# 柯西积分定理

$\oint_\gamma f(z)\,dz = 0.$

U单连通的条件，意味着U没有“洞”，例如任何一个开圆盘$U=\{ z: |z-z_{0}| < r\}$都符合条件，这个条件是很重要的，考虑以下路径

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

$\oint_\gamma \frac{1}{z}\,dz = \int_0^{2\pi} { ie^{it} \over e^{it} }\,dt= \int_0^{2\pi}i\,dt = 2\pi i$

$\int_\gamma f(z)\,dz=F(b)-F(a).$

## 参考文献

• Arfken, G. "Cauchy's Integral Theorem." §6.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 365-371, 1985.
• 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.