一道典型复围线积分的探讨

A Typical Problem on the Complex Contour Integration

摘要: 复变函数是理工科数学教学的一门重要基础课。复围线积分是其中最核心的内容之一，因此掌握复围线积分是最重要的能力。本文通过一道经典复积分题目的求解，对此问题进行解剖分析，举一反三，以促进对复变函数最核心的知识点——柯西积分定理，柯西积分公式，高阶导数公式，复合闭路定理及留数定理的理解和掌握，最后也给出Matlab对此类问题解决的简单方法。

Abstract: Complex function is an important basic course in mathematics teaching for science and engineering students. Complex contour integral is one of the core contents, so mastering complex contour integral is the most important ability. This article dissects and analyzes this problem through solv-ing a classic complex integral, and draws inferences to promote the understanding and mastery of the core knowledge points of complex function, such as Cauchy integral theorem, Cauchy integral formula, higher order derivative formula, composite closed circuit theorem and residue theorem. Finally, it also gives a simple method for solving such problems in Matlab software.