# 柯西－黎曼方程

（重定向自柯西－黎曼方程）

(1a)     ${\displaystyle {\partial u \over \partial x}={\partial v \over \partial y}}$

(1b)    ${\displaystyle {\partial u \over \partial y}=-{\partial v \over \partial x}.}$

## 注释和其他表述

### 共形映射

(2)    ${\displaystyle {i{\partial f \over \partial x}}={\partial f \over \partial y}.}$

${\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}},}$

### 复共轭的独立性

(3)    ${\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0}$

${\displaystyle {\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right).}$

### 复可微性

${\displaystyle f(z)=u(z)+iv(z)}$

${\displaystyle \lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {f(z_{0}+h)-f(z_{0})}{h}}=f'(z_{0})}$

${\displaystyle \lim _{\underset {h\in \mathbb {R} }{h\to 0}}{\frac {f(z_{0}+h)-f(z_{0})}{h}}={\frac {\partial f}{\partial x}}(z_{0}).}$

${\displaystyle \lim _{\underset {ih\in i\mathbb {R} }{h\to 0}}{\frac {f(z_{0}+ih)-f(z_{0})}{ih}}=\lim _{\underset {ih\in i\mathbb {R} }{h\to 0}}-i{\frac {f(z_{0}+ih)-f(z_{0})}{h}}=-i{\frac {\partial f}{\partial y}}(z_{0}).}$

f沿着两个轴的导数相同也即

${\displaystyle {\frac {\partial f}{\partial x}}(z_{0})=-i{\frac {\partial f}{\partial y}}(z_{0}),}$

### 物理解释

${\displaystyle {\bar {f}}={\begin{bmatrix}u\\-v\end{bmatrix}}}$

${\displaystyle {\frac {\partial (-v)}{\partial x}}-{\frac {\partial u}{\partial y}}=0.}$

${\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial (-v)}{\partial y}}=0.}$

### 其它解释

${\displaystyle {\frac {\partial u}{\partial s}}={\frac {\partial v}{\partial n}},\quad {\frac {\partial u}{\partial n}}=-{\frac {\partial v}{\partial s}}}$

${\displaystyle {\partial u \over \partial r}={1 \over r}{\partial v \over \partial \theta },\quad {\partial v \over \partial r}=-{1 \over r}{\partial u \over \partial \theta }.}$

${\displaystyle {\partial f \over \partial r}={1 \over ir}{\partial f \over \partial \theta }.}$

## 非齐次方程

${\displaystyle {\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}=\alpha (x,y)}$
${\displaystyle {\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}=\beta (x,y)}$

${\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=\phi (z,{\bar {z}})}$

${\displaystyle f(\zeta ,{\bar {\zeta }})={\frac {1}{2\pi i}}\iint _{D}\phi (z,{\bar {z}}){\frac {dz\wedge d{\bar {z}}}{z-\zeta }}}$

## 推广

### Goursat定理及其推广

f = u+iv为复函数，作为函数f : R2R2可微。则柯西积分定理（柯西－古尔萨定理）断言f在开域Ω上解析当且仅当它在该域上满足柯西-黎曼方程（Rudin 1966，Theorem 11.2）。特别是，f不需假定为连续可微（Dieudonné 1969，§9.10, Ex. 1）。

f在整个域Ω上满足柯西-黎曼方程是要点。可以构造在一点满足柯西-黎曼方程的连续函数，但它不在该点解析（譬如，f(z) = z5/|z|4）。只满足柯西-黎曼方程也是不够的，（需额外满足连续性），下面的例子表明了这一点：（Looman 1923，p.107）

${\displaystyle f(z)={\begin{cases}\exp(-z^{-4})&\mathrm {if\ } z\not =0\\0&\mathrm {if\ } z=0\end{cases}}}$

• f(z)在开域Ω⊂C上局部可积，并以弱形式满足柯西-黎曼方程，则f和Ω上的一个解析函数几乎处处相等。

### 多变量的情况

${\displaystyle {\bar {\partial }}}$

${\displaystyle {\partial f \over \partial {\bar {z}}}=0}$,

${\displaystyle {\partial f \over \partial {\bar {z}}}={1 \over 2}\left({\partial f \over \partial x}-{1 \over i}{\partial f \over \partial y}\right).}$