# 柯西积分定理

## 定理

${\displaystyle \Omega }$复平面${\displaystyle \mathbb {C} }$的一个单连通的开子集${\displaystyle f\;:\;\Omega \;\rightarrow \;\mathbb {C} }$是一个${\displaystyle \Omega }$上的全纯函数。设${\displaystyle \gamma }$${\displaystyle \Omega }$内的一个分段可求长的简单闭曲线（即连续而不自交并且能定义长度的闭合曲线），那么：

${\displaystyle \oint _{\gamma }f(z)\,dz=0.}$[1]:52

### 单连通条件的必要性

${\displaystyle \Omega }$单连通表示${\displaystyle \Omega }$中没有“洞”，例如任何一个开圆盘${\displaystyle D=\{z:|z-z_{0}|都符合条件，这个条件是很重要的，考虑中央有“洞”的圆盘：${\displaystyle D_{h}=\{z:0<|z-z_{0}|<2\))$，在其中取逆时针方向的单位圆路径：

${\displaystyle \gamma (t)=e^{it}\quad t\in \left[0,2\pi \right)}$

${\displaystyle \oint _{\gamma }{\frac {1}{z))\,dz=\int _{0}^{2\pi }{ie^{it} \over e^{it))\,dt=\int _{0}^{2\pi }i\,dt=2\pi i}$

### 等价叙述

${\displaystyle \gamma _{1}(0)=\gamma _{2}(0),\quad \gamma _{1}(1)=\gamma _{2}(1),}$

${\displaystyle \int _{\gamma _{1))f(z)\,dz=\int _{\gamma _{2))f(z)\,dz.}$

### 推广

${\displaystyle \oint _{\gamma }f(z)\,dz=0.}$[1]:59

## 证明

${\displaystyle \oint _{\gamma }f(z)\,dz=\oint _{\gamma }(u+iv)(dx+i\,dy)=\oint _{\gamma }(u\,dx-v\,dy)+i\oint _{\gamma }(v\,dx+u\,dy)}$

${\displaystyle \oint _{\gamma }(u\,dx-v\,dy)=\iint _{D_{\gamma ))\left(-{\frac {\partial v}{\partial x))-{\frac {\partial u}{\partial y))\right)\,dx\,dy\;,\qquad \oint _{\gamma }(v\,dx+u\,dy)=\iint _{D_{\gamma ))\left({\frac {\partial u}{\partial x))-{\frac {\partial v}{\partial y))\right)\,dx\,dy}$

${\displaystyle {\partial u \over \partial x}={\partial v \over \partial y}\;,\qquad {\partial u \over \partial y}=-{\partial v \over \partial x))$

${\displaystyle \iint _{D_{\gamma ))\left(-{\frac {\partial v}{\partial x))-{\frac {\partial u}{\partial y))\right)\,dx\,dy=\iint _{D_{\gamma ))\left({\frac {\partial u}{\partial y))-{\frac {\partial u}{\partial y))\right)\,dx\,dy=0}$
${\displaystyle \iint _{D_{\gamma ))\left({\frac {\partial u}{\partial x))-{\frac {\partial v}{\partial y))\right)\,dx\,dy=\iint _{D_{\gamma ))\left({\frac {\partial u}{\partial x))-{\frac {\partial u}{\partial x))\right)\,dx\,dy=0}$

${\displaystyle \oint _{\gamma }f(z)\,dz=0}$[2]:420-421

## 推论

${\displaystyle \forall b\in \Omega ,\;\;F(b)=\int _{\gamma _{a}^{b))f(z)\,dz=\int _{a}^{b}f(z)\,dz}$

## 参考来源

### 脚注

1. 郑建华. 《复变函数》. 清华大学出版社. 2005. ISBN 9787302096931.
2. George B. Arfken, Hans J. Weber. Mathematical Methods for Physicists. Elsevier Academic Press（第6版）. 2005. ISBN 0-12-088584-0 （英语）.

### 参考文献

• Kaplan, W. "Integrals of Analytic Functions. Cauchy Integral Theorem." §9.8 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 594-598, 1991.
• Knopp, K. "Cauchy's Integral Theorem." Ch. 4 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 47-60, 1996.
• Krantz, S. G. "The Cauchy Integral Theorem and Formula." §2.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 26-29, 1999.
• Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 363-367, 1953.
• Woods, F. S. "Integral of a Complex Function." §145 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 351-352, 1926.