Applications of Cauchy-Riemann Equations on Several Examples

Abstract

Cauchy-Riemann equations is always a heated topic in mathematics. It is widely used to solve both mathematic problems and daily issues. In this paper, basic knowledge which is needed for the study of Cauchy-Riemann equations has been explained carefully. This includes the understanding of analytic function, derivatives, and partial derivatives, as well as the method to derive the final equations. The method mainly used to obtain the equations in the paper is to calculate the limits of a function with complex variables along two axes. The application that Cauchy-Riemann equations are most frequently applied to is to find derivatives of complex functions, and some relevant examples are included. In addition, there is an extension of the basic use of Cauchy-Riemann equations in functions that are in Cartesian forms. This study can help to inspire the green hands in the field of complex analysis and let them have a good command of knowledge of the most superficial level of this field.