##### 全纯函数 - 全纯函数 - **全纯函数**是指[[开集]]上处处可[[导数|复微分]]的[[复函数]], 或者说满足[[柯西-黎曼方程]]. 复函数的全纯性要比实函数的可微性具有更好的性质, 存在无穷阶复微分, 与[[解析函数]]等价, 复导数可写成[[雅可比矩阵]], 其几何意义是旋转和放缩, 不改变夹角, 并且复导数模的平方等于雅可比行列式. 设 $U$ 是复平面 $\mathbb{C}$ 中的一个开集, 如果复函数 $f: U \to \mathbb{C}$, $f(z) = u(x, y) + i v(x, y)$ 在 $U$ 内处处可复微分, 那么称 $f$ 在 $U$ 上全纯 - $\displaystyle f'(z) = \lim_{\Delta z \to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}$ - $\displaystyle f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}$ - $\displaystyle |f'(z)|^2 = \left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial v}{\partial x} \right)^2$ - $\displaystyle J = \begin{bmatrix}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{bmatrix} = \begin{bmatrix}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\-\frac{\partial v}{\partial y} & \frac{\partial v}{\partial y}\end{bmatrix}$ - $\displaystyle |J| = \frac{\partial u}{\partial x} \frac{\partial v}{\partial y} - \frac{\partial u}{\partial y} \frac{\partial v}{\partial x} = |f'(z)|^2$