Complex Tests

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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 )

    !!!