复变函数与积分变换复习重点.pdf

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1、名师精编优秀资料复变函数复习重点(一)复数的概念1.复数的概念：zxiy，,x y是实数,Re,Imxzyz.21i.注：一般两个复数不比较大小，但其模（为实数）有大小.2.复数的表示1）模：22zxy；2）幅角：在0z时，矢量与x轴正向的夹角，记为Arg z（多值函数）；主值arg z是位于(,中的幅角。3）arg z与arctanyx之间的关系如下：当0,xargarctanyzx；当0,argarctan0,0,argarctanyyzxxyyzx；4）三角表示：cossinzzi，其中arg z；注：中间一定是“+”号。5）指数表示：izz e，其中arg z。(二)复数的运算1.加减

2、法：若111222,zxiyzxiy，则121212zzxxi yy2.乘除法：1）若111222,zxiyzxiy，则1212122112z zx xy yi x yx y；112211112121221222222222222222xiyxiyzxiyx xy yy xy xizxiyxiyxiyxyxy。2）若121122,iizz ezz e,则名师精编优秀资料121212iz zz z e；121122izzezz3.乘幂与方根1）若(cossin)izziz e，则(cossin)nnninzzninz e。2）若(cossin)izziz e，则122cossin(0,1,21)

3、nnkkzziknnn（有n个相异的值）（三）复变函数1复变函数：wfz，在几何上可以看作把z平面上的一个点集D变到w平面上的一个点集G的映射.2复初等函数1）指数函数：cossinzxeeyiy，在z平面处处可导，处处解析；且zzee。注：ze是以2 i为周期的周期函数。（注意与实函数不同）3）对数函数：ln(arg2)Lnzzizk(0,1,2)k（多值函数）；主值：lnlnargzziz。（单值函数）Lnz的每一个主值分支ln z在除去原点及负实轴的z平面内处处解析，且1lnzz；注：负复数也有对数存在。（与实函数不同）3）乘幂与幂函数：(0)bbLnaaea；(0)bbLnzzez注：

4、在除去原点及负实轴的z平面内处处解析，且1bbzbz。4）三角函数：sincossin,cos,t,22cossinizizizizeeeezzzzgzctgzizzsin,coszz在z平面内解析，且sincos,cossinzzzz注：有界性sin1,cos1zz不再成立；（与实函数不同）文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C

5、7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG

6、4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B

7、8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编

8、码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A

9、10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L

10、8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA

11、6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4名师精编优秀资料4）双曲函数,22zzzzeeeeshzchz；shz奇 函 数，c h z是 偶 函 数。,s h z c h z在z平 面 内 解 析，且,s h zc h zc h zs h z。（四）解析函数的概念1复变函数的导数1）点可导：0fz=000limzfzzfzz；2）区域可导：fz在区域内点点可导。2解析函数的概念1）点解析：fz在0z及其0z的邻域内可导，称fz在0z点解析；2）区域解析：fz在区域内每一点解析，称fz在区域内解析；3）若()f

12、z在0z点不解析，称0z为fz的奇点；3解析函数的运算法则：解析函数的和、差、积、商（除分母为零的点）仍为解析函数；解析函数的复合函数仍为解析函数；（五）函数可导与解析的充要条件1函数可导的充要条件：,fzu x yiv x y在zxiy可导,u x y和,v x y在,x y可 微，且 在,x y处 满 足CD条 件：,uvuvxyyx此时，有uvfzixx。2函数解析的充要条件：,fzu x yiv x y在区域内解析,u x y和,v x y在,x y在D内 可 微，且 满 足CD条 件：文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5

13、A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10

14、L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 H

15、A6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8

16、C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 Z

17、G4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5

18、B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档

19、编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4名师精编优秀资料,uvuvxyyx；此时uvfzixx。注意：若,u x yv x y在区域D具有一阶连续偏导数，则,u x yv x y在区域D内是可微的。因此在使用充要条件证明时，只要能说明,u v具有一阶连续偏导且满足CR条件时，函数()f zuiv一定是可导或

20、解析的。3函数可导与解析的判别方法1）利用定义（题目要求用定义，如第二章习题1）2）利用充要条件（函数以,fzu x yiv x y形式给出，如第二章习题 2）3）利用可导或解析函数的四则运算定理。（函数fz是以z的形式给出，如第二章习题3）（六）复变函数积分的概念与性质1复变函数积分的概念：1limnkkcnkfz dzfz，c是光滑曲线。注：复变函数的积分实际是复平面上的线积分。2复变函数积分的性质1）1ccfz dzfz dz（1c与c的方向相反）；2）,cccfzg z dzfz dzg z dz是常数；3）若曲线c由1c与2c连接而成，则12cccfz dzfz dzfz dz。3复

21、变函数积分的一般计算法文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H

22、5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文

23、档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS

24、5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E1

25、0L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1

26、HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D

27、8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4名师精编优秀资料1）化为线积分：cccfz dzudxvdyivdxudy；（常用于理论证明）2）参数方法：设曲线c：()zz tt，其中对应曲线

28、c的起点，对应曲线c的终点，则()cfz dzf z tz t dt。（七）关于复变函数积分的重要定理与结论1柯西古萨基本定理：设fz在单连域B内解析，c为B内任一闭曲线，则0cfz dz2复合闭路定理：设fz在多连域D内解析，c为D内任意一条简单闭曲线，12,nc cc是c内的简单闭曲线，它们互不包含互不相交，并且以12,nc cc为边界的区域全含于D内，则cfz dz1,knkcfz dz其中c与kc均取正向；0fz dz，其中由c及1(1,2,)ckn所组成的复合闭路。3 闭路变形原理：一个在区域D内的解析函数fz沿闭曲线c的积分，不因c在D内作连续变形而改变它的值，只要在变形过程中c不

29、经过使fz不解析的奇点。4解析函数沿非闭曲线的积分：设fz在单连域B内解析，G z为fz在B内的一个原函数，则212112(,)zzfz dz G zG zz zB说明：解析函数fz沿非闭曲线的积分与积分路径无关，计算时只要求出原函数即可。5。柯西积分公式：设fz在区域D内解析，c为D内任一正向简单 闭 曲 线，c的 内 部 完 全 属 于D，0z为c内 任 意 一 点，则002cfzdzifzzz文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L

30、8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA

31、6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C

32、7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG

33、4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B

34、8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编

35、码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A

36、10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4名师精编优秀资料6高阶导数公式：解析函数fz的导数仍为解析函数，它的n阶导数为0102(1,2)()!nncfzidzfznzzn其中c为fz的解析区域D内围绕0z的任何一条正向简单闭曲线，而且它的内部完全属于D。7重要结论：12,010,0()ncindznza。（c是包含a的任意正向简单闭曲线）8复变函数积分的计算方法1）若fz在 区 域D内 处 处 不 解

37、析，用 一 般 积 分 法cfz dzf z tz t dt2）设fz在区域D内解析，c是D内一条正向简单闭曲线，则由柯西古萨定理，0cfz dzc是D内的一条非闭曲线，12,z z对应曲线c的起点和终点，则有2121zczfz dzfz dzF zF z3）设fz在区域D内不解析曲线c内仅有一个奇点：0001022()!cnncfzdzi fzzzfzidzfzzzn（()f z在c内解析）曲线c内有多于一个奇点：cfz dz1knkcfz dz（ic内只有一个奇点kz）或：12Re(),nkkcfz dzis f z z（留数基本定理）文档编码:CS5A10E10L8S1 HA6B1D8C

38、7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG

39、4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B

40、8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编

41、码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A

42、10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L

43、8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA

44、6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4名师精编优秀资料若被积函数不能表示成1()nofzzz，则须改用第五章留数定理来计算。（八）解析函数与调和函数的关系1调和函数的概念：若二元实函数(,)x y在D内有二阶连续偏导数且满足22220 xy，(,)x y为D内

45、的调和函数。2解析函数与调和函数的关系解析函数fzuiv的实部u与虚部v都是调和函数，并称虚部v为实部u的共轭调和函数。两个调和函数u与v构成的函数()f zuiv不一定是解析函数；但是若,u v如果满足柯西黎曼方程，则uiv一定是解析函数。3 已知解析函数fz的实部或虚部，求解析函数fzuiv的方法。1）偏微分法：若已知实部,uu x y，利用CR条件，得,vvxy；对vuyx两边积分，得uvdyg xx（*）再对（*）式两边对x求偏导，得vudygxxxx（*）由CR条件，uvyx，得uudygxyxx，可求出g x；代入（*）式，可求得虚部uvdyg xx。2）线 积 分 法：若 已 知

46、 实 部,uu x y，利 用CR条 件 可 得vvuudvdxdydxdyxyyx，文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:

47、CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10

48、E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S

49、1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B

50、1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M8 ZG4C4H5B8S4文档编码:CS5A10E10L8S1 HA6B1D8C7M