The additional proof and application of Cauchy-Riemann Equation

Abstract

As far back as the seventeenth century, Newton and Leibniz introduced and refined the idea of calculus, and the concepts of derivative and differentiation gradually entered the mathematical world as powerful tools. However, after the appearance of the theory of complex functions in the 18th century, the conventional methods used to analyze the properties of complex functions were somewhat limited. Thus, mathematicians Cauchy and Riemann came up with the idea of the “C-R equation” and provided the necessary and sufficient conditions in derivable complex functions based on the research done by d'Alembert. This equation laid the foundation for the analytic function. This paper introduces the idea of the limit of two variables function and shows both the concept of and graphical meaning partial derivative, paving way for the concept of complex function and that of C-R equation. In addition, the paper specifically proves the C-R equation by a much more rigorous method with the routes from the linear equation and applies this important equation to determine the number of and the locations of the differentiable points in a complex plane. It is also one of the factors to determine the analyticity of complex functions. At the end of the paper, an example will be provided to illustrate the application of the C-R equation.