第4章2幂级数

1、1 ( ) n fz 1( ), fz 2( ), fz .,(.), n fz 1( ) fz 2( ) fz .( ) n fz ( ) n fz 1n ( ) n sz 1( ) fz 2( ) fz . 0 () n szn 0 ()s z 0 () n szlim n 0 ()s z 0 z 0 z 0 z 0 z ( ) n fz 1n 0 ()s z ( )s z 1( ) fz 2( ) fz . ( )s z( )s z ( ) n fz()n n cza ( ) n fz n n c z .( ) n fz . 2 0n n c 0 c 1( )c za 2 2( )c z

2、a .()n n c za . ()nza 0n n c 0 c 1 c z 2 2 c z . n n c z . 0n n z1 z 2 z . n z . |z1 n zlim n n z 0 0n n z |z1 ( ) n sz1 z 2 z 1 . n z 1z 1 n z n 1z 1 0n n z|z1 1z 1 |z1 0n n z1 z 2 z . n z . 1z 1 3 0n n n c z 0 c 1 c z 2 2 c z . n n c z 0 0z 0 n n c z 0n 0 z 0 z 0 z 0 z o 0 |z 0 z z z 0n n n c z 0

3、n n c z 0n z 0n n n c z 0n ()n n cza a 0 z z za 0n ()n n cza . 0 za 0 z 0 z | | | | | | | | | 4 0n (1)n n cz 1 1 z 0n (1)n n cz 2 2 0 z 3 0n (2)n n cz 1z 11 | | | | 5 or z 0n n n a z 0n n n a z 0n n n a z z z 0n n n a z z r | | z r | | zr | | z r | | 6 o r z 0n n n a z 0 z 0 n n a z 0n n n a z 0 0 n

4、 n a z n n a z 0n 0n n n a z 0n n n a z z 0 z | | | 7 0n n c 0n n c 0n n n c z o1 z z 0n n c 0n n n c z 0n n n c z 0n 0 n n c z 0 z 0n n n c z 0n | n c | z 1 | | 0 z1 | | z 0 z | | | 0n n n c z 0n n n c z 0n z 1 | | n c | | 8 0n n a n a 1n a lim n l 0, r n a n lim n l 0, r 1 l 1 l n z 9 1n n a p n n

5、 z 1 p n 1n n a a lim n lim n 1 r 1n n n 2 ( !)n n z n a n n 2 ( !)n 1n n a a 1 (1)nn (1)!n n n !n n n (1)nn (1)n (1)n 1 (1)n n lim n lim n lim n lim n r 1 0 (1)!n !n p n (1) p n 10 n n (1)ni 0n n a (1)ni n z lim n 2() n n 2 lim n lim n r 2 2 1n i n e n z n a i n e cossini nn 1 n a 1 n lim n r 1 2 1

6、 11 0n n n a z 0 a 1 a z 2 2 a z . n n a z . ( )f z ( )f z 1 | | zr 0n n n b z 0 b 1 bz 2 2 b z . n n b z . ( )g z ( )g z2 | | zr | | z 12 ,r r() nn ab n z 0n 0n n n a z 0n n n b z ( )f z( )g z 0 (a 1 a z 2 2 a z . n n a z ). 0 (b 1 bz 2 2 b z . n n b z ). 0 a 0 b ()z 0 1 a b 1 0 a b 2 ()z 0 2 a b 1

7、 1 a b 2 0 a b . () n z 0 n a b 11n a b 1 1 . n ab 0n a b . min 12 d z a z 0n ()n n aza 0 a 1( )a za 2 2( )a za ( )f z ( )f z |zar .()n n aza . ( )f z|zar ( )f z ( )f z 1 a 2 2 ()a za 1 .()n n naza . |zar 1n n na ( )f z ( )f z 0( )aza 2 1 1 () 2 a za 1 1 .() 1 n n aza n . 0n 1 1 n a n 1 ()nza 1 ()nz

8、a |zar 13 0n n n a z 0 a 1 a z 2 2 a z . n n a z . ( )f z ( )f z| | zww www f( )g z 0 a 1 a 2 2 a . n n a | . ( )g z ( )g z( )g z ( )g z 0n n z1 z 2 z . n z . 1 1z | | z w ww w w1 1z 1 . | | z w | | z 1 2z 1 2 1 2 | 2 z 0n 0n 1 2 n 2 z r r 1 1 z 2 z .( 1) nn z 2 1 1 2 z 1 () n( 1) n n z 14 0n n z 2 1. n zzz . 1 1z | | 1z w ww www 1 z 0n n c 0 ()nzz 1 z 1 1w w 0 ()g zz 1 z 1 0 z 0 1 z 0n 0 ()nzz 0 1 z 0n 0 1 ()nz 1 0 1 ()nz 0 ()nzz 0 |zz 0 |z 1 0 z 0 ()zz 0 1 1 z 0 ()zz 15 16 lim n 1n n a a 0n n a, n z 1 n a n n na , n z n z 0n 0n lim n 1n n a a 1n n a