# 柯西積分公式

## 定理

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

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

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

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

## 证明

${\displaystyle {1 \over 2\pi i}\oint _{\gamma }{f(z) \over z-a}\,dz+{1 \over 2\pi i}\oint _{C_{0}^{-}}{f(z) \over z-a}\,dz=0.}$

${\displaystyle {1 \over 2\pi i}\oint _{\gamma }{f(z) \over z-a}\,dz={1 \over 2\pi i}\oint _{C_{0}^{+}}{f(z) \over z-a}\,dz.}$

{\displaystyle {\begin{aligned}\left|\oint _{C_{0}^{+}}{f(z) \over z-a}\,dz-2\pi if(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}ri\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{aligned}}}

${\displaystyle {1 \over 2\pi i}\oint _{\gamma }{f(z) \over z-a}\,dz={1 \over 2\pi i}\oint _{C_{0}^{+}}{f(z) \over z-a}\,dz={1 \over 2\pi i}2\pi if(a)=f(a).\qquad \Box }$[2]:168-169

## 例子

${\displaystyle g(z)={\frac {z^{2}}{(z-z_{1})(z-z_{2})}}.}$

${\displaystyle g}$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：${\displaystyle C_{1}}$${\displaystyle C_{2}}$，应用柯西积分定理可知，所要求的积分

${\displaystyle \oint _{C}g(z)\,dz=\oint _{C_{1}}g(z)\,dz+\oint _{C_{2}}g(z)\,dz.}$

${\displaystyle \oint _{C_{2}}g(z)\,dz=\oint _{C_{2}}{f_{1}(z) \over z-z_{2}}\,dz=2\pi if_{1}(z_{2}).}$

${\displaystyle \oint _{C_{1}}g(z)\,dz=\oint _{C_{1}}{f_{2}(z) \over z-z_{1}}\,dz=2\pi if_{2}(z_{1}).}$

{\displaystyle {\begin{aligned}\oint _{C}g(z)\,dz&=\oint _{C_{2}}g(z)\,dz+\oint _{C_{1}}g(z)\,dz.=2\pi if_{1}(z_{2})+2\pi if_{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{aligned}}}

## 参考来源

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