In recent years has become increasingly in-depth study of closed integral, closed by space closed curve integral to space has a lot of closed surface integral solution emerged, such as gauss formula, green's theorem, etc. To solve this kind of method of contour integral has a wide range of applications. Thus we can easily know the contour integral research significance, so we determine the research direction in this paper.In this paper, the developing course of contour integral is studied and summarized, then the cauchy integral theorem of contour integral using the new method are verified, and then to promote theorem, cauchy integral theorem is studied in the promotion of significance. We will then go on to cauchy integral theorem application in definite integral and the mathematical equation, studied the use of residue theorem and cauchy integral theorem to solve the problems of the mathematics and physics method, and gives the corresponding examples, make on the application of the theorem can be more simple to be understood.At the same time, this article will also be residue theorem and cauchy integral theorem and the combination of series expansion, study their use in solving practical problems, this paper studies the multiple use of Fourier series, and using series and cauchy integral theorem to solve the corresponding integral problem.This article main research content is around the residue theorem and cauchy integral theorem, so in this paper, the study of contour integral can be converted into the study of double integral.