解析函数的几个等价命题及其应用

摘要

解析函数是复变函数论研究的主要对象,它具有重要的性质和广泛应用.本文在教材内容的基础上,进一步探讨解析函数的几种等价条件,文章首先给出了解析函数的四种不同的定义,然后运用柯西积分定理、泰勒定理及递推的方法,证明了这四种不同定义彼此相互等价；探讨了它们的一些应用,从而使学生能够用从多种方法判断函数是否解析,为以后应用解析函数打下了坚实的基础.%Analytic function is the primary object of complex function theory, which has important properties and widespread applications. Based on the content of teaching material, this paper further discussed several equivalent conditions of analytic function and their proofs. First the analytic functions of four different definitions was given, and then the Cauchy integral theorem, the Taylor's theorem and the Recurrence algorithm are used to prove the four different definitions are equivalent. Then it discusses their app licat ion, so that students can judge whether the functions are parsed in many different ways and they will lay a solid foundation for the future application of analytic functions.