##### 柯西-黎曼方程 - 柯西-黎曼方程 - **柯西-黎曼方程**是两个[[偏微分方程]], 也是[[复函数]]成为[[解析函数]]或[[全纯函数]]的充分必要条件, 设复函数 $f(z)=u(x,y)+iv(x,y)$, 则复导数的存在要求极限与自变量的趋近路径无关, 考虑实轴和虚轴两种特殊路径, 它们必然相等 - $\displaystyle f'(z) = \lim_{h \to 0, h \in \mathbb{R}} \frac{f(z + h) - f(z)}{h} = \frac{\partial f}{\partial x} = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}$ - $\displaystyle f'(z) = \lim_{k \to 0, k \in \mathbb{R}} \frac{f(z + i k) - f(z)}{i k} = \frac{1}{i} \frac{\partial f}{\partial y} = -i \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}$ - $\displaystyle \frac{\partial f}{\partial x} = -i \frac{\partial f}{\partial y}$ $\begin{cases} \dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y} \\[2mm] \dfrac{\partial u}{\partial y} = -\,\dfrac{\partial v}{\partial x} \end{cases}$