## Rouche定理及其应用

Rouche Theorem and The Application of Rouche Theorem
Abstract: In this paper, firstly, it shows readers classical Rouche theorem and simple application of this theorem. Secondly, by means of analysing the conditions among the simple application of the classical Rouche theorem, we realize the punitive of conditions above classical Rouche theorem and the limitations of application. thus, in order to enhance the applicability of Rouche theorem, we weaken or change the conditions of this theorem properly. So, parts of the generalizations of Rouche theorem are summarized and proof of them is given. Then, researching the distribution of the root about functions of real variable in some areas, compared with the root of the existence theorem in mathematical analysis, the superiority of Rouche theorem is promulgated. Finally, we discuss emphatically the application of Rouche theorem and parts of the generalizations of Rouche theorem in three investigative aspects.
Key words: Argument principle; Rouche theorem; Analytic function; Number of zeros 源￥自%六^^维*论-文+网=www.lwfree.cn

1.知识预备 3
2.Rouche定理及其部分推广形式.. 5
2.1 Rouche定理. 5
2.2 Rouche定理的部分推广形式. 6
3.Rouche定理的优越性.. 10
4.Rouche定理的应用 13
4.1 Rouche定理在解决方程在一些区域内根的个数上的应用. 13
4.2 Rouche定理在确定解析函数零点个数上的应用... 15
4.3 Rouche定理在证明一些解析函数定理上的应用... 17
5.结束语... 19

Rouche定理及其应用引言
我们在复变函数论的学习中，会学到一个著名的定理，那就是儒歇（Rouche)定理. 该定理是辐角原理的一个推论，也可称作零点个数比较定理. 在考察函数零点的分布状况以及零点的个数上，运用儒歇定理能够使问题解决更加地方便和高效. 除此以外，该定理在另一些领域也有着十分重要的应用.

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