复变函数的积分课件

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1、 3.1:复变函数的积分复变函数的积分 3.2:柯西柯西-(古萨古萨)积分定理积分定理3.3:复合闭路定理复合闭路定理3.4:科西积分公式科西积分公式3.5:解析函数的高阶导数解析函数的高阶导数3.6:几个重要的定理几个重要的定理3.7:解析函数与调和函数解析函数与调和函数本章补充新题型本章补充新题型本章小节本章小节本章测试题本章测试题 (1)柯西积分定理柯西积分定理(单、复连通区域单、复连通区域);(4)调和函数的应用调和函数的应用；(2)柯西积分公式柯西积分公式(单、复连通单、复连通,无界区域无界区域);(3)高阶导数公式及其应用高阶导数公式及其应用;LLLLPPQQQPLLLLLL()(

2、,)i(,)f zu x yx ywvLLLab011,kknz zzzz1 (1,2,.,)kkzzknk1()nnkkkSfz1kkkzzz1kkkszz1maxkk ns 0nS()dLf zz01()dlim()nkkLkf zzfz x y 0za 1kz kz k nzb 图 3.1()dLfzz()f z()dLfzz()dLfzzn0(,)kk()f z,u v()d(,)d(,)d i (,)d(,)d LLLf z zu x y xx y yx y x u x y yvv()dLf zz()df zz(i)(did)uxyv()dddi(dd)f zzu xyxu yvv(

3、)fz()dLf zz()dCfzz根根据据复复变变函函数数积积分分和和曲曲线线积积分分之之间间的的关关系系以以及及曲曲线线积积分分的的性性质质，不不难难验验证证复复变变函函数数积积分分具具有有下下列列性性质质【3】，它它们们与与实实变变函函数数中中定定积积分分的的性性质质相相类类似似：()f zLL1L2L12()d()d()dLLLf z zf z zf z zk()d()dLLkf zzkf zz1212()()d()d()dLLLf zf z zf z zf z z()d()dLLf zzf zz LL()d()d()dLLLf z zf zzf zSdS22d(d)(d)Sxy111

4、()()()nnnkkkkkkkkkfzfzfS,kkzSL1kkzz()d()d()dLLLf zzf zzf zSLLL()f z()f z()(0)f zMM()dLf zzMll()f zL()f zM()dddLLLf zSM SMSMl()d()dLLf zzf zS()dLf zzMldCzz3,4,01xt ytt()3 i4,01z tttt 11222001d(3 4i)d(3 4i)d(3 4i)2Cz zt tt td(i)(did)ddiddCCCCz zxyxyx x y yy x x ydCz zC21(3 4i)200ddiddddiddiddddxyxyzxy

5、sxyzyxsyxleenee0d(,)(,)dd (,)d(,)dd(,)(,)dd (,)d(,)dxyxyxyxysu x yx yxyu x yxx yysu x yx yyxx yxu x yyvvvv0P l=eeeeP neeee00()(,)i(,)d(i)(,)d(,)d+i(,)d(,)ddidLLLLLLf z dzu x yx yxyu x yxx yyx yx u x y yssvvvP lP n()dLfzz0dLsP lPLL0dLsP n L D 图 3.2 l G()f z()f zDDDl()d0Lf zz()d0lf zz()(,)i(,)f zu x y

6、x y v()fz()dddidd()d di()d dLLLDDf z zu xyx u yuux yx yxyxyvvvv,uuxyxy vv()d0Lf z z DGl,LlDG()d0lf zz D(,),(,)P x y Q x y(,)d(,)dd dLDQPP x y x Q x y yx yxyDD()f zD0z1z12,C CD0z0z0z1zz1z1z1120()d()d()dzCCzf z zf z zf z z0()dzzf0()()dzzF zf()F z定理定理 3.2.4 如果如果 在单连在单连通域通域 内处处解析，则内处处解析，则 在在D内也解析，并内也解析，

7、并且且()(,)i(,)f zu x yx y v()F zD()()F zf z0()()dzzF zf0000(,)(,)(,)(,)ddiddx yx yxyxyu xyxu yvv00(,)(,)(,)ddx yx yP x yu xyv00(,)(,)(,)ddx yx yQ x yx u yv()(,)i(,)F zP x yQ x y(,)P x y(,)Q x y(),()G zH z()f z()()0G HGHf zf z()()G zH zC2zDD()G z()f z1z110012()d()()()zzzzf z zG zG zG z1idzzz212zz12122i

8、i11d|1(i)122z zz【解法【解法1】在整个复平面上解析，且在整个复平面上解析，且运用复积分的牛顿莱布尼兹公复积分的牛顿莱布尼兹公式式有2tziz1t 1z 1t 12111ii111dd()d()1(1)12222zttz z【解法解法2】换元积分法 令，则当，有，有；当；当，有，有所以所以 2|2d1zz zz2zt122|2|2|2dd1d12ii11212ztttz ztztt【解法【解法1】作积分变换得：作积分变换得：?因而积分与路径无关，可用分部积分法得因而积分与路径无关，可用分部积分法得i0sin dzz zsinzzii00ii00sin dd(cos)(cos)(c

9、os)dzz zzzzzz z ii1icosi sinii(cosi isini)iiee【解】【解】由于由于 在复平面内处处解析，在复平面内处处解析，L 1C D 图 3.4 2C kC 不失一般性，取不失一般性，取n1进行证明进行证明.有下述定理：有下述定理：定理定理3.3.2 设设 L和和 为复连通区域内的两条为复连通区域内的两条简单闭曲线，如图简单闭曲线，如图3.5所示，所示，在在L内部且彼此内部且彼此不相交，以不相交，以 和和L为边界所围成的闭区域为边界所围成的闭区域 全全含于含于D则对于区域则对于区域D内的解析函数内的解析函数 有有1C1C1C1D()f z 1()d0LCfzz

10、 1()d()dLCf zzf zz L 1C A B 1D D 图图 3.5()d()dCCf zzf zz ()d()dCCf zzf zzC()d()dCCf zzf zz 0z 1D 3D L L a b c d kD 图图 3.6,L L0zL1DLLL()d()d()d()d()d()d()d()d0abcdabbccddaabcdbcdaf z zf zzf zzf zzf zzf zzf zzf zz,bc da()d()d0bcdaf zzf zz()d()d()dabcddcf z zf z zf z z()d()dLLfzzfzz0z0z0d2iLzzzl00dd2iLl

11、zzzzzzd(i)(3)CzzzC2z 1()(i)(3)f zzz2z iz()f z111()3ii3f zzzd111d(i)(3)3ii3CCzzzzzz11113ii3i312i2i03i3iCCdzdzzz()fz0z001()()d2iLf zf zzzz00()()ddLKf zf zzzzzzz0000000()()()dd()()2i()dKKKf zf zf zzzzzzzf zf zf zzzz()f z0z0000()d2i()Lf zzf zzz0000()()()()ddd2KKKf zf zf zf zzzSzzzzR 0z D L R K 图 3.8 i()

12、zf zeiiiii 1d2i2 iizzzzezeez 2()5zf zz2z i2z 222|2i25dd(5)()i1 2i53zzzzzzzzzzizzz【解解】根据柯西积分公式，得到 22|52371()d2i(371)|2i(371)zzf zzzz故得到故得到 ()2i(67)fzz 1 i()|(1i)2i6(1i)7=122i zfzf 0z D R K 图图 3.9 1C L 12120000001()()d2i1()()=dd2i()()dd nnL CCCLCCCf zf zzzzf zf zzzzzzzf zf zzzzzzz (3.4.3)|,()0zf z()f

13、z|,()0zf z0z001()()d2iLf zf zzzz 0z RC R 图 3.10 L RR()f zRCRCR0001()1()()dd2i2iRCLf zf zf zzzzzzzR R12RCRCzMzR0M 000|zzzzRz 000()2 2|d|1|/RCf zRzMMzzRzzR|,()0zf z|()|f z0()d0RCf zzzzR 0()d0RCf zzzz001()()d2iLf zf zzz z0zL0zLL()f zL()0f z|z 1222211d()(3)()(3)dd()(3)11 2i|2i|()(3)()(3)11i 2i2i2(2)(2)(

14、4)4Lllz azazza zaza zaIzzzazazazaza zaza zaaaaaa|,()zf z()f z0z001()()d()2iLf zf zzfzzRC0z()f zL0001()1()()dd2i2iRLCf zf zf zzzzzzz()f z0|()()|f zf()f 0000001()|d()|2i1()1()|dd|2i2i1|()()|1|d|2|2|2|1 RRRRCCCCf zzfzzf zfzzzzzzf zfzRzzzRzR0z0|lim0RzRR01()limd()2iRCf zzfzz001()()d()2iLf zf zzfzz|z ()0f

15、 z()0f 001()()d2iLf zf zzzz()f x()010!()()d (1,2,)2i()nnCnf zfzznzzC()f zDD0z0z D 图 3.11 d z C z 1n 0201()()d2i()Cffzz0z0z z z z0000()()()limzf zzf zfzz 001()()d2iCff zz001()()d2iCff zzzz0000()()1()()dd2iCCf zzf zffzzzzz 001()d2i()()Cfzzz220001()()dd2i()()()CCfzfzzzz从而有从而有I220000()d()|d|()()CCz fSzf

16、Izzzzzz0MCC()fMd0z0|zdzz0z2dz002dzzzz 32M LIzd 0z 0I 000200()()1()()limd2i()Czf zzf zff zzz ()()ff0302!()()d2i()Cffzz()f z0zzR2i0001()(Re)d2f zf z()f z0zLi0,(02)z zRe i2i000i00(Re)1()1()d(Re)2i2iReCf zf zf zzd zzz2i001(Re)i2if zd2i0001()(Re)d2f zf z()f z0zzR()f zM()0!()(1,2,)nnn MfznR()010!()()d (1,

17、2,)2i()nnCnf zfzznzz()0110()!()d222nnnnCf znnMn MfzzRRRzz0z 在整个复平面上解析的函数称为在整个复平面上解析的函数称为整函数整函数例如多项例如多项式，式，,coszez及及sinz都是整函数，常数当然也是整函数应用柯西不都是整函数，常数当然也是整函数应用柯西不等式可得到关于整函数的刘维尔定理等式可得到关于整函数的刘维尔定理()f z()f zDDDC()d0Cfzz max|()|,min|Mfz0|()|2nnM Lf z0L10|()|()2nLf zMn10()12nL|()|f zM()fz1n|2d2 i1zzz2n1()1n

18、f zz2 i (1,2,)knkzekn31212123111()()()nnnnaaaazz z z zz zz zz zz zz z12321312111()()()()()()()()()()()nnkkkna z zz zz za z z z zz za z z z zz zz zz z111111dd2 2i20 ikknnkkCCkkkknnkkkkazazzzzzaa注意：注意：到推导中已使用到推导中已使用|211dd2ikzCkkzzzzzz2n kCkz|21dd11knnnzCkzzzzkC11()()nmmmkh zzz1|2111()11dd1()()dd()()kn

19、mmmkkknnnzCkkkzznnCCkkkkzzzzzh zzzzzzz1111 =2i()2i0()nnnkkkkmmm kh zzz1110()nnkkmmmkzz2|2|d|1|zzIz2|zzzi2ze|d|z:|2Cz 02i2,(02)zeid2idi dd|d|i d|2d2()izezzzzz2|2,|4zz|2|22|22d2dii(1)(1)(1)(1)2d i(1)(|)zzzzzzIzzzzzzzzz|2|22dd2ii(1)(4)(1)(4)zzzzzzzz|212i2i4 2id114zzzz4 3例例 3 3.6 6.5 5 求 100|98.51d()zkz

20、Izk 99100100100|98.5|98.51110010011dd()()dd()()zzkkCCkkzzIzkzkzzzkzk 9910010099199111()()dd(99)(100)kkkCCzkzkzzzz 10099119911(99)(100)1198 972 1(1)99!1198198!99!99!99 97!kkkkk ,2211k nnk knkn22222122122|1|1|10021d()dd 2innnknkknknnzzzkknnzzC zzCzzzzC222n22|101d()i2cosd2innnnzzzCzz 222222202(2)!2(2)!cosd222(2)!2(!)nnnnnnCnnnnnn