复变函数级数

1、nkkwwww210nkknwS0SSnnlim0kkwS0kkw？0Nn npnSSkkba 0kka0kkb 0 ,rrwkkrwwkkk 1rwkkk r1时级数发散；时级数发散；r=1时不一定。时不一定。BAbabakkkkkkk000Aakk0Bbkk0BAbabakllklkkk 0000 00000 ,nnkknknnlklkkllkbababanlkn012 nnknkknnlklkkllkbababanlk ,00n-1010, , 0lkbalk zwzwzwzwnkk210DNn DzzSzSn ,DD0kka Dzazwkk zwkk0D zwk zwk zwzSk z

2、wk zwk lklkkkldzzwdzzwdzzw00 knknnnzzwzzSdddd zwk zwk 0nnnazazSa收敛收敛发散发散RrRazrazRaz0nnnaza0zz Razaz0azrraz0 aRr0zRazaz000nnazaznnnnnnazazMazazazaaza00000nnnazaMazann0aRr0znnnnazrMazazMaza00azrraz0 aRr0z1zz azaz1aR1zLR11limlim11azLazaaazaannnnnnLannnlim1limnnnaaR1nnnz11 , 0nn11limlim1nnaaRnnnn1iennn1

3、1 ,nnn0R121211nnnnaR22zR 002222nnnznnnz212211nnnnaaR Razazazazfnnn00 , !)(nafannaRr0zrc rczfzfdi21z1aRr0zrc1 , 11101aazaazaazaznnn rrcnnncnnnafazaazfzfdi21 di211001 afaaz0 , afaaz1 , ! ,)(nafaaznn Razaznafzfnnn ,!0)( 0nnnazazf1|e|)e ()0(00)()(zzznznf.! 2! 11e02nnnznznzzz0z!)!1(limlim1nnaaRnnnne|e|)e

4、 () 1 (11)()(zzznznf.!1e !1! 21! 111ee02nnnznznzzz1z!)!1(limlim1nnaaRnnnn1 zeeee1z02! ! 2! 11ennnnn00!1e!eennnnznzn0|)(sin ;) 1(|)(sin0)2(0)12(znnznzz.)!12() 1( )!12() 1(! 5! 3! 1sin0121253nnnnnznnzzzzzzzfsin)(1zzfcos)(202242)!2() 1( )!2() 1(! 4! 21cosnnnnnznnzzzz211zzzz11dd)1 (1201102) 1( dd11dd)1

5、(1nnnnnnznnzzzzzz02111nnzzzzzzf 1)(i20e11kzz-1- zfzzzf111 zfzzf211 zfznzfnn111)( 1!1!111!0)(nnfnn 1 ,11110zznnznzfnnnnnzzfln)(i21lnln1kzz 11 ,1111110zzzznnn11 , 1lnlnd11zzzzz11 ,111lnd11lnln111zznzzznnnztz1 dDzzgdzzfzF , ,Dd 1 ,111 ,0zzzzzFnn1az0nnkaza0nnka0收敛收敛发散发散RrRazRrraz ,a收敛收敛发散发散RrRazRaznnkaz

6、aRazraRr Razrazazfnnn ,Razr cnnafadi211aRrc1Rc 12di21di21RRcczfzfzf2Rcazaaza011nnnaazz011nnnazazaRr2R1Rc 110112di21 di21nncnnncnazafazafzfRR cncnafafkRdd11aRr2R1Rc nncnazafzfdi211 !)(di21)(1nafafncnRaz 3211zzzzf110 z 1221112121 1 ,212 , 012 ,321dd!1102 , 0 d1321i21nnnnznncnnzzfnnnzzznna12c 321211213

7、211zzzzzzzf11 ,1210zzznn21 ,123102zzznnn 12020121 121121nnnnnnnnzzzzzf110 z|1z11)(2zzf12022222111 111111)(nnnnzzzzzzzf11)(2zzf1z11211121) 1)(1(1)(zzzzzf21 21) 1(412111412112111210zzzzznnn210 1211211121)(12102zzzzzfnnnnnnnn0z)1(2e)(zzxzf 121!1e ;21!1e0102121nnzxllxzzxnxzlzxxzzf12121ee)( zzxJzxnnmzxnl

8、zxnxzlmmmmmnnmnlnnlnlnlnnlzxxz0 , 2!1 2!1 12!12!1ee02000012121mnl z0z0 邻域邻域ie)(zzzfz22Re)(xzzfzzfsin1)(0sinz10 ,1kkz0z Rzzzzazfnnn000 ,mnan 0 3211zzzzf 110 ,12112zzzfnnn 0!1nnznzf zzf1emnan 0 azRazazfnnn , 5zzf 020 ,!121sinnnnznzzzzf czfzz0lim zfzz0lim 0lim00czfzzmzz zfzz0lim1. z 沿正实轴沿正实轴0 时，时，1/z , 故故 e1/z 2. z 沿负实轴沿负实轴0 时，时，1/z , 故故 e1/z .1! 2111e21zzz3. z 沿虚轴，按沿虚轴，按i/(2n) 0 时，时，e1/z 14. z 按序列)arg2i(|ln1ln1AnAAzn)( ,elim 1/为任意复常数有AAnzn 0nnz Razazazfmnnn ,kkkazf)()(Arg1i000e |zzmmmzzzzkmkkzzazf/0)()