Cauchy–Riemann equations

柯西-黎曼方程

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For f=u+iv on a plane domain, the Cauchy–Riemann equations are u_x=v_y and u_y=-v_x. Under C^1 regularity they characterize holomorphicity.

对平面区域上的 f=u+iv，柯西-黎曼方程为 u_x=v_y、u_y=-v_x；在一阶偏导连续的条件下，它们等价于全纯性。

Example

For $z^2$, $u=x^2-y^2$ and $v=2xy$, so $u_x=v_y=2x$ and $u_y=-v_x=-2y$.

Usage and distinctions

Equations satisfied at an isolated point without sufficient differentiability do not establish holomorphicity on a neighborhood.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Complex Analysis§1, equations (1.28)–(1.38) and Proposition 1.6 · Checked 2026-10-14

Translation sources

1. 肇庆学院 —《复变函数》课程教学大纲第二章，第一节 解析函数概念与柯西-黎曼方程，printed p.5 · Checked 2026-10-14

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