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פונקציות מרוכבות ־ תקציר הגדרות ומשפטים:2014 באפריל15 1

Complex Analysis

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  • :

    51 4102

    1

  • 1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    1.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.2.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2.2.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    3.2.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    4.2.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    3.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01

    3 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.3 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    4 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    1.4 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    2.4 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    3.4 , : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    5 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    1.5 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

    6 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    1.6 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    2.6 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    7 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    1.7 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    8 )tneruaL(: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    9 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    1.9 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03

    2.9 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    1.01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    2.01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    3.01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    4.01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    1.4.01 : . . . . . . . . . . . . . . . . . . . . . . . 04

    5.01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    11 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    21 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    31 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    41 )meroehT gnippaM nnamieR( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    51 ) ...( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    2

  • 1 :

    1 :

    z.fed

    1.1 : C z = ib +a z ib a = 2.1

    C v ,z :1. ib +a = z 2b + 2a = z z ) " (.

    2. R )z + z(.3. z = )z(.

    4. v + z = v + z.

    5. v z = v z.(zv

    )z =

    v6. 0 =6 v

    7. 0 =6 z 1z = 1z.

    3.1 :

    C z = ib +a :1. z zR = )z( eR = a.

    2. z zI = )z( mI = b.

    4.1 zIi + zR = z :

    z + z = zR = zR2 = )zIi zR( + )zIi + zR( = z + z2

    z z = zI = zI i2 = )zIi zR( )zIi + zR( = z zi2

    5.1 :

    C z z z = 2|z| 2b + 2a = z z = |z|, " z.

    6.1 :

    C v ,z :1. |v| |z| = |v z|.

    .z v

    2. 0 =6 v |v||z| = 3. |z| = |z|.

    4. |z| |zR| |z| |zI|.5. |v| +|z| |v + z| ||v| |z||.

    2 = 2|v z| + 2|v + z|(2|v| + 2|z|

    )6.

    3

  • 1 :

    7.1 :

    C z = iy +x z )y ,x( ] ,[ soc |z| = a nis |z| = b :

    )( si |z| =: )nisi +soc( |z| = z

    z " :

    =

    natra

    (yx

    )0 > x

    natra

    (yx

    )0 y , 0 < x +

    natra

    (yx

    )0 < y , 0 < x +

    pi0 > y , 0 = x 2

    0 < y , 0 = x 2pi0 = y , 0 = x etanimretednI

    8.1 :

    C z z zgrA z ] ,(. z }Z k | k2 + zgrA{ = zgra 0 = k .

    9.1 :

    : )( si = )( si : :

    )2 + 1( si |2z1z| = )2( si |2z| )1( si |1z| = 2z 1z

    N n :

    ))n( si( n|z| = n))( si |z|( = nz

    01.1 :

    R nis i +soc = ie. ie ie = )+(ie " = nz.

    ie |z| = )( si |z| = z. nie n|z| = n)ie |z|(

    11.1 :

    n 1 = nz N n. n n 0=k1n}k{. :

    e = kk pi2i

    si = n

    (k2

    n

    )1 n k 0 :

    4

  • 2 :

    2 :

    1.2 :

    1.2 :

    C E C E : f )z( vi +)z( u = )z( f f ))z( f( R = )z( u f ))z( f( I = )z( v.

    2.2 v ,u R 2R :

    : v ,u{R }E z = yi +x| 2R )y ,x(

    3.2 :

    mil = a 0 > 0 > 0zz

    C E C E : f E 0z )z( f E z < |0z z| < |a )z( f|.

    4.2 :

    mil 0 > 0 > R z

    1. C C : f C c = )z( f)R ,0( B\C z < |c )z( f|.

    mil 0 > M 0 > R )R ,0( B\C zz

    2. C C : f = )z( f M > |)z( f| )M,0( B\C )z( f.

    mil 0 > M 0 > ) ,0z( B z0zz

    3. C C : f = )z( f M > |)z( f| )M,0( B\C )z( f.

    5.2 :

    mil = 0zz

    mil = a " )z( u0zz

    C E C E : f E 0z )z( fmil = i + = a.

    0zz)z( v

    6.2 :

    mil = )0z( f.0zz Ez

    C E C E : f E 0z )z( f

    7.2 ) (:

    C E C E : f E 0z " )z( u , )z( v )0y ,0x( .

    5

  • 2 : 2.2 :

    2.2 :

    1.2.2 :

    8.2 o: C C : f )0z z( o :

    mil0zz

    0z z)z( f 0 =

    C }0z{ \ )r ,z( B : 0 0zz )z( )0z z( )z( = )z( f.

    9.2 R C: R )z ,0z( B : u 0yi + 0x = 0z R B,A :

    )0z z( o + )0y y( B+ )0x x( A+ )0z( u = )z( u

    u :

    u = A

    u = B , )0z( x

    )0z( y

    01.2 C C: C )r ,0z( B : f 0z 0z

    R 2R.

    11.2 :

    C )r ,0z( B : f f 0z :

    mil0zz

    )0z( f )z( ff = 0z z

    C )0z(

    f :

    C )0z(

    f + )0z( f = )z( f

    )0z z( o + )0z z( )0z(

    2.2.2 :

    : C C : f, )z( vi + )z( u = )z( f 0yi + 0x = 0z 0y 0yi +x = z 0x x 0x x = 0z z :

    )0z( f )z( f= 0z z

    ))0y ,0x( vi )0y ,0x( u( ))0y ,x( vi + )0y ,x( u(0x x

    =))0y ,0x( u )0y ,x( u(

    0x x))0y ,0x( v )0y ,x( v( i

    0x xf :

    )0z(

    f

    mil = )0z(0zz

    )0z( f )z( f= 0z z

    u

    xi + )0z(

    v

    x)0z(

    6

  • 2 :2.2 :

    0x yi + 0x = z 0y y )0y y( i = 0z z :)0z( f )z( f

    = 0z z))0y ,0x( v )y ,0x( v( i ))0y ,0x( u )y ,0x( u(

    )0y y( i=

    )y ,0x( v )y ,0x( vi 0y y

    )0y ,0x( u )y ,0x( u0y y

    21.2 :

    f 0z :

    )0z(

    f

    = )0z(u

    xi + )0z(

    v

    x= )0z(

    v

    yui )0z(

    y)0z(

    f :

    u

    x=

    v

    y,

    u

    yv =

    x

    fi :x

    f =y

    if

    xi =

    (u

    xi +

    v

    x

    )i =

    u

    xv x

    R.Ci =

    v

    y+u

    y=

    u

    yi +

    v

    y=

    f

    y

    f.

    f =x

    1 =if y

    fi =y

    31.2 :

    C )r ,0z( B : f C 0z " ))z( f( R = )z( u ))z( f( I = )z( v 0z 0z.

    41.2 :

    R : f 0z : 2R 2

    ted = )0z( fJ(

    ux

    )0z(uy

    )0z(vx

    )0z(vy

    )0z(

    )ted =

    (u

    u )0z( x

    )0z( y

    v

    v )0z( x

    )0z( y

    )

    f :

    )0z(

    ted = )0z( fJ(u

    u )0z( x

    )0z( y

    v

    v )0z( x

    )0z( y

    )ted =

    (u

    v )0z( x

    )0z( x

    u

    u )0z( x

    )0z( x

    )

    =(u

    )0z( x2)

    +(v

    )0z( x2)

    =0 2)0z( f = 2)0z( xvi + )0z( xu

    C )r ,0z( B : f 0 < )0z( fJ 0z f 0z.

    51.2 : R 2R : f = f .

    f22x

    +f22y

    61.2 )(:

    7

  • 2 :2.2 :

    C C : f fR fI .

    3.2.2 :

    R 2R : f y ,x )s ,t( x = x )s ,t( y = y ) y ,x s ,t( 2R :

    (ftfs

    )=

    (xt

    xs

    yt

    ys

    )(

    fxfy

    )

    . :

    :

    (xr

    xr

    y

    y

    )=

    ( nis soc

    soc r nis r)

    : soc r = x nis r = y

    f

    r=

    u

    ri +

    v

    r=

    1

    r

    (v

    ui

    )=

    1

    ri

    (iv

    +u

    )=

    1

    ri

    f

    :

    (xz

    yz

    xz

    yz

    )=

    (12

    1i2

    1i21 2

    )= x i2zz = y

    z+z : 2

    f

    z=

    1

    2

    (u

    xv y

    )+

    i

    2

    (v

    x+u

    y

    )

    f .z

    C C : f " 0 =

    71.2 z z: C C : f :

    f

    z=

    1

    2

    (u

    x+v

    y

    )+

    i

    2

    (v

    xu y

    )f

    z=

    1

    2

    (u

    xv y

    )+

    i

    2

    (v

    x+u

    y

    )

    ( :f

    z+f

    z

    )=

    u

    xi +

    v

    x=

    f

    x

    i

    (f

    zf

    z

    )=

    v

    xi +

    v

    y=

    f

    y

    :

    x=

    (

    z+

    z

    ),

    yi =

    (f

    zf

    z

    )

    f.z

    f =x

    f =

    f z

    f 0 =z

    f =

    f "z

    " 0 =

    8

  • 2 :2.2 :

    81.2 )(:

    C C : f ) v ,u ( 0z z :

    = )z( f)0z( f

    z)0z( f + )0z z(

    z)|0z z|( o + 0z z

    4.2.2 :

    91.2 ) 2(:

    g ,f 0z C , g + f, g f gf ) 0 =6 )0z( g( 0z :

    )g + f(

    f = )0z(

    g + )0z(

    )0z(

    f = )0z( )g f(

    g + )0z( g )0z(

    ()0z( f )0z(f

    g

    )= )0z(

    g )0z( f

    )0z( f )0z( g )0z()0z( 2g

    02.2 :

    C )r ,0z( B : f C ) ,0( B : g )0z( f = 0 f 0z g 0 ))z( f( g = )z( )f g( 0z, 0z :

    g = )0z( )f g(

    f ))0z( f(

    )0z(

    9

  • 2 :3.2 :

    3.2 :

    12.2 )oitnuF ihpromoloH(:

    C , C : f f z.

    22.2 :

    C yi +x = z )y nis i + y soc( xe = ze.

    32.2 ) 2(:

    1. C z 0 =6 ze.2. C z ze = ze.

    3. C z Z k ze = kpi2+ze.4. C =6 0 = ze .

    5. C 2z ,1z 2ze 1ze = 2z+1ze.

    42.2 :

    C z :1. i2ziezie = )z( nis }Z k | k{ )z( soc.

    2. 2zie+zie = )z( soc }Z k | k + 2pi{ )z( nis .= )z( nat . 1+)z2(soc2 = )z( 2ces.

    )z(nis3. )z(soc

    4. )z(nis)z(soc = )z( toc . 1)z2(soc2 = )z( 2s

    5. 2zeze = )z( hnis }Z k | ki{. )z( hsoc.

    )z( hnis.({

    pik + 2

    )6. 2ze+ze = )z( hsoc }Z k | i

    1 .)z(2hsoc

    =(

    2ze+ze

    2)7. )z(hsoc)z(hnis = )z( hnat .

    .(

    2zeze

    2)8. )z(hnis)z(hsoc = )z( htoc .

    52.2 ) 2(:

    C C : f 0 = )z( f z f .

    62.2 ) 2(:

    C )r ,0z( B : f 0z )z( f )r ,0z( B 0z .

    01

  • 3 :

    3 :

    1.3 : R ]b ,a[ C ]b ,a[ : C )t( yi + )t( x = )t( . )t( " " )t( y , )t( x .

    2.3 :

    C ]1b ,1a[ 1, C ]2b ,2a[ : 2 ]2b ,2a[ ]1b ,1a[ : h :1. 2a = )1a( h 2b = )1b( h .

    2. H , " .

    3. ]1b ,1a[ t )t( 1 = ))t( h( 2.

    3.3 :

    C ]b ,a[ : C ]b ,a[ : " )t b +a( = )t( .

    4.3 :

    C ]b ,a[ : : .

    x = )t(

    yi + )t(

    1. ]b ,a[ t )t(2. b = nt < ... < 1t < 0t = a

    ]1+it ,it[ .

    5.3 :

    C ]b ,a[ : )( :

    pus = )(

    {1n0=i

    }b = nt < ... < 1t < 0t = a{ = T| |)it( )1+it( |} M M |)z( f| z )( M zd )z( f

    31

  • 4 :1.4 :

    4.4 :

    zd)z( f

    xam)( zd |)z( f|z

    |)z( f|

    " ":

    xam = |)z( f| : z z

    |)z( f|

    " ":

    tsno = )z( fgra : z

    " f . " f f

    .

    5.4 :

    C ]b ,a[ : C : f :

    = zd)z( f

    b

    a

    ))t( ( f

    td )t(

    6.4 :

    F

    C C : f . C : F F )z( f = )z( z ) F f(. ]b ,a[ : , :

    ))a( ( F ))b( ( F = zd )z( f

    7.4 .

    8.4 " :

    C ]b ,a[ : . C : nf " :

    N n. zd)z( f

    C : f zd)z( nf

    miln

    = zd)z( nf

    zd)z( f

    9.4 5:

    41

  • 4 :2.4 :

    }h )z( I 0 | C z{ =: P. C P : f P

    P

    }h < )z( I < 0 | C z{ =:

    0 > 0 > M P z M |)z( R| < |)z( f|

    .

    . xd )hi +x( f

    xd )x( f

    2.4 :

    01.4 :

    C ]1 ,0[ : / 0z 0z :

    = )0z ,( dni1

    i2

    1

    zd 0z z

    11.4 0z. .

    21.4 2 ,1 2 1 )2 1( mI / 0z :

    )0z ,2( dni + )0z ,1( dni = )0z ,( dni

    31.4 :

    C ]1 ,0[ : / 0z Z )0z ,( dni.

    :

    )( mI\C . B,A A B )A( dni + 1 = )B( dni A B 1+)B( dni = )A( dni. ) 09 (.

    41.4 :

    C , ]b ,a[ : C : f . }b = nt < ... < 0t = a{ = T ]b ,a[ C ]b ,a[ : T ]1+jt, jt[| T )jt( = jz

    )1+jt( = 1+jz. 0 > 0 > T < ) T( :

    zd)z( f

    T

    zd)z( f

    51.4 T| 0 > .

    51

  • 4 :3.4 , :

    61.4 :

    .

    0zz1 ipi21 = )z( f 0z =6 z. C ]1 ,0[ : / 0z C ]1 ,0[ : n ) ( :

    = |)0z ,n( dni )0z ,( dni|

    1

    i21 zd 0z z

    n

    f1

    i21 zd0z z

    0 n n1 < )0z ,( dni n )0z ,n( dni =

    n Z )0z ,n( dni N n Z )0z ,( dni .

    3.4 , :

    71.4 meroehT largetnI yhuaC tasruoG-yhuaC.

    C,B,A . C ]3 ,0[ : C,B,A ]B,A[ = ]1,0[|, ]C,B[ = ]2,1[| ]A,C[ = ]3,2[|.

    ]A,C,B,A[ = ". 81.4 :

    .

    f = 0 = zd )z( f

    91.4 : C E , f E D D C f D.

    02.4 :

    :

    1. : D,C,B,A )DCA( )CBA( = . f :

    0 = zd)z( f

    2. : f )R ,0z( B :

    )R,0z(S

    0 = zd)z( f

    61

  • 4 :3.4 , :

    12.4 ) (:

    f )R ,0z( B )R ,0z( B a :

    = )a( f1

    i2

    )R,0z(S

    )z( f

    zda z

    )R ,0z( S .

    22.4 C C : f. )r ,0z( B )r ,0z( B a :

    = )a( f1

    i2

    )r,0z(S

    )z( f

    zda z

    71

  • 5 :

    5 :

    1.5 :

    . :

    0=n

    C 0=n}nz{ nz

    = NS .

    1N0=n

    1. nz

    = NS .1N0=n

    2. |nz|

    nz. 2.5 0 n

    3.5 :

    " 0 > N 0n 0n n > m :0=n

    nz

    = |)z( nS )z( mS|1mn=k

    kz

    <

    .

    1mn=k

    kz

    1mn=k

    4.5 |nz|

    5.5 :

    0=n})z( nu{ C E, :

    . "

    0=n

    E 0z )0z( nu0=n

    1. )z( nu

    E.

    0=n

    E 0z )z( nu

    . "

    0=n

    E 0z |)0z( nu|0=n

    2. )z( nu

    E.

    0=n

    E 0z |)z( nu|

    = )z( nS )1 n( " E1n0=k

    " E )z( ku

    0=n

    3. )z( nu

    )z( f. 0 > N 0n 0n n < |)z( f )z( nS| E z.

    81

  • 5 :1.5 :

    6.5 ":

    0=n})z( nu{ C E. R 0=n}na{ N 0n 0n n " E.

    0=n

    )z( nu

    0=n

    E z na |)z( nu|. na

    7.5 " :

    0=n})z( nu{ C E, :

    " E E.

    0=n

    1. nu n )z( nu

    " :

    0=n

    2. nu n E )z( nu

    0=n

    = zd)z( nu

    0=n

    zd )z( nu

    1.5 :

    8.5 :

    C na n.0=n

    C 0z 0z n)0z z( na

    9.5 n)0z z( na = )z( nu C n.

    01.5 )lebA( :

    0z 0 > r 0=n}nrna{ . r < 0 0=n

    n)0z z( na

    " ) ,0z( B.

    11.5 :

    + R 0 " ) ,0z( B 0=n

    n)0z z( na

    R < )R ,0z( B / z. )z( S )R ,0z( B.

    21.5 R }dednuob si 0=n}nrna{ | 0 r{ pus = R.

    31.5 :

    1. " .

    2. " )R ,0z( B )R ,0z( B " )R ,0z( B )R ,0z( B.

    91

  • 5 :1.5 :

    41.5 :

    = R.

    (pus miln

    |na|1n

    1)

    0=n

    n)0z z( na

    51.5 :

    .

    0=n

    " 1n)0z z( nan0=n

    n)0z z( na

    61.5 :

    C , C : f , 0z K K 0z. C : F K K w 0z :

    = )0z( F )w( F

    zd)z( f

    F f K.

    71.5 :

    C C : ku . K . : K 0z .

    1. )0z( ku 0=k

    = )z( f " K.

    u 0=k

    2. )z( k

    F.

    " K )z( F )z( f = )z(

    )z( ku 0=k

    81.5 .

    91.5 :

    =: )z( F 0 > R. )R ,0z( B

    0=n n)0z z( nc

    F.

    =: )z( f " )z( f = )z(

    1=n1n)0z z( ncn

    02.5 " )z( F )z( f )R ,0z( B F.

    . )z( f = )z(

    02

  • 6 :

    6 :

    1.6 :

    C , C : f 0z )r ,0z( B )r ,0z( B z :

    0=n

    )z( f = n)0z z( na

    2.6 :

    .

    C )r ,0z( B C )r ,0z( B : g )r ,0z( B 0 = d )( g

    g )r ,0z( B )r ,0z( B z :

    = )z( G

    ]z,0z[

    d )( g

    3.6 " :

    C , C : f " .

    4.6 :

    f 0z, R < r < 0 :

    = na1

    i2

    )r,0z(S

    )z( f

    1+n)0z z(zd

    R R < r < 0 " 0z f ) 0z (. , R R < r

    r .

    5.6 )r ,0z( S , , 0z.

    6.6 :

    = )z( f )r ,0z( B. f :

    0=n

    n)0z z( na

    = )z( )m( f

    m=n

    mn)0z z( na )1 +mn( )1 n( n

    12

  • 6 :

    7.6 f )r ,0z( B r < < 0 :

    = ma)0z( )m( f

    !m=

    1

    i2

    ),0z(S

    )z( f

    1+m)0z z(zd

    ) ( f 0z:

    = )0z( )n( f!n

    i2

    ),0z(S

    )z( f

    1+n)0z z(zd

    ) ,0z( S , , 0z.

    8.6 :

    C C : f f .

    9.6 :

    C C : f , 0z z :

    = )z( f

    0=n

    )0z( )n( f

    !nn)0z z(

    .

    01.6 :

    C : f C : F f. F f .

    11.6 :

    C , C : f )r ,0z( B. C )r ,0z( B : F F, F f .

    )r ,0z( B z )z( f = )z(

    21.6 areroM:

    C C : f . 0z 0zB 0z . f .

    0zB 0 = zd )z( f

    31.6 " :

    C C : nf " K f " )k( f

    )k(C : f. f . k C : n

    .

    22

  • 6 : 1.6 :

    1.6 :

    41.6 :

    C ]b ,a[ : C , C : f :1. f ) ,z(.

    2. C : ) ,z( f .= )z( F F :

    d) ,z( f

    F

    = )z(

    f

    zd) ,z(

    51.6 :

    )( = ) ,z( f , " C / z z

    :

    = )z(

    )(

    d z

    z .

    61.6 :

    C C : f . 1=n}nz{ a N n 0 = )nz( f. 0 = )z( f z.

    71.6 A f A f .

    81.6

    C : g ,f C nz )nz( g = )nz( f n g f .

    91.6 :

    C C : nf ,... ,1f 0 nf 1f . j 0 jf . 02.6 nf 1f .

    : nz 0z , 0 = )nz( nf 1f n. j 0 = )nz( jf n, j 0 = )0( jf 0 = )nz( nf 1f n. N 0n j 0n n 0 = )nz( jf 1=n}n+0nz{

    0z 0 = )nz( jf n 0 jf .

    12.6 8:

    C )1 ,0( B : f )z( f = )z( f f 0 .

    : )z( f = )z( f. R )1 ,0( B z =: A. A z z = z )z( f = )z( f f A )0( f . n

    32

  • 6 :2.6 :

    )0( )n( f . A z, f A :

    f

    mil = )z(n

    f(1 + z

    n

    )z( f )1n

    mil =n

    f(1 + z

    n

    )z( f )1n

    mil =n

    (f(1 + z

    n

    )z( f )1n

    ))z( f =

    f .

    f A )0(

    N n )n( f A )0( )n( f n, .

    22.6 :

    C C : f R )x( f R x )z( f = )z( f z.

    : R x )x( f = )x( f = )x( f ) x = x( )z( f )z( f . , .

    2.6 :

    32.6 :

    C C : f C.

    42.6 ellivuoiL:

    C C : f f .

    52.6 :

    1. f z f ) 3(

    2. f 2z ,1z |2z| = |1z| )2z( f = )1z( f. f ) 3(.3. f 2z ,1z )2z( gra = )1z( gra )2z( f = )1z( f. f ) 3(.

    4. f |)z( f| . f ) 3(.5. f 0 > R R > |z| |ze| C |)z( f| C c zec = f.

    6. f ) ( f ) 7(.

    7. f )i + z( f = )1 + z( f = )z( f. f ) 7(.

    8. f |z| b +a < |)z( f| C z 0 > . f )(.

    62.6 :

    f 0 )m( f f 1 m.

    72.6 ) 7(:

    mil .z

    C C : f . )z( f

    42

  • 7 :

    7 :

    1.7 :

    C I : 1 ,0 , 1 ,0 I I : :1. ) I I(.

    2. I t )t( 0 = )t ,0( I t )t( 1 = )t ,1( . " Is}s{ )t ,s( = )t( s.

    2.7 :

    C I : 1 ,0 ))0( 1 = )0( 0 )1( 1 = )1( 0(. 0 1 I I : I s

    0z = )0 ,s( 1z = )1 ,s( .

    3.7 :

    C , C : f I : 1 ,0 :

    0

    = zd)z( f

    1

    zd)z( f

    1.7 :

    4.7 :

    2R . 5.7 .

    6.7 :

    .

    f 0 = zd)z( f

    7.7 :

    f f .

    8.7 : f . :

    0 = zd )z( f

    =: )z( F

    ]z,0z[

    ud )u( f

    f ) F(.

    . " :

    1. f f f .

    2. f f f .

    3. f f .

    52

  • 7 :1.7 :

    9.7 a a :

    = )a ,( dni1

    i2

    zd

    Z a z

    01.7 :

    f . \ 0z :

    1 = )0z ,( dni )0z( fi2

    )z( f

    zd 0z z

    0 n :

    !n = )0z ,( dni )0z( )n( fi2

    )z( f

    1+n)0z z(zd

    62

  • 8 )tneruaL(:

    8 )tneruaL(:

    1.8 :

    C 0z C Zn}na{ :

    = )z( L

    =n

    n)0z z( na

    :

    = )z( +L0=n

    .... + )0z z( 1a + 0a = n)0z z( na

    = )z( L1

    =n

    1a = n)0z z( na+ 0z z

    2a

    2)0z z(.. +

    L z +L L z :

    )z( L + )z( +L = )z( L

    L L )traP lalpinirP(.

    2.8 :

    = R r 1r

    L +R +L.

    =: )(

    1=n

    na= n

    1=k

    kak

    +R < R 0 L " :

    }+R < |0z z| < R | C z{ = )+R ,R ,0z( A

    +L L " . L z :

    }+R |0z z| R| C z{ = A / z

    :

    pus mil = Rn

    |na|1= +R , n

    (pus miln

    |na|1n

    1)

    3.8 .

    72

  • 8 )tneruaL(:

    4.8 :

    L C 0z. L )+R ,R ,0z( A. +R < r < R :

    = na1

    i2

    )r,0z(S

    )z( L1+n)0z z(

    Z n zd

    , , 0z.

    5.8 :

    2R < 1R 0 )2R ,1R ,0z( A =: A. f A A a :

    = )a( f1

    i2

    )2R,0z(S

    )z( f

    zda z1

    i2

    )1R,0z(S

    )z( f

    zda z

    6.8 :

    2R < 1R 0 f )2R ,1R ,0z( A =: A f . 7.8 2R < 2r < 1r < 1R )2r ,1r ,0z( A a:

    1 = )a( +Li2

    )2r,0z(S

    )z( f

    zda z

    1 = )a( Li2

    )1r,0z(S

    )z( f

    zda z

    8.8 :

    f )2R ,1R ,0z( A L 0z 2R < r < 1R :

    1 = na : Z ni2

    )r,0z(S

    )z( f

    1+n)0z z(zd

    .

    9.8 )r ,0z( S , A.

    01.8 :

    f )r ,0 ,a( A = }a{ \ )r ,a( B = )r ,a( B )r ,a( B z f r < < 0 .

    = )z( L 1

    11.8 n)a z( na =n 0 = R .

    1R

    82

  • 9 :

    9 :

    1.9 :

    f )r ,0z( B . 0z f 1 m 0 = )0z( )k( ff. f 0z = )z( f

    1 m k 0 0 =6 )0z( )m(.)0z()m(f

    0 =6 !m

    m=n)0z()n(f

    n)0z z( !n

    2.9 :

    1. 0z m )r ,0z( B )z( m)0z z( = )z( f = )z( .

    m=n

    )0z()m(f0 =6 ma = )0z( . " mn)0z z( !m

    2. 0 f )r ,0z( B.3. f , 0 > 0 =6 )z( f }0z{ \ ) ,0z( B z

    0z n nz 0z =6 nz 0 = )nz( f n 0 f.

    3.9 f C 0z f 0z :

    = )z( )0z,f(G

    1=n

    n)0z z( na

    }0z{ \C.

    4.9 :

    C 0z f 0 > r f =: )r ,0z( B= )z( f B :

    }0z{ \ )r ,0z( B. " f n)0z z( na =n

    1. 0z )elbavomeR( 0 = na 0 < n ) f B(.

    2. 0z 0 > m 0 =6 ma 0 = na m < n ) f (.3. 0z )laitnessE( ) (

    5.9 , f .

    6.9 :

    C 0z f, :1. 0z f.

    2. C )r ,0z( B : f )z( f = )z( f )r ,0z( B z.B ) 0z f (.

    (,0z

    r2

    )3. f

    xam = )( M.),0z(Bz

    fni mil |)z( f|0

    4. 0 = )( M

    92

  • 9 :1.9 :

    7.9 )(:

    mil " C c.0zz

    0z f 0z " )z( f

    8.9 m:

    C 0z f, :1. 0z 1 m f.

    2. g )r ,0z( B 0 =6 )0z( g m)0zz()z(g = )z( f )r ,0z( B z.

    3. 0 > )z(f1 }0z{ \ ) ,0z( B ) ,0z( B m 0z.

    mil ,0.0zz

    4. )z( f m)0z z(

    9.9 )(:

    C 0z f 0z f C. C )nz( f.

    0z n nz n

    01.9 ) 9(:

    mil .0zz

    0z f 0z f " = |)z( f|

    11.9 )(:

    xam = )f( M. 0z =|0zz|

    C 0z f 0z. }|)z( f|{f " N m :

    pus mil

    = )f( Mm

    1.9 :

    21.9 ' ) (:

    mil .0zz

    )z(f= )z(g

    f

    )z(

    g)z(

    g ,f C 0z 0 = )0z( g = )0z( f

    31.9 '

    .

    41.9 )(:

    fg

    g ,f f 0z, 0 =6 )0z( f g 0z. \ m\ 0z " g1 \ m\ 0z.

    03

  • 9 :2.9 :

    51.9 ) (:

    h ,g 0z h m 0z, :

    = f m 0z.gh

    1. 0 =6 )0z( g 2. 0 = )0z( g 0z n g m < n hg = f n m 0z m n

    0z f.

    61.9 )(:

    g ,f 0z m f n g 0z m+n gf.

    = f n)0zz(2h = g. 1h

    )0zz(: 2h ,1h 0z 0z m

    = gf 2h1h 0z 0z 0z m+n gf.2h1h

    )0zz( m+n

    2.9 :

    71.9 :

    f 0 > R f }R > |z| | C z{. B , f

    (1 ,0

    R

    1 ( f = )z( g }0{ \ )z

    )

    f .(1z

    )

    81.9 ) (:

    f :

    1. " f .

    2. m " f m.

    91.9 ) (:

    f , f " m " f m " f .

    mil :z

    02.9 )z( f

    1. " f " f .

    f (1z

    )mil 0 = z

    0zf(1z

    )2. " f =

    f " f .

    mil .z

    " 72.6 f , )z( f

    13

  • 01 :

    01 :

    1.01 ':

    C ]b ,a[ : ' ) )2t( = )1t( " a = 1t b = 2t(.

    2.01 ' ) (:

    C ' = \C C , C = = . , z ) ( 1 = )z ,( dnI z 0 = )z ,( dni.

    3.01 1 = )z ,( dni .

    4.01 :

    = )z( f f )r ,0z( B.

    f )r ,0z( B n)0z z( na =nseR.0z=z

    f 0z 1a = f

    5.01 :

    1. 0z f 0z f ) 1a(. 0z f f

    0z f 0z .

    2. 0 = 1a ) n)0z z( }1{ \Z n( f " )

    (.

    6.01 0z r < < 0 Z n :

    = na1

    i2

    ),0z(S

    )z( f

    1+n)0z z(zd

    :

    seR

    0z=z= f

    1

    i2

    ),0z(S

    zd )z( f

    23

  • 01 :1.01 :

    1.01 :

    7.01 ) (:

    1. g ,f 0z C b ,a :

    seR

    0z=zseRa = )gb + fa(

    0z=zseRb + f

    0z=zg

    2. 0z 1 m f m)0zz()z(g = )z( f :

    seR

    0z=z= f

    1

    g!)1 m(= )0z( )1m(

    1

    0zzmil !)1 m(1md

    1mzd])z( f m)0z z([

    :

    seR

    0z=zmil = f

    0zz)0z( g = )z( f )0z z(

    = )z( f g 0z :)z(g

    )0zz(3. m

    seR

    0z=z= f

    1

    g!)1 m()0z( )1m(

    h :

    4. h ,g 0z 0 = )0z( h 0 =6 )0z(

    seR

    0z=z

    g (h

    )=

    )0z( g

    )0z( h

    8.01 ) (:

    C C : ,f 0z 1 f 0z :

    seR

    0z=zseR )0z( = ) f(

    0z=zf

    9.01 .

    01.01 ) 11(:

    g 0z g :

    seR

    =z

    )z( g

    )0z( R = 0z z

    R g .

    33

  • 01 : 2.01 :

    2.01 :

    11.01 f 0z f 0z :

    = )z( )0z,f(G

    1=n

    n)0z z( na

    }0z{ \C. 21.01 )meroehT eudiseR(:

    C f mz ,... ,2z ,1z. mz ,... ,1z :

    i2 = zd)z( f

    m1=k

    seR )kz ,( dnikz=z

    f

    ' mz ,... ,1z 1 = )iz ,( dnI i :

    i2 = zd)z( f

    m1=k

    seR

    kz=zf

    31.01 : C ) ( \ A . f A\ :

    i2 = zd )z( fAz

    seR)z ,( dnizf

    / z 0 = )z ,( dni :

    i2 = zdf

    Az

    seR)z ,( dnizf

    41.01 ) 11(:

    )z(p = )z( f . )z( q , )z( p 1 +p ged > q ged )z(q

    R q

    : m = qged. m m 0 >

    B :(R ,0

    )mz ,... ,1z ) (

    f fo ytiralugnis z

    seR

    zf

    = m1=k

    seR

    kz=z

    (p

    q

    = )1

    i2

    )R,0(B

    )z( p

    )z( qzd

    1

    2

    )R,0(B

    )z( q)z( p)R,0(Bzxam R2 21 zd

    )z( q)z( p

    = )z( q " 1. :m

    zkb 0=k= )z( p k

    lzka 0=k

    k

    )z( q/ )z( p

    mlzlmz =

    ()z( p

    )z( q

    )=

    mz(lmz

    ))z( p

    )z( qmz=

    z1la + laz0a +.... + 1

    m

    mz0b +... + 1zmb + mbla z

    mb1 =

    43

  • 01 :3.01 :

    1 lmz

    )z(pz 0 > k. )z(q

    0 z k 0 > A . :

    0 > R R R = |z| lmRA < )z(q)z(pm1=k

    seR

    kz=z

    (p

    q

    )R,0(Bxam R ))z( q)z( p

    0 R 1lmRA = lmRA R < R " .

    3.01 :

    :

    )1 ,0( B C : R )1 ,0( B z. :

    =: I

    pi2

    0

    d ) nis , soc( R

    R. :(1 +z

    z

    , 21 z

    z

    i2

    ) ie = z ]2 ,0[ )1 ,0( B ))( nis , )( soc( R =

    )1,0(B

    1

    ziR

    (1 + z

    z

    2,1 z

    z

    i2

    )= zd

    pi2

    0

    1

    ieiR

    (1 + ie

    ie

    2,1 ie

    ie

    i2

    )d )ie(

    =

    pi2

    0

    1

    iei= diei ))( nis , )( soc( R

    pi2

    0

    d ))( nis , )( soc( R

    :

    = I

    )1,0(B

    1

    ziR

    (1 + z

    z

    2,1 z

    z

    i2

    )zd

    R )1 ,0( B :

    i2 =: I

    )1,0(Bz

    1=seR )z ,( dni

    zi2 = f

    )1,0(Bz

    seR

    zf

    1 = )z( f f )1 ,0( B.ziR(1 +z

    z

    , 21 z

    z

    i2

    )

    : 0 > R }R < |z| 0 )z( I | C z{ = )R ,0( +B R. B = R R. :

    )R ,0( +

    =: R(]R,R[ R = )]R,R[ )R ,0( +B

    R\R = ]R,R[.

    53

  • 01 :3.01 :

    :

    )x( Q , )x( P 2 + Pged Qged ) ( Q

    =: I Q .

    )x( P . xd)x(Q

    P . :Q

    Q,P

    mil = IR

    R

    R

    )z( P

    )z( Qmil = zd

    R

    R

    )z( P

    )z( Q zd

    R

    )z( P

    )z( Qzd

    :

    milR

    R

    )z( P

    )z( Q0 = zd

    R A . : 2 + Pged Qged R R = |z| 2RA < )z(Q)z( P

    R

    )z( P

    )z( Qzd

    0 R RA = 2RA R = 2RA )R( < :

    mil = IR

    R

    )z( P

    )z( Qmil i2 = zd

    R

    Rz

    seR

    z

    (P

    Q

    )

    ( R. PQ

    Q )

    R " :

    i2 = I

    zR

    seR

    z

    (P

    Q

    )

    1.R R

    :

    )x( P = )x( f Qged < Pged Q R, 0 > a : )x(Q

    =: I

    xdxaie )x( f

    63

  • 01 :3.01 :

    51.01 ' ) 11 (:

    C C : f +H :

    milR

    (xam

    R=|z|,+Hz|)z( f|

    )0 =

    R R 0 > a :

    milR

    R

    0 = zdzaie )z( f

    0 < a .

    :

    )z(Q)z( P Qged < Pged R R = |z| 2RA < )z(Q)z( PmilR

    R

    )z( P

    )z( Q0 = zdzaie

    ]R,R[ R = R :

    mil = IR

    R

    i2 = zdzaie )z( f+Hz

    seR

    z

    (zaief

    )

    zaie )z( f +H ) )z( Q (.

    73

  • 01 : 4.01 :

    4.01 :

    61.01 :

    f C f . 71.01 :

    f ' f, :

    f

    )z(

    )z( f) fP fZ( i2 = zd

    fZ ) ( f fP ) (.

    81.01 :

    1. .

    2. f f.

    , f C f

    f 91.01 : zd

    f f f . :

    1

    i2

    f

    )z(

    )z( f= zd

    1

    i2

    f

    1

    z)0 , f( dni = zd

    .

    02.01 )ehuoR(:

    g ,f C '. :

    |)z( f| < |)z( g )z( f|

    z ) 0 =6 )z( g 0 =6 )z( f z "( :

    }ytiilpitlum gnidulni ni g fo sorez{ # = }ytiilpitlum gnidulni ni f fo sorez{ #

    12.01 :

    1. .

    2. f g 0 g f

    = .

    : ' . g ,f = z |)z( g| +|)z( f| < |)z( g )z( f|. f g ) (.

    83

  • 01 :4.01 :

    22.01 ) 31(:

    kzka 0=kn= )z( p 1 n p n ) (.: : " p 1.

    pz = )z( g, :

    )z( )z(p

    = )z( gz1a + + 1nz1na)1 n( + nzn

    = 0a + + 1nz1na + nz1na)1n( +n

    z1a +... +

    1nz

    1na + 1z

    0a ++... +nz

    )z( g 0 > R 21 < |n )z( g| R |z|. n z

    p )R ,0( B :

    = pZ1

    i2

    )R,0(S

    p

    )z(

    )z( p= zd

    1

    i2

    )R,0(S

    )z( g

    z= zd

    1

    i2

    )R,0(S

    n

    z+ zd

    1

    i2

    )R,0(S

    n )z( gz

    zd

    1 + )0 , )R ,0( S( dni ni2

    )R,0(S

    n )z( gz

    +n = zd1

    i2

    )R,0(S

    n )z( gz

    zd

    R :1

    i2

    )R,0(S

    n )z( gz

    zd

    1

    2xam R2

    )R,0(Sz

    zn )z( g21 |n )z( g| )R,0(Szxam R1 R =

    21 |n pZ| " n = pZ, .

    1 2

    zna = )z( f )R ,0( B =: 1 R ) : n

    xam > R(. z :{|na|1 ,1

    1n|ka| 0=k

    }

    = |)z( p )z( f|1n0=k

    zkak

    1n0=k

    = k|z| |ka|1n0=k

    kR|ka|

    1n0=k

    |)z( f| = nR|na| < 1nR|ka|

    :

    n = }0 = nz | )R ,0( B z{ = |}0 = )z( p | )R ,0( B z{| = pZ

    n n C )R ,0( B R ".

    93

  • 01 :4.01 :

    1.4.01 :

    32.01 m ) 31(:

    f 0z 0z m f. U 0z ] U[ f = )z( f m U.

    1 .f

    42.01 m

    : f ) 0z( " 0z . 0z f f . 0 > f ) ,0z( B ) 0z 0 f (. 0z

    f f (. : f ) ,0z( B ) f 0

    =: 1

    2fni

    ),0(Bz0 > |)z( f|

    }0z{ \ ) , )0z( f( B = )z( f m ) ,0z( B. )z( f = )z( . ) ,0z( B z :

    |)z( f| 2 < < || = |)z( )z( f|

    = )z( f 0 = )z( f ) ,0( B " m. }0{ \ ) ,0( B = )z( f m ) ,0( B.

    0z f ) ,0z( B 1.

    52.01 " ) 31(:

    C C : nf " " f f ".

    : f . f f ". 2z =6 1z a = )2z( f = )1z( f. ' a =6 )z( f z 2z ,1z.

    :f

    n

    anf

    f yllamrofinu

    nf aff yllamrofinu

    1 = |}a = )z( nf : z{|i2

    f

    )( n

    da )( nf1 n

    i2

    f

    )(

    |}a = )z( f : z{| = da )( f

    .

    n :

    2 |}a = )z( f : z{| = |}a = )z( nf : z{|

    nf " .

    62.01 ) 21(:

    C C : nf " f . ' f . nf f

    nf .

    04

  • 01 :5.01 :

    5.01 :

    72.01 :

    C C : f f .

    82.01 )elpinirP suludoM mumixaM(:

    f C . |)z( f| z. .

    z z |)z( f| > )z( f 92.01 | f| .

    : + : 0z, 0z | f|. 0 > ) ,0z( B. :

    = |)0z( f|

    1

    i2

    ),0z(S

    )z( f

    zd 0z z

    1

    2

    ),0z(S

    0z z)z( f zd

    =1

    21

    ),0z(S

    zd |)z( f|1

    2

    pi2xam )) ,0z( S(

    ),0z(Szxam = |)z( f|

    ),0z(Sz|)z( f|

    0z | f| :

    xam |)0z( f|),0z(Sz

    |)0z( f| = |)z( f|

    :

    1

    2

    ),0z(S

    |)0z( f| = zd |)z( f|

    " ) ,0z( S z |)0z( f| = |)z( f| ) f ) ,0z( S (. " < r < 0 z |0z z| |)0z( f| = |)z( f| |)z( f| ) ,0z( B. f ) ,0z( B f ) ,0z( B

    .

    03.01 : f 0z :1

    i2

    )r,0z(S

    zd)z( f

    |)z( f| )r,0z(Szxam )( = |)z( f )0z z(| )r ,0z( S. " 2R r 1R |)z( f )0z z(| )2R ,1R ,0z( A

    )z( f )0z z( .

    14

  • 01 :5.01 :

    13.01 ) 21(:

    1. C f . | f| .2. C f . fR fI .

    :

    . 1. f1 f1

    0z | f|. 0z z |)0z( f| |)z( f| .

    )0z(f1 )z(f1 . 0z f1 2. )z(fe, :

    )z(fIie )z(fRe = )z(fIi+)z(fRe = )z(fe

    fR 0z. 0z z e 0z

    = )0z(fRe f)0z( fR )z( fR )z(fRe )0z(fRe. )z(fe

    . )z(fie, :

    )z(fIe )z(fRie = )z(fI+)z(fRie = )z(fie

    0z )z( fI )z(fIe = )z(fie

    fie . )z(fIe )z(fie

    23.01 :

    C f , . |)z( f| ) (.

    33.01 f |)z( f| .

    43.01

    C f , , f . " f .

    53.01 )zrawhS(:

    f )1 ,0( B =: D 0 = )0( f D z 1 |)z( f| :1. D z |z| |)z( f|.

    .

    2. 1 )0( f 1 2 C 1 = || z = )z( f D z.

    24

  • 11 :

    11 :

    C ) 1=n}n{ (.

    1.11 ) (:

    C. }{ C = : C C C }{ V C V 0 > R V }0 > R| R > |z|{.

    C C . C 1=n}nz{ C a ) C

    (. 2.11 :

    b+za = )z( M, \C : M d+zc

    d {c

    d cb a C } C d ,c ,b ,a 0 =6 cb da =

    a = )( M. C C : M cM

    d (c

    ) C =

    a+zcbzd = )z( 1M .

    3.11 C:

    .

    d cb a C d ,c ,b ,a 0 =6 cb da =

    4.11 :

    C 2d ,1d ,2c ,1c ,2b ,1b ,2a ,1a 1d+z1c1b+z1a = )z( 1M 2d+z2c2b+z2a = )z( 2M 2M 1Mb+za = )z( M :

    ( d+zcb ad c

    )=

    (1b 1a1d 1c

    ()2b 2a2d 2c

    )=

    (1b2d + 1a2b 1b2c + 1a2a1d2d + 1c2b 1d2c + 1c2a

    )

    5.11 :

    C C : M M .

    6.11 :

    1. 3 .

    . .

    2.

    .

    3

    ) (.

    7.11 oitaR ssorC :

    C 4z ,3z ,2z ,1z jz =6 iz j =6 i oitaR-ssorC :

    = ]4z ,4z ,2z ,1z[3z1z/4z1z

    3z2z4z2z

    C 4z ,3z ,2z ,1z ]4z ,3z ,2z ,1z[ .

    34

  • 11 :

    8.11 :

    C 3z ,2z ,1z jz =6 iz j =6 i C 3 ,2 ,1 j =6 i j =6 i i = )iz( M 3 ,2 ,1 = i. M )z( M = :

    2/3

    2131

    = ]3z ,2z ,1z ,z[ = ]3 ,2 ,1 ,[ =2zz/3zz

    2z1z3z1z

    9.11 M , :

    ]4z ,3z ,2z ,1z[ = ])4z( M, )3z( M, )2z( M, )1z( M[

    01.11

    .

    .

    11.11 4 :

    C 4z ,3z ,2z ,1z " R ]4z ,3z ,2z ,1z[.

    : = 4z ,3z ,2z ,1z , M 0 = )1z( M, 1 = )2z( M 2 = )3z( M. )4z( M

    3z ,2z ,1z. :

    R ])4z( M,2 ,1 ,0[ = ]4z ,3z ,2z ,1z[

    " R .

    = . M 4z ,3z ,2z 0 ,1, :

    ],0 ,1 , )1z( M[ = ])4z( M, )3z( M, )2z( M, )1z( M[ = ]4z ,3z ,2z ,1z[

    =)1z(M

    /)4z(M01= 1

    )1z( M

    1

    )1z( M =

    R )1z( M 4z ,3z ,2z ,1z .

    21.11 :

    }1 < |z| | z{ =: D M D = ]D[ M za1az = )z( M D a 1 = ||.

    31.11 1S = D 1S .

    41.11 ) 31(:

    }0 > )z( I | C z{ =: H }1 < |z| | z{ =: D, :b+za = )z( M R d ,c ,b ,a 0 > cb da.

    1. M H = ]H[ M d+zc

    az = )z( M 1 = || H a.2. M D = ]H[ M az

    44

  • 11 :

    51.11 :

    H[ M 1+zi+zi = )z( 1M +H = ]D[ 1M. iz = )z( M D = ]+

    1. i+z

    1 D H H D.2. z+1z1 = )z( M :

    }0 > zR : C z{ = ]D[ M

    1+z1+z = )z( 1M :

    D = ]}0 > zR : C z{[ 1M

    1 }0 < )z( R : z{ D D }0 < )z( R : z{.

    M.[Dc]D = ]D[ M D =

    c1 = )z( M

    z3.

    .

    61.11 " :

    f.

    C C : f " 0z 0 =6 )0z(

    71.11 :

    C C : f " ][ f : 1f .

    f 0z

    81.11 f 0z 0 =6 )0z( f ", . "

    .

    91.11 :

    D D : h " )D = ]D[ h( h .

    54

  • 21 :

    21 :

    : C z ) ,( zgrA z :

    }Z k | k2 + zgrA{ = )z( gra

    z = e C z ,. 0 =6 z i +|z| nl = )z( gra . 0 =6 0z )0z( gra 0 0i +|0z| nl = 0. " 0 0. e = )( f " C 0 0 =6 0e = )0( f C 0. L U )0( f = 0z 0 = )0z( L U z z = )z(Le. = ze

    :

    )z( i +|z| nl = )z( L

    z )z( gra )z( U z. 0z 0 = )0z( :

    0i +|0z| nl = 0 = )0z( L

    1.21 z nl: 1 = 0z 0 = 0 z 1 = 0z :

    )z( grAi +|z| nl = )z( L

    ) ( xnl )0>R x( ]0 ,( \C. xnl }0{ \C ) ( 0 < x :

    mil0>)z(Ixz

    i +|x| nl = )z( L

    mil0

  • 21 :

    3.21 nl :

    C ]1 ,0[ : 2 ,1 1 }0{ \C 0z Z k :

    ki2 = )0z( 2gol )0z( 1gol

    4.21

    0 =6 0z 0z = e i +|0z| nl = )0z( gra . ]0 ,( \C 0 1 0z :

    0

    1

    t)0z( grAi +|0z| nl = )0z( L = td

    1 1 z k :

    1

    1

    ttd

    0

    1

    tk i2 = td

    :

    1

    1

    t)k2 + )z( grA( i +|z| nl = ki2 + )z( grAi +|z| nl = td

    =

    tdt

    e 1 0z 0z =

    5.21 nl: C / 0. nl L z = )z(Le z.

    6.21 nl :

    C / 0 0z. 0 0z = 0e L nl

    + 0 = )z( L 0z z.z

    0z

    d

    0 = )0z( L "

    7.21 :

    C C / 0. z f )z(Le = )z( f z. L nl .

    8.21 :

    p = Z q ,p, 2 q 1 = )q ,p( dg. C / 0. :q

    1. f qp z pz = q))z( f( z.z 0 = )0z( f.

    pz = 0q )z( f q

    p2. 0z 0 0

    z .p3. q q

    74

  • 51 ) ...(

    31 :

    K X . ]1 ,0[ C C ]1 ,0[

    xam = )2f ,1f( d]1,0[x

    }|)x( 2f )x( 1f|{

    1.31 : ]1 ,0[ C K :1. K : 0 > M M < |)x( f| ]1 ,0[ x K f.

    2. K : 0 > 0 > ]1 ,0[ 2x ,1x K f < |2x 1x| < |)2x( f )1x( f|.

    2.31 :

    C K . K ) ( K }nf{ }knf{ .

    3.31 " 0 > M N N > k K M > |)z( knf| K z.

    4.31 ) (:

    K . C b =6 a b ,a =6 )z( f K f z K .

    5.31 :

    M ) K KM KM |)z( f| K z M f(. M .

    6.31 " :

    C D : nf ". 1=k}knf{ " ".

    41 )meroehT gnippaM nnamieR(

    1.41 :

    C ( 0z . D : f :1. f , " D.

    f ) (.

    2. 0 = )0z( f 0 > )0z(

    51 ) ...(

    f C f . .

    f

    f zd

    84