The Application Of Regular Covering Surface In Complex Integral

In this paper,we take a new method to prove the first Cauchy integral theorem, through the regular covering surface of the Riemann surface. We can derive that the simply connected domain of variable upper limit integral is a single valued function.This paper mainly includes the following parts:The first part introduces the researching results and the significance of domestic and foreign mathematicians in this field simply, at the same time introduces some def-initions related with this paper, and introduces the lemmas and theorems related with this paper.The second part proves in this chapter:f is analysis in the D domain, a fixed pointa within the D, and for every p there is a path of the continuous mapping: l:[0,1]â†'D,l(0)=a,l(1)=p, the integral along this path:∫aâ†'pl f(z)dz, the or-dered pair written (p,∫aâ†'plf(z)dz)=p, all the ordered pairs written D, then D is a Riemann surface, and the Riemann surface D is the regular covering surface of D. And we can get two relevant inferences:(1)/is analytic function in the D, a fixed starting point, a, if there is a continuous mapping,l:[0,1]â†'D,l(0)=a,l(1)=p, then for F(p)=∫aâ†'pl f(z)dz, each point,p has the same number of values;(2)if D is a simple connected domain,F(p)=∫aâ†'p f(z)dz is a single function.The third part summarizes and prospects.