英语翻译摘要近些年来对闭合积分的研究已经越来越深入，闭合积分由空间闭曲线积分到空间闭曲面积分已经有很多的解决方法出现了，

英语翻译
摘要
近些年来对闭合积分的研究已经越来越深入，闭合积分由空间闭曲线积分到空间闭曲面积分已经有很多的解决方法出现了，如高斯公式，格林公式等，对于解决围道积分的此类方法都有很广泛的应用。由此我们很容易知道围道积分的研究意义，所以我们确定了本文的研究方向。
本文对围道积分的发展历程进行了研究和概括，接着我们对围道积分中的柯西积分定理运用了新方法进行证明，而后对定理进行了推广，研究了柯西积分定理的推广意义。而后，我们将柯西积分定理在定积分和数理方程上进行应用，研究了利用柯西积分定理和留数定理解决数学和物理问题的方法，并给出相应的例子，使这些定理在应用上可以更简单的被理解。
同时，本文也将柯西积分定理和留数定理与级数展开相结合，研究他们在解决实际问题上的使用，本文具体研究了多元傅里叶级数的使用，并利用级数和柯西积分定理求解相应的积分问题。
本文的主要研究内容是围绕柯西积分定理和留数定理展开的，所以本文对围道积分的研究可以转化为对复积分的研究。

Summary in recent years,research has become more of a closed score,points from
closed curves in space points to the closed space closed surface integral has
many solutions appear,such as the Gauss formula green formula for solving
contour integral which have a very wide range of applications.It is easy to
know the significance of contour integral,we determine the direction of this
article.This contour integral conducted research and provides an overview of
the history,then our contour integrals of Cauchy integral theorem using new
methods to prove that,then the theorem was promoted significance research on
generalization of the Cauchy integral theorem.Then,we will be in definite
integrals and Cauchy integral theorem of mathematical equations to be applied,
studied using residue theorem and Cauchy integral theorem to solve problems of
mathematics and physics,and gives you the appropriate examples,making these
theorems in the application can be simple to understand.At the same time,also
the residue theorem and Cauchy integral theorem combined with the series
expansion,study their use in solving the practical problems,this article
examined the use of multiple Fourier series,and used progression and integrals
of Cauchy integral theorem problem.Main research contents of this article are
organized around the residue theorem and Cauchy integral theorem,so this study
on contour integral can be converted into a study of complex integration.