Fundamental theorems in Complex analysis

Abstract

This article will first introduce the conception of complex numbers and then deal with some famous results in complex numbers. In the beginning, we prove the Cauchy-Goursat theorem and calculate the integral along a closed path in a domain on which the given function is analytic. We want to check the relation between a function being analytic, having primitive, and its integral along a closed path equals zero. In conclusion, for a given analytic function on a simply connected domain, integral along any closed path is zero. Using this conclusion, we can prove the residue theorem and the Cauchy integral formula. To prove the existence and uniqueness of Laurent expansion, we first prove the Laurent decomposition by the Cauchy integral formula and Liouville’s theorem. By Laurent’s decomposition theorem, we change the case into dealing with two analytic functions. But one of them is analytic at infinity. The definition of the analytic function is applied to get the result. It’s easy to notice that every singularity of a meromorphic function on a subset of the complex number is isolated. Since meromorphic functions are widely used in variable aspects, it’s important to study isolated singularities. Then by studying the unique Laurent expansion in the neighborhood of the isolated singularity, we can classify isolated singularity.