柯西积分定理

Cauchy-Goursat Fundamental Theorem 柯西—古萨基本定理

柯西积分定理实际给出积分与路径无关的充分条件

设函数 $f (z)$ 在单连通区域 $D$ 内解析，闭区域 $\overset{―}{D} = D + C$ 上连续，则在 $D$ 内沿任意一条简单闭曲线 $C$ 的积分为零：

\begin{array}{r} \int_{C} f (z) d z = 0 \end{array}

\int_{C} f (z) d z = \int_{C} u (x, y) d x - v (x, y) d y + i \int_{C} v (x, y) d x + u (x, y) d y

闭曲线运用格林公式:

\iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d x d y = \oint_{L} P d x + Q d y

在区域内解析满足 C-R方程：

\begin{aligned} \frac{\partial u}{\partial x} & = \frac{\partial v}{\partial y} \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} \end{aligned}

综上有柯西积分定理：

\begin{aligned} \int_{C} f (z) d z & = \int_{C} u d x - v d y + i \int_{C} v d x + u d y \\ = - \iint_{G} (\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}) d x d y + i \iint_{G} (\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}) d x d y \\ = 0 \end{aligned}