柯西积分定理

1、第二节 柯西积分定理,一、单连通区域的柯西积分定理 二、复函数的牛顿莱布尼兹公式 三、多连通区域上的柯西积分定理,一、单连通区域的柯西积分定理,1. 问题的提出,此时积分与路线无关.,观察上节例2,观察上节例1,由于不满足柯西黎曼方程, 故而在复平面内处处不解析.,由以上讨论可知, 积分是否与路线有关, 可能决定于被积函数的解析性及区域的连通性.,2. 单连通区域的柯西积分定理,定理3.2.1(Cauchy积分定理),证,注：,定理3.2.2(CauchyGoursat积分定理),Goursat,例1,解,根据柯西古尔萨(Cauchy-Goursat)定理, 有,例2,解,根据柯西古尔萨(Ca

2、uchy-Goursat)定理, 有,例3,解,根据柯西古尔萨(Cauchy-Goursat)定理得 及上节例2知,定理3.2.2,定理3.2.3(推广的Cauchy积分定理),定理3.2.4,证,定理3.2.5,二、复函数的牛顿莱布尼兹公式,1. 原函数,定义3.2.1,注：,定理3.2.6,证,推论3.2.1,2. 牛顿莱布尼兹公式,定理3.2.7,证,Newdon,Leibniz,另证,例4,解,由牛顿-莱布尼兹公式知,例5,解,(使用了微积分学中的“凑微分”法),例6,解,此方法使用了微积分中“分部积分法”,三、多连通区域上的柯西积分定理,1. 问题的提出,根据本章第一节例2可知,由此

3、希望将柯西积分定理推广到多连通域中.,2. 多连通区域上的柯西积分定理,定理3.2.8(多连通区域上的柯西积分定理),证,例7,解,根据多连通区域上的柯西积分定理得,例8,证明,根据多连通区域上的柯西积分定理得,例9,解,依题意知,根据多连通区域上的柯西积分定理得,例10,解,Edouard Goursat,Edouard Goursat,1858 - 1936,Edouard Goursat was a French mathematician who is best known for his version of the Cauchy-Goursat theorem stating th

4、at the integral of a function round a simple closed contour is zero if the function is analytic inside the contour.,Sir Isaac Newton,Sir Isaac Newton,1643 - 1727,Isaac Newton was the greatest English mathematician of his generation. He laid the foundation for differential and integral calculus. His

5、work on optics and gravitation make him one of the greatest scientists the world has known.,Gottfried Wilhelm von Leibniz,Gottfried Wilhelm von Leibniz,1646 - 1716,Gottfried Leibniz was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit