E ‘‌‘›—¨— £¤‘...

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Transcript of E ‘‌‘›—¨— £¤‘...

  • E

  • 1

    z : ( ) ( )4Re 2Re , 1z zz

    + =

    . z . . ( )Re 0z , :

    i. 4w zz

    = + 4 4w .

    ii. 3 4c z i= + + .. , c .

    . 1 2,z z 3z (1) ,

    1 2 2 3 3 1 1 2 32z z z z z z z z z+ + = + +

    2

    3( ) 4( ),zz z z i z z z C+ = + (1) : . z

    . z 1z

    . , R , 1z 21 0

    4z z + + =

    . 0z (1), 2012

    20120 4 3 5 , 55

    z i w iw i

    +=

    ,

    w (0,5) 2 1 = .

    30

    + 10

  • 3

    z C , 1z = 1z a+ = , a R . : . 0 2a

    . 2 2( )2

    aRe z =

    . 2 21 3z z a + =

    . 2133 14

    a z z a +

    4

    ,z w 2 2| | 1,| | 3z zw w zw+ = + = . . | | 2z w+ = . . z w , . . z w . . ,z w .

    5

    2 2(cos ) (5 4sin ) 0, [0, ]z t z t t + = . : . 1 2,z z

    . 1 2| |z z

    . 1 2| |z z+

    6

    ( 1) , [0,1]z t t i t= + . . z . | |z

    2 2( 2) ( 1) ,w k k i k= + + , : . w . | |w . | |z w . | |w | |z w [0, 4]k

  • 7

    . 2 1 0w w+ + = . 1 2,z z

    2 21 1 2 2 0z z z z+ + =

    i. : 1 2z z=

    ii. : 1 2 1 2z z z z+ = =

    iii. *N 1 2 0z z + , 1 2

    1 2

    z zuz z

    =

    + .

    8

    1 2 3, ,z z z , ,A B ,

    : 1 2 32 3z z z+ = 1 3 21, 2z z z= = =

    . 1 2Re( ) 0z z =

    . i. 2 2 2

    1 2 1 2z z z z = +

    ii. OAB .. 2 3Re( )z z 1 3Re( )z z. i. , ,A B .

    ii. A B

    9

    1 2,z a bi z c di= + = + , , ,a b c d 1 2 1z z= = .

    2 1 22 2 0x z z x + = 1 2,x x . :

    . 1 2,x x .

    . 1 2 2x x= =

    . 2 2

    1 2 1 24 8x x z z + =

    . 1 22 1

    x xwx x

    = + .

  • 10

    ( ) ( )6 8 , 0z t t i t = + + + .. 6 8z i . . z . . z . 0t , z 1 3

    . 0t = R , 11

    w zz

    = + ++

    .

    11

    2( )(1 )x y iz

    x yi+ +

    =+

    *x, y R .

    . ( )2 2

    2 2

    22 x y xyRe zx y+ +

    = +

    ( )2 2

    2 22x yIm zx y

    =+

    *x, y .

    . z . . z . . z .

    12

    z 2| 1| | 9 20 |z z z = + | 4 | , 0z = > . . :

    i. 2 2| | 16

    4zz z + + =

    ii. 2 2 2 2(1 ) | | (5 1)( ) 25 1z z z + + = iii. | | 2z

    . z . . z .

    13

    ,z w C 2 22 2(3 4) 5 0, (4 3 ) 5 0z i iz w i iw+ + = + + = .

    . 2 2(4 3 ) 5 ( ) 0z w i i z w+ + + + = .

    . q 1 1qz

    = + .

  • 14

    ,z w C 1zw = .

    , , ,a b c d R 2 2 2 2 0a b c d+ + > ( ) ( ) ( 1) ( 1) 0a z z ib z z c zz d zz+ + + + + = , : . z . z , w . . z , w

    .

    15

    , *a C 1 2,z z 2 2 0z az + + = . :

    . | | | | 1a = = 1| | 2z 2| | 2z .

    . 1 2| | | |z z= , a

    .

    . 2a R , 1

    2

    zz

    , 1 2| | | |z z=

    16

    *22( 2 ),z z Cz

    = (1)

    . 1z 2z (1)

    . v , 1 2 0v vz z+ =

    . x y ,

    2011 162 21 2

    1 1 1( )i ix yi z z

    + = + ++

    . z , 2 41 2z z z z =

    . 0z .

    . 7z i+ .

  • 17

    ( ) ( )z k t k t i = + t R 1k > . . z .

    . w ( 1)y x k= , k 5 2| | 12min

    z w =

    . k () z | |z z . k () | 3 4 |w i min + . u ( 1 ) ( 1 )u m t m t i = + + + , m u . . ,k m () () , | |z u .

    18

    z, w 2 24 | | 2 1,| | 2 3z zw w zw = = . . | 2 | 2z w = . . z w , . . | 6 |z w+ . . z w .

    19

    z ( )| 2 | | 1|

    if zz z

    =

    . z ( )f z . . | ( ) | 1f z . ( )f z i= , :

    i. N | 1| ( ) 1z Re z + =ii. ( )Re z .iii. z , z

    .i.

    20

    . ( 3 )( 4 2 )z i z i+ + +. 2 (1 3 ) 14 2 0z i z i + = (1). 1 2,z z (1) 1Re( ) 0z > , ,A B 1 2,z z 3 3z i= + . AB . M( )z z

    2 2 2( ) 2( ) 2( ) 30MA MB M+ = +

  • 21

    1 2,z z 12

    z zz

    = 1 2, ( ) 0z Im zz

    + = .

    . z . . *N 2 21 2z z

    = .

    . 95 94 ... 1 0z z z+ + + + = . 2z 2 1y x= + , 1z , . . OAB , , ,O A B 1 20, ,z z . . OAB () .

    22

    {2 }z C i 3 8( )

    2z if zz i+

    =

    . 2z i , 2( ) 2 4f z z iz= +

    . (1 )f i+

    . 2( ) 2 4f z z iz= +

    . ( ) 2 6f z iz z= +

    . 1z = , ( )f z 7 =

    23

    z ( ) ( )2005 2008 1 1z z = .. z

    . 2z z= . (1)

    . z ( )Im 0z > , (1) zwz

    +

    =

    R .

    i. w

    ii. , w

  • 24

    1 2,z z 1( ) 0Im z > 1 2 2 1| | | | 40(1)z z z z+ = 1 2 25(2)z z = .

    . 1z 2z .

    . 1w z i= , .

    . *N 1 2z z z = . 1 2,z z z

    .

    25

    2 0, ,z z R + + = 12zi

    = 2z

    . , R 2z

    . v R , 1 2 16v vz z i =

    . z

    2 2

    1 2 16z z z z + = (1)

    . z (1) , 4 4z i

    26

    , 0z w 0zw zw+ =

    . 2010z

    w

    .

    . ,z w .

    . z w z w = + .

    . 2z w iw z+ =

    i. zw

    ( )1,0 .

    ii. 2012z

    w

    .

    27

    z : 2 2 2 21 2 | | | 1| 2 | 1|z z z+ = + + +. z . . 2 2( 1) ( 1) 0z z zz+ + + =

  • . 2 1 0z z+ + = . z . . 2012 2014 2013A z z= + +

    28

    z 2z i 2 4( )

    2zf zz i+

    =

    (1)

    . Im( (1 ))f i+ . z , ( )f z R

    . ( ) 2f z z i= + . z , ( 5 ) ( ) 10f z i f z i + + = (2) . z (2) ,

    . 1z 2z (2) , 1 28 10z z

    29

    1 2z , z 2z z+9=0 , R 1 2z , z R .

    . .

    . ( )17 171 2z z R+ . . 1 2z , z .

    . 1 22 1

    z z 2z z

    + = .

    . =0 ( )1Im 0z > z 1 2z z 4 z z = + .

    30

    . , , : 3 23 4 2 0z z z + =

    . , : 3 24 3 2z z z+ = + 10 32 32 0z z + =

  • 31

    f , [ ],a b , 0a > ( ),a b . : ( ) ( )1 2z a if a z b if b= + = +. 1 2 1 2z z z z+ = , ( )1 ,x a b , ( )1 0f x =. A B , A B , : 1 2 1 2 100Az z Bz z+ = . :

    i. 1 2z z

    ii. ( ) ( )f a f ba b

    =

    iii. ( ),ox a b , ( )( )o

    oo

    f xf x

    x =

    iv. fC

    32

    f : [ , ] R ( ) 0f > > , ( )( )

    ifzif

    +

    =

    . :

    . ( )f x x= ( , ) . 0 ( , )x 0( ) 1f x z .

    . i. f z , ii. .

    .

    Bolzano f [ 3, 3]

    . ( ) 23 ( 2 1)3

    w f z i= + + , w

  • 34

    z x yi= + , 22012 ( ) 1 2012 2012 3 0

    21Im z i

    z+ + =

    :

    . ( , )M x y z . . z ; . z , ( ) 0Im z < .

    35

    f R , ( ) 0f x x R 1

    ( ) 0z

    f x dx = f(1) = 1 .. ( ) 0f x > . z .

    . ( )( )

    3

    2

    3lim

    3xz z x xz z x x+ +

    +

    . f 'x x 0x = 1x = 2z z+ ,

    20

    ( ) 3 6 6x

    f t dt x x= + (0,1)

    36

    [ ], f ( )2z if = + ( )2w if = 0 ( ) ( ) 0f f . w z w z+ <

    ( ) ( ) ( )f f f < < . . ( )0 ,x ( )0 0f x =. ( )1 ,x ( ) ( )1f x f =

    37

    . 1 2,z z 1 2 1 2z z z z+ .

    1 2w z z= .

    . ( )1 1f xz i= + ( )( )2 1z f x i= + +

    () f R ( )0 0f = ( )0 0f . 2 3< <

  • . g ( ) 1 2( )g x Im z z= Rolle [ ], ( )

    ( )( )( )

    11

    f

    f

    fefe

    +=

    +, ( ) 1f

    38

    f : R R (0, 2)A .

    ( ) ( )z f x f x i= + 2( ) ( )w f x f x i= 2( 1)xz e= +

    . f . . z x R . . ( ) Re( )g x z w= .

    39

    :f R R 12

    z C

    2 2( ) 2 ( )f x x xf x+ =

    x 0

    ( ) | 2 |,|

    m2 1|

    lix

    f x zl lx z

    = =

    .

    . : i. | 2 | | 2 1|z z = ii. z .

    . 20( )lim

    x

    f xx x

    .

    . ( ) ( )g x f x x= ( ,0) (0, )+ . f .

    . 3(| 3 4 | 5) 10z i x x+ + = + [ ]1,2

    40

    ( ) ( )1 3 3 3z x x i = + + + , [ )0,2x .. M z ( )C .

    . x z .

    z .

  • . 1 2,x x x z

    1 2,M M z . :f R R

    1 2,M M . f