Cauchy-Riemann 方程

Cauchy-Riemann 方程（柯西-黎曼方程），简称为 C.-R. 方程，是判断复变函数在某点（区域）解析的必要条件，即如果复变函数在某点（区域）内解析，那么必然满足 C.-R. 方程，不满足该方程的点或区域上该函数都不解析。

函数$w = f(z)$在定义域内一点$z = z_0$可微的充要条件是在点$z = z_0$可微且满足 C.-R. 方程。函数$w = f(z)$在区域$D$上解析的充要条件是在区域$D$上可微且满足 C.-R. 方程。

内容

直角坐标形式

设定义在点集$E$上的复变函数$w = f(z) = u(x, y) + \text{i} v(x, y)$，称如下的方程组为对应于$z = x + \text{i} y$的 C.-R. 方程 $$\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y}, \quad \dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}.$$ 此时复变函数的导数 $$f'(z) = \dfrac{\partial u}{\partial x} + \text{i} \dfrac{\partial v}{\partial x}.$$

极坐标形式

设定义在点集$E$上的复变函数$w = f(z) = u(r, \theta) + \text{i} v(r, \theta)$，称如下的方程组为对应于$z = r \text{e}^{\text{i}\theta}$的 C.-R. 方程 $$\dfrac{\partial u}{\partial r} = \dfrac{1}{r} \dfrac{\partial v}{\partial \theta}, \quad \dfrac{\partial v}{\partial r} = -\dfrac{1}{r} \dfrac{\partial u}{\partial \theta}.$$ 这时 $$f'(z) = \dfrac{1}{2} \text{e}^{-\text{i}\theta} \left( \dfrac{\partial f}{\partial r} - \dfrac{\text{i}}{r} \dfrac{\partial f}{\partial \theta} \right).$$

共轭分解形式

$f(z) = u(x, y) + \text{i} v(x, y) = g(x, y) = g(\dfrac{z + \overline{z}}{2}, \dfrac{z - \overline{z}}{2\text{i}})$。

我们做如下形式定义 $$\dfrac{\partial}{\partial z} = \dfrac{1}{2} \left( \dfrac{\partial}{\partial x} - \text{i} \dfrac{\partial}{\partial y} \right), \qquad \dfrac{\partial}{\partial \overline{z}} = \dfrac{1}{2} \left( \dfrac{\partial}{\partial x} + \text{i} \dfrac{\partial}{\partial y} \right).$$ 可以验证有 $$\mathrm{d} f(z) = \dfrac{\partial f}{\partial z} \mathrm{d} z + \dfrac{\partial f}{\partial \overline{z}} \mathrm{d} \overline{z}.$$ 于是复变函数$w = f(z)$的 C.-R. 方程是 $$\dfrac{\partial f(z)}{\partial \overline{z}} = 0.$$

共轭分解的性质

共轭分解还有如下性质：设复变函数$f(z) = u + \text{i}v$在区域$D$上可微且连续到边界，那么 $$\begin{vmatrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\ \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} \\ \end{vmatrix} = \left| \dfrac{\partial f}{\partial z} \right|^2 - \left| \dfrac{\partial f}{\partial \overline{z}} \right|^2$$ 特别地，当$f(z) = u + \text{i}v$在$D$上解析时，有 $$\begin{vmatrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\ \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} \\ \end{vmatrix} = \left| \dfrac{\partial f}{\partial z} \right|^2.$$

参考资料

1. 钟玉泉, 《复变函数论（第五版）》, 高等教育出版社, 北京, 2021-03, ISBN 978-7-0405-5587-5.