8/9/2019 Solution 14

1/3

1

)20243(

14-2010"

1

- .7.12

f - 1 2

;i f - 0- .i-

1n

,{ }\ 0n Z, , 1

lim ( )n

z z

,

. = z 0

f .

2

- f - ,

.

f 1 . z 1 ,

1 11sin sin 1 cos1 sin( 1) sin1 cos( 1)1 1

z z z

z z = + =

,

sin cos (1 ) .cos1

3

1( )( )

g z f z

=.

f z 0 ( ) z 0 g ,0C \{ }- g ( ) z z . z 0

g z ( ) = z 0 .= z 0-

g - , -

( 7 ):6.6 0

( ) nnn

g z a z

== n nr a r .< 0r

1n 0nr

r , r - 0na = 1n , g

. g ( ) z z z 0

, = g z 0 ( ) , z ( )1-( )

g z z

=.

, f .

8/9/2019 Solution 14

2/3

2

4

. 1

nn

na

=

, .> R

- , = z (2 )1

.< z R 1 2

, 3 f :2 ( > z 1 2 , )

> > z 3 2 1 .< z 3 2

, .-3

. ) ) ( 1)( 3) ( ) g z z z f z + = C-

1 ,3 g

. f g :

( )

( ) ( )

1 12

1

2 1 2 1 1

( ) ( 1)( 3) ( 2) ( 2) 2( 2) 3 ( 2)

2 3 ( 2) 2 ( 2)

n nn n

n n

nn n n

n

g z z z a z z z a z

a a a z a a a z

= =

=

= + = + =

= + + + +

g = z 2 ,0-

( )2 1 1( ) 2 ( 2) g z a a a z = + + .

- ,( )2 1 1 1 22 ( 2)( )

( 1)( 3) ( 1)( 3)

a a a z a z a f z z z z z

+ + += =+ +

.

5

,7.3 f = z 0 :

2

11 10

( )( )1 1 1[ ] ( )2 2 2 2

i in nn n n

z r z r

f z f z M a dz dz z re f re d z r r z

++ += == = = =

8/9/2019 Solution 14

3/3

3

6

) )( )( ) ( ) 1 ( ) 1 g z f z f z = . ,

= g z 0 ( ) ( = z 1

, , 0i

z e

,= ). , ( = g z 0 ( ), )8.2 . z 1

0 1 z z 1 -

0( ) 1 f z . - , , f .

7

13( ) ( ) z g z z f .= g z ( ) z <