复杂积分问题提供理论依据，同时也为解决生活实际中的积分问题提供理论方法.

【关键词】留数 留数定理 复积分 实积分 极点 零点 广义积分

Application of Residue in Regulation Caculating

Abstract

Integral computation is not only the main contents in higher mathematics, or other subjects in dealing with real life need to solve a major problem. Some integrand is often difficult to find out the function, at this moment, we need to use new method for calculating integral residue.Residue is the important tool of integral calculation, the general integral calculation we can use Newton, leibniz formula,Cauchy integral

theorem, higher-order derivative formula, change element method and other methods but relatively complex integral calculation requires new methods of operation, so the residue and residue theory will play a crucial role. Summarized in this paper, first of all, the system residue in limited singularity at infinity and place, the definition of residue theorem and the related theory and the method for calculating the residue; Second specific residue theorem is introduced in the application of the definite integral calculation, mainly including trigonometric function rational expression of integral calculation, rational function integral calculation, rational function by trigonometric function integral calculation, the generalized integral calculation of two kinds of special and Poisson integral is used as the auxiliary function calculation the Frensnel integral; Finally, this article has carried on the summary.

In this paper, the application of residue theory are analyzed and summarized, aimed to provide theoretical basis for solving the problem of complex integral, as well as provide theoretical method to solve the integral problem in actual life.

【Key Words】 Residue The residue theorem Complex function integral Real integral The pole Zero Generalized integral

目 录

一、引言 ............................................................... 1 二、留数的定义及相关定理 ......................................... 1

（一）定义 ............................................................... 1 （二）主要定理及证明 ..................................................... 1

三、留数的求法及应用 ............................................... 4

（一）留数的求法 ......................................................... 4 （二）应用留数求复积分 ................................................... 6

四、应用留数计算定积分 ............................................ 8

（一）三角函数有理式积分 ................................................. 8 （二）有理函数积分 ....................................................... 9 （三）三角函数乘有理函数积分 ............................................ 12 （四）两类特殊路径上的广义积分 .......................................... 15 （五）利用函数ecz计算积分 ............................................... 19

2五、小结 .............................................................. 21 参考文献 .............................................................. 22 致谢 ................................................................... 23