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math exercises in complex analysis with their solutions

### Transcript of Complex Tests

• 2014-2015

:

1

. 1 2 1 2 1 2, ,z z z z z z = (10 )

. ; (5 )

. () ().

) z w 2 2 0z w+ = 0z w= =

) 0z 0 1z =

) { }, , 0z i = + 1 1 1 iz = +

) (0,0).

) 2 2z z= z .

(10 )

z : ( )1 2 11 2 2 5i z i + = (1)

) 3 4 2z i = (z).

( 8 )

) z . ( 6 )

) 1 2,z z (1)

1 2 4z z = 1 2z z+ . (5 )

) w 2 2w w = +

i) N(w); (3 )

ii) z w ; (3 )

• 2014-2015

:

( ) ( )w z i z i= + + .

) 1z = (5 )

) 2 22 3 3 2 1z z z z + = + (5 )

) 2 32 3

ziv

zi+

=+

. (5 )

) ( )

5

5

z iu

z i+

=+

. (5 )

) b 2b z b z + + (5 )

z 3z = (1)

) 9

w zz

= + (5 )

z w .

(2 )

) 1 2,z z (1)

1 2 3z z = , 1 2z z+ . (6 )

) ( ) ( )2 ,u i = +

(6 )

) z u (6 )

!!!

• 2014-2015

:

1

. 1 2 1 2 1 2, ,z z z z z z = ( . )

. ; ( . )

. ----

) ( ) ( ) ( )1 2 11 2 2 5 1 2 11 2 2 5i z i i z i + = =

( ) ( )11 21 2 2 5 1 2 3 4 2 51 2

ii z i z ii

= + =

( ) ( )5 3 4 2 5 3 4 2z i z i + = + =

(z)

(3,4) =2.

) ( ) 2 23 4 5OK = + =

min ( ) 5 2 3max ( ) 5 2 7

z OKz OK

= = =

= + = + = 3 7z

) 1 2,z z (1)

1 2 4 2z z = = ,

.

1 2 2 2 5 10z z+ = + = = =

) w=x+yi : ( ) ( )2 2 2 2w w x yi x yi = + + = + +

( ) ( )2 22 22 2 .... 0x y x y x + = + + =

N(w) x=0 , yy.

H z w yy.

( )min , 3 2 1z w d K yy = = =

• 2014-2015

:

) ( ) ( ) ( ) ( )w I w w z i z i z i z i = + + =

1 1zz iz iz zz iz iz+ + = + + +

22 2 1 1zz z z= = =

) 2 11 1 1z z z z z

z= = = =

2 2 22 3 2 3 2 3z z z z z z + = + = + =

222

22 2

3 2 11 1 3 2 12 3 3 2 1z zz z

z zz z z z

+ + + = = = +

) 2 32 3

ziv

zi+

=+

2 22 32 31 1 1 2 3 2 3 2 3 2 3

2 3 2 3zizi

v zi zi zi zizi zi

++= = = + = + + = +

+ +

( )( ) ( ) ( )2 3 2 3 2 3 2 3zi zi zi zi+ = + +

4 6zi 6zi+ 9 4 6zz zz zi+ = + 6zi 9 5 5 1zz z+ = = ().

) ( )

( )( ) ( ) ( )

( )( )

5 555 5 5 52

5

5 55 2 5 55

5

1 1

11

izi i izz i i i z i zz zu uizz i i iz i zi i z

izz

+ + = = = = = = = + +

u .

) z b b z =

( ) ( ) 2 2 2b z b z z b b z z b b z z z + + = + + + + = = =

) 2 93 9 9z z zz z

z= = = =

9 2 ,w z z z x z x yiz

= + = + = = +

w

• 2014-2015

:

3 3x 6 2 6 6 6x w

w xx (-6,0) (6,0) .

z w ( ) 3z w z z z z z = + = = = (.)

) 1 23 32 2 3 3

2z z+ = + = = =

( ) ( ) ( ) ( )2 2 2 2 9 2794 4

OK OA AK OK= = =

.

( ) 3 32

OK =

) ( ) ( )2 ,u i = +

, ,u x yi x y= +

( )

22 2

2

2 2

212 4 4

x xx x y

y y y

+= = = + =

= = =

u xx ,

=2 =1 ( 2 2 3a = = )

) ( )min 3 2 1z u A = = = =

( )max 3 2 5z u A = = + = + =

1 5z u

• 2014-2015

:

2

. 1 2 1 2 1 2, ,z z z z z z+ = + (9 )

. 0oz z = > , zo ; (3 )

. 1 2z z z z = , z1 , z2 ; (3 )

. () ().

) 1 2 1 2 1 2, ,z z z z z z C +

) 0 0 0z z w w + = 0 oz z w w= =

) 2 2z z z z I =

) yy.

) *,i i = (-3) 4.

(10 )

z : 2 5 4z z = (1)

) 2 1z = (z).

( 6 )

) 4 5 4 6z i + . ( 6 )

) 12 , 2

2w z z

z=

(5 )

z w

(3 )

) 1 2,z z (1)

2 4 0 ,x x k k + = , 1 2,z z k.

(5 )

• 2014-2015

:

11 32 2

z i= + 2 0 , ,z z + + =

) 1 = = (5 )

) z2 : 2 3 3

1 2 1 2, 1z z z z= = = (6 )

) 20151 20142

1 1 0zz

+ + = (5 )

) w 1 32 1 32 2

w w+ + + = ,

(w) 1

,02

K

3

2 = (5 )

) 2 2

1 2K w z w z= + , w

(4 )

z 2z = (1)

) 8

, 0w z zz

= + (5 )

z w .

(2 )

) 1 2,z z (1)

1 2 2 2z z = , 1 2z z+ . (6 )

) 1 2,w w w

1 2 12w w = , 1 2w w+ . (6 )

) 1 2,u u (u1) (z)

(u2) (z) 1 2 1 24 2u u u u < .

(6 )

!!!