Proof of equation of complex variable function

The analysis of complex variable function is mainly composed of two functions, two theorems and one series. They are holomorphic function, meromorphic function, Cauchy integral theorem, residue theorem and Laurent series. The theorems of meromorphic functions and complex analytic functions are the most important, especially complex analytic functions. These functions are defined on the complex plane, and their values are complex and differentiable. The commonly used theories, formulas and methods include Cauchy integral theorem, Cauchy integral formula, Residue theorem, Laurent series expansion and so on. As a research field of modern analysis, complex analysis was established in the 19th century. The main founders were Cauchy, Riemann and Weierstrass. This article mainly study and summarize holomorphic functions and Cauchy integral theorem. Cauchy's integral theory is the pioneer of complex analysis. It mainly describes the reason and extension of the independence of integral and path.