# 柯西積分公式

## 定理

$\Omega$复平面$\mathbb{C}$的一个单连通开子集$f \; : \; \; \Omega \; \rightarrow \mathbb{C}$是一个$\Omega$上的全纯函数。设$\gamma$$\Omega$内的一个简单闭合的可求长曲线（即连续而不自交并且能定义长度的闭合曲线），那么函数$f$$\gamma$内部的点$a$上的值是：

$f(a) = {1 \over 2\pi i} \oint_{\gamma} {f(z) \over z-a}\, dz.$

$f^{(n)}(a) = {1 \over 2\pi i} \oint_{\gamma} {f^{(n)}(z) \over z-a}\, dz.$

$f^{(n)}(a) = {n! \over 2\pi i} \oint_{\gamma} {f(z) \over (z-a)^{n+1}}\, dz.$

## 证明

${1 \over 2\pi i} \oint_{\gamma} {f(z) \over z-a}\, dz + {1 \over 2\pi i} \oint_{C_0^-} {f(z) \over z-a} = 0.$

${1 \over 2\pi i} \oint_{\gamma} {f(z) \over z-a}\, dz = {1 \over 2\pi i} \oint_{C_0^+} {f(z) \over z-a} \, dz .$

\begin{align} \left| \oint_{C_0^+} {f(z) \over z-a}\, dz - 2\pi i f(a) \right| &= \left| \oint_{C_0^+} {f(z) - f(a) \over z-a}\, dz \right| \\ &= \left| \int_{0}^{2\pi} {f(a + r\cdot e^{it}) - f(a) \over a+ r\cdot e^{it} - a} r i \cdot e^{it}\, dt \right| \qquad (z = a + r\cdot e^{it}) \\ &= \left| \int_{0}^{2\pi} \left[ f(a + r\cdot e^{it}) - f(a)\right] i \, dt \right| \\ &\leqslant \int_{0}^{2\pi} \left| f(a + r\cdot e^{it}) - f(a)\right| \, dt \\ &\leqslant 2\pi \max_{0 \leqslant t < 2\pi } \left| f(a + r\cdot e^{it}) - f(a)\right| \xrightarrow[r\to 0]{} 0. \end{align}

${1 \over 2\pi i} \oint_{\gamma} {f(z) \over z-a}\, dz = {1 \over 2\pi i} \oint_{C_0^+} {f(z) \over z-a} \, dz = {1 \over 2\pi i} 2\pi i f(a) = f(a). \qquad \Box$[2]:168-169

## 例子

$g(z)=\frac{z^2}{(z-z_1)(z-z_2)}.$

$g$在两个极点附近趋于无穷。在两个极点周围各作一个小圆圈：$C_1$$C_2$，应用柯西积分定理可知，所要求的积分

$\oint_{C} g(z) \, dz = \oint_{C_1} g(z) \, dz + \oint_{C_2} g(z) \, dz.$

$\oint_{C_2} g(z)\, dz = \oint_{C_2} {f_1(z) \over z - z_2} \, dz = 2\pi i f_1(z_2).$

$\oint_{C_1} g(z)\, dz = \oint_{C_1} {f_2(z) \over z - z_1} \, dz = 2\pi i f_2(z_1).$

\begin{align} \oint_{C} g(z) \, dz &= \oint_{C_1} g(z) \, dz + \oint_{C_2} g(z) \, dz. = 2\pi i f_1(z_2) + 2\pi i f_2(z_1) \\ &= 2\pi i \left( \frac{z_2^2}{z_2 - z_1} + \frac{z_1^2}{z_1 - z_2}\right) = 2\pi i \frac{z_1^2 - z_2^2}{z_1 - z_2} = 2\pi i \left( z_1 + z_2 \right) \\ &= -4\pi i \end{align}

## 参考来源

1. ^ Reinhold Remmert. Theory of Complex Functions. Springer (GTM122). 1991. ISBN 9780387971957 （英文）.
2. ^ 2.0 2.1 2.2 S.D. Joglekar. Mathematical Physics: The Basics. Universities Press. 2005. ISBN 9788173714221 （英文）.