关于Cauchy—Riemann方程的注记

单值复变函数的可微性,不但要求复变函数的实部及虚部都可微,而且要求复变函数的实部及虚部所确定的两个实函数必需满足Cauchy—Riemann方程.因此,Cauchy—Riemann方程是判断复变函数可微和解析的主要条件.在直角坐标和极坐标下的Cauchy—Riemann方程,在复变函数论教程或专著中,都已论述.为了解析函数理论和应用上的需要,本文研究复平面上任意两个正交向量上的Cauchy—Riemann方程,并得到有关定理.

NOTES ON CAUCHY-RIEMANN EQUATIONS

First of all, the author studies the Cauchy-Riemann Equations on any two orthogonal vectors on the complex plane, and then proves some theorems and lemmas that have to do with the Cauchy-Riemann Equations, finally gives their applications.