复变函数练习题习题3.3.pdf

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1、习题 3、31.计算下列积分,其中积分闭路取正向、(1)解:(4)解:(6)解:(8)解:被积函数有 6个奇点,只有在圆得内部,于就是函数在闭圆域上解析,则由 Cauchy 积分公式得4、用 Cauchy 积分公式计算函数沿正向圆周得积分值,然后利用圆周得参数方程证明下面积分(1)解:函数得奇点在积分路径得内部,而函数在闭区域上解析,于就是由 Cauchy 积分公式得(2)证明:圆周得参数方程为,在它上有于就是比较等式两边得虚部得又所以7、由下面所给调与函数求解析函数(2)解:对 u 求偏导数有解法 1:由 CauchyRiemann条件得对第一式两边对x 积分得两边对 y 求导,并且与上面所

2、得比较有于就是得即其中c 为任意实常数、从而,即由于代入上式得所以文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T

3、6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU

4、5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T

5、6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU

6、5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T

7、6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU

8、5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3解法 2:由 CauchyRiemann 方程与解析函数得求导公式可得于就是其中 c 为任意实常数、由于代入上式得所以(4)解:对 v 求偏导数有解法 1:由 CauchyRiemann条件得对第二式两边对y 积分得两边对 x 求导,并且与上面所得比较有于就是得即其中c 为任意实常数、从而,即

9、,文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ

10、6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P

11、3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ

12、6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P

13、3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ

14、6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P

15、3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3由于代入上式得所以解法 2:由 CauchyRiemann 方程与解析函数得求导公式可得于就是其中 c 为任意实常数、由于代入上式得所以10、设与在简单闭路C 上及其内部解析,试证:(1)若在 C 上及其内部处处不为零,则有(2)若在 C 上有则在 C 得内部有证明:(1)因为在简单闭路C 上及其内部解析并且处处不为零,则在简单闭路C 上及其内部处处解析,

16、于就是由 Cauchy 积分定理得(2)若对于 C 上得任意一点有由于与在简单闭路 C 上及其内部解析,则对于 C 得内部得任意一点,由 Cauchy 积分公式得文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ

17、6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P

18、3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ

19、6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P

20、3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ

21、6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P

22、3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3所以在 C 得内部有文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W

23、8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6

24、ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W

25、8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6

26、ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W

27、8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6

28、ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3文档编码:CU5W8Q2V9N7 HZ6U7A3G2T6 ZX5W9H3I6P3