第4章5孤立奇点

1、10z( )f z0z0z 0|zz 0 0z( )f z2(1)zz 10,1.sinz1k (k3(sin) z 0,1, 3,. 12, 1 01ze 11ze 0 z Ln12i k 02i 4i 4i 2i k 2i (k( )f z0z0|zz 0 0z( )f z2( )f z0|zz 0 0z( )f z0z( )f z0z( )f zze1 212!z . | | z 414!z z 313!z 1 12! . 0z 13! z2zz1ze 2zze0z 21z 1z 12! 214!z . e1z0z 1 . 1z 2z 12! 30z zsinzsinzz 515!z .

2、 | | z 313!z 717!z 1 213!z 415!z . 3zsinz0z 21z 13! 215!z . sin1z0z 1z 313!z 515!z 2z1sinz0z z 113!z 315!z 4( )f z0|zz 0 0z( )f zzln(1) z 1 213z 314z . 12z 0z 0limz0limz ( )f z0limzz0z( )f z0limz0limz 1 1, z1ze ze0z zln(1) z 11z 5( )f z0|zz 0 0z( )f z2(1)zz 10,z 1z | | z0 1 21z 11z 21z .) 3z .nz (12

3、z 2z 1z z 1 . .nz z 0z 2z10limz? 0z( )f z( )f z0limzz (1) z 6( )f z0|zz 0 0z( )f ze1z a 1 1()za 12! 2()za . za sin1za 1()za 31()3!za 51()5!za za . 70z( )f z( )f z0limzz. e1z0z 0limx e1x 0,0limx e1x , e1z0limz. 81sin1 z 1z e1zz 1z 1e e 11 z e1zz 0z 9( )f z mc . 0c 1c . . 10()zz 0()mzz mc 0z( )f z0()m

4、zz mc 1.c 0c 1c . . ( )g z0z0()g zmc 0z( )f z( )g z0z0()g z0 ( )f z 0, 0z0()mzz ( )g z0()mzz 1c 10()zz 0()zz nc0()nzz 10()mzz 0()mzz 10()mzz nc0()nmzz 0, 100z( )f z( )f z 0()mzz ( )g z( )g z0z0()g z0 1z 5(1)z 2z 51 2(1)z (1)z 3z2z z 1 11z 21z 12(1)z z(1)ze z 1z 1i (21)ik 3z122(1)z z 3z 20i sinz1z k

5、k11nz 2nzei21kn 111e11 z 1ze 1,z z 2i k e11 z 1ze 1limz 1z e11 z 1ze 2limzi k 2zi k 120z( )f z( )g z0z( )f z,mn 0z,mn 0z( )f z0()mzz 1( ) z ( )g z0()nzz 2( ) z 1( ),z 2( ) z 0z10()0,z 20()0z ( )( )f zg z 0()mzz 1( ) z 0()nzz 2( ) z 2( ) z 0()m nzz 1( ) z 2( ) z 1( ) z 1( ) z 0()n mzz 2( ) z ( )g z (

6、 )g z( )f z( )g z( )f z,mn 0z( )g z( )f z0z 3sinzz13( )f z 3(sin) z ( )f z0,z 1, 3, 4,. 1z ( )f z2z ( )f z0,z 2, 3, 4, 5,. ( )f z2(1)z 3(2)z 1z 21z 2z 3(2)z 3(sin) z 2, 140z ( )f z 2z sinz1ze 1.0z 1ze 2z sinz0z ( )f z 4z sinz1 2.e2z0z 21ze 4z sinz15(1)2(1)z (1)ze z,zi 3 , i 5 , i 7 ,.i zi 3 ,zi 5 , i 7 ,.i (2)2z (1)ze 10,z 2, i 4, i 6,.i 0z z 2, i 4, i 6,.i 21z 0 z i 1ze 0 z , i 3 , i 5 , i 7 ,.i 16(4)(5)1nz 2nz10,nz nz ze 2kin k 1 ,ie ( )f z 0,1,2,.,1n ,zk tanz1zk 2zk 2k