-

, 2014

. ' -

3 9 9 9

.

, 2014 - www.dimoshopoulos.gr

, 1, 94.

z = (+ 3i)(2 i) z = 2i + 6i3i2 z = (2+ 3)+ (6)i .

) z ! Im(z) = 0 6 = 0 = 6 .

) z Re(z) = 0 2+ 3 = 0 2 =3 =

32

.

' , 2, 94.

)

x +y = 3 x y = 1

.

2x = 2 x = 1 . x = 1 , 1+y = 3 y = 2 .

x = 1 , y = 2 .

) ' ,

3x 2 + x 6 0 x ,1 73

6

1+ 736

, +

.

,

3x 2 + x 6 = 2

x 2 3 = 1

3x 2 + x 6 = 4

x 2 = 4

3x 2 + x 10 = 0x = 2 x =2

x =2 x =53

x = 2 x =2

x =2 .

.

1+ 736

, 2

1+ 736

12 >1+ 73 11 > 73 112 > 732

121 > 73 , .

x =2 .

)

3x + 2y = 9 y = 27

.

y = 27 ,

3x + 2 27 = 9 3x = 954 3x =45 x =15 . x =15 , y = 27 .

-

2

' , 3, 95.

1+ i (1,1) .

! "!!

= (1,1) .

1 (1,0) .

! "!!

= (1,0) .

i (0,1) .

! "!!

= (0,1) .

2i (0,2) .

! "!!

= (0,2) .

3 + 4i (3,4) .

! "!!

= (3,4) .

3 4i (3, 4) .

! "!

= (3, 4) .

5 (5,0) .

! "!!

= (5,0) .

0 (0,0) , .

.

' , 4, 95.

z = x +yi (x,y !) , :

) x = 0 , y'y.

) y = 0 , x'x.

) x = y , y = x (- ).

-

3

' , 5, 95.

) (4 + 6i)+ (72i) =4 + 6i + 72i = 3 + 4i .

) (32i)(6 + 4i) = 32i6 4i =36i .

) (3 + 4i)+ (87i)+ (5 + 3i) = 3 + 4i87i + 5 + 3i = 0 = 0 + 0i . . 0, + i.

) (3 + 2i)(4 + 5i) = 12 +15i + 8i +10i2 = 2 + 23i .

) 3i(6 + i) = 18i + 3i2 =3 +18i .

) (4 + 3i)(43i) = 42 + 32 = 25 = 25 + 0i . . 25, + i.

) i(3 + i)(2 i) = i(63i + 2i i2) = i(7 i) = 7i i2 = 1+ 7i .

' , 6, 95.

)

11 i

=1 (1+ i)

(1 i)(1+ i)=

1+ i12 +12

=1+ i

2=

12

+12i .

) i6 = i 41+2 = i2 = 1 = 1+ 0i . . 1, + i.

) i2 + 2i +1 =1+ 2i +1 = 2i = 0 + 2i . . 2i, + i.

) (1+ i 3)2 = 12 + (i 3)2 + 2i 3 = 1+ 3i2 + 2 3i =2 + 2 3i .

)

3 + i2 i

=(3 + i)(2 + i)(2 i)(2 + i)

=6 + 3i + 2i + i2

22 +12=

5 + 5i5

=5(1+ i)

5= 1+ i .

)

6 i 2

1+ i 2=

(6 2i)(1 2i)

(1+ 2i)(1 2i)=

66 2i 2i + 2i2

12 + 22

=47 2i

3=

43

7 23

i .

-

4

' , 7, 95.

) (32i)2(x + iy) = x yi 9 + 4i212ix yi = x yi (52x)12i = 0 . 12 0 , , x ,y ! , .

) ' , x + iy 0 x 0 y 0 .

1+ i1 i

2

+1

x + iy= 1+ i

i(1 i)1 i

2

+1

x +yi= 1+ i i2 +

1x +yi

= 1+ i

1+

1x +yi

= 1+ i1

x +yi= 2 + i x +yi =

12 + i

x +yi =2 i

(2 + i)(2 i)

x +yi =

2 i22 +12

x +yi =2 i

5 x +yi =

25

15i x =

25

, y =15

.

) (32i)(2x iy) = 2(2x iy)+ 2i1 (32i)(2x yi)2(2x yi) =1+ 2i

(2x yi)(32i2) =1+ 2i (2x yi)(12i) =(12i) 2x yi =1 .

2x =1 y = 0

x =

12

, y = 0 .

' , 8, 95.

) : i6 = i 41+2 = i2 = 1 .

i16 = i 43+4 = i 4 = 1 .

i26 = i 46+2 = i2 = 1 .

i36 = i38+4 = i 4 = 1 .

i 46 = i 411+2 = i2 = 1 .

i56 = i 413+4 = i 4 = 1 . i6 + i16 + i26 + i36 + i 46 + i56 =1+11+11+1 = 0 .

) : i11 = i 42+3 = i3 = i .

i 41 = i 410+1 = i1 = i .

i75 = i 418+3 = i3 = i .

-

5

i1023 = i 4255+3 = i3 = i .

1i11

1i 41

+1i75

1i1023

=1i

1i

+1i

1i

=2i

=2ii2

= 2i .

, 9, 95.

) z =5 + 7i z=5 + 7i z = 57i .

) z =49i z =49i z =4 + 9i .

) z = 4i z = 4i z =4i .

) z = 11 z = 11 z = 11 .

) z =i z =i z = i .

) z = 0 z = 0 z = 0 .

' , 10, 96.

z (x ,y) .

z = x yi , (x ,y) , x'x.

z z x'x.

z =x yi , (x ,y) , .

z z .

z =x +yi , (x ,y) , y'y.

z z y'y.

-

6

' , 11, 96.

5 + 9i7 4i

=59i

7 + 4i=

59i7 + 4i

, z2 = z1 , :

z1 + z2 = z1 + z1 = 2Re(z1) ! (z1 + z2) ! .

z1z2 = z1z1 = 2 i Im(z1) (z1z2) .

' , 12, 96.

) z = x +yi (x,y !) . z z = 6i 2 i Im(z) = 6i Im(z) = 3 y = 3 .

y = 3 .

) z 2 = z 2 z 2z 2 = 0 (z z )(z + z ) = 0 z z = 0 z + z = 0 z = z z =z z ! z .

( x'x) ( y'y).

) z 2 =z 2 z 2 + z 2 = 0 2Re(z 2) = 0 Re(z 2) = 0 (1)

z = x +yi (x,y !) , z2 = (x +yi)2 = x 2 +y 2i2 + 2xyi z 2 = (x 2y 2)+ 2xyi .

, (1) x2y 2 = 0 x 2 = y 2 x = y x =y .

y = x (

) y =x (

).

) z = 2z z + z = 2 2Re(z) = 2 Re(z) = 1 . z = x +yi (x,y !) , x = 1 .

x = 1 .

-

7

' , 13, 96.

) = (3)

2 4 1 2 = 98 = 1 > 0 ,

,

x =

3 12

=3 1

2 x =

3 +12

x =

312 x = 2 x = 1 .

) = (2)

2 4 1 3 = 412 =8 < 0 ,

,

x =

2 8i2

=2 2 2i

2=

2(1 2i)2

x = 1+ 2i x = 1 2i .

) ' , x 0 .

x +

1x

= 1 x 2 +1 = x x 2x +1 = 0 , -

= (1)2 4 1 1 = 1 4 =3 < 0 ,

,

x =

1 3i2

x =12

+32

i x =

12

32

i .

' , 14, 96.

, , , , 32i .

z1 = 3 + 2i , z2 = 32i ,

z1

+ z2

=2

z1z

2=

2

2(z

1+ z

2) =

2z1z

2=

, :

z1 + z2 = 3 + 2i + 32i z1 + z2 = 6 .

z1z2 = (3 + 2i)(32i) = 32 + 22 = 9 + 4 z

1z

2= 13 .

2 6 = 2 13 =

= 12 , = 26 .

-

8

' , 1, 96.

' , + i 0 0 0 .

z =

+ i + i

, z =

(+ i)( i)( + i)( i)

=i + ii2

2 + 2

z =

( + )+ ()i2 + 2

z = + 2 + 2

+2 + 2

i .

: z ! Im(z) = 0

2 + 2

= 0 = 0 = .

' , 2, 96.

z =

1 i 32

z =12

32

i .

, z2z = z(z 1) .

z 1 =12

32

i 1 = 12

32

i z 1 = 12

+32

i

.

,

z 2 z = z(z 1) =12

32

i

12

+32

i

=

12

2

+32

2

=

14

+34

z 2z =1

1z 2z

=11

1z 2z

=1 .

' , 3, 96.

: (1+ i)

20 = [(1+ i)2 ]10 = (1+ i2 + 2i)10 = (2i)10 .

(1 i)20 = [(1 i)2 ]10 = (1+ i22i)10 = (2i)10 = (2i)10 .

, (1+ i)20 (1 i)20 = (2i)10 (2i)10 = 0 .

-

9

' , 4, 96.

= i + i , = i +

1i

.

4, = 4 + , !* ,

: 0 1 2 3, :

) = 0 , = i0 +

1i0

= 1+11= 2 .

) = 1 , = i1 +

1i1

= i +1i

= i +ii2

= i i = 0 .

) = 2 , = i2 +

1i2

=11=2 .

) = 3 , = i3 +

1i3

=i1i

=iii2

=i + i = 0 .

.

' , 5, 96.

) z = x +yi (x,y !)

z = z2 x yi = (x +yi)2 x yi = x 2 +y 2i2 + 2xyi x yi = (x 2y 2)+ 2xyi .

x = x 2 y 2 (1)y = 2xy (2)

.

(2) 2xy =y 2xy +y = 0 y(2x +1) = 0 y = 0 2x +1 = 0

y = 0 2x =1 y = 0 x =

12

.

. y = 0 , (1)

x2x = 0 x(x 1) = 0 x = 0 x 1 = 0 x = 0 x = 1 .

, z = 0 z = 1 .

. x =

12

, (1)

12

= 12

2

y 2 y 2 =14

+12 y 2 =

34 y =

34 y =

32

y =

32

.

, z =

12

+32

i z =

12

32

i .

-

10

) z = x +yi (x,y !) ,

z = z3 x yi = (x +yi)3 x yi = x 3 + 3x 2yi + 3xy 2i2 +y 3i3

x yi = (x33xy 2)+ (3x 2yy 3)i .

x 3 3xy 2 = x (1)

3x 2y y 3 = y (2)

.

(1) x33xy 2 = x x 33xy 2x = 0 x(x 23y 21) = 0

x = 0 x23y 21 = 0 x = 0

x 2 = 3y 2 +1 (3)

. x = 0 , (2)

y3 =y y 3y = 0 y(y 21) = 0 y = 0 y

21 = 0 y = 0 y2 = 1

y = 0 y = 1 y =1 .

, z = 0 z = i z =i .

. x2 = 3y 2 +1 , (2)

3y(3y2 +1)y 3 =y 9y 3 + 3yy 3 +y = 0 8y 3 + 4y = 0 y(8y 2 + 4) = 0

y = 0 , 8y2 + 4 > 0 , y ! .

y = 0 , (3) x2 = 1 x = 1 x =1 .

, z = 1 z =1 .

' , 6, 96.

w =

zz

+zz

, w =

zz

+zz

= 2Rezz

! , w .

u =

zz

, w = u +u w = 2Re(u) .

,

2

zz

+zz 22w 22 2Re(u) 21Re(u)1 .

, .

z = x +yi (x,y !) ,

u =

zz

=x +yix yi

=(x +yi)2

(x yi)(x +yi)=

x 2 +y 2i2 + 2xyix 2 +y 2

=x 2y 2 + 2xyi

x 2 +y 2

u =

x 2 y 2

x 2 +y 2+

2xyx 2 +y 2

i .

-

11

, 1

x 2y 2

x 2 +y 21 .

z 0 , x +yi 0 x 0 y 0 x2 +y 2 > 0 .

,

1

x 2y 2

x 2 +y 21(x 2 +y 2) x 2y 2 x 2 +y 2 x 2y 2 x 2y 2 x 2 +y 2

x 2 y 2 x 2 y 2

x 2 y 2 x 2 +y 2

x 2 x 2

y 2 y 2

2x 2 0

2y 2 0

, .

.

' , 7, 96.

z = + i i =i(+ i) i =iz .

(+ i)10 + (i)10 = z10 + (iz)10 = z10 + i10z10 = z10 + i 42+2z10 = z10 + i2z10 = z10 z10 = 0 .

' , 8, 96, 97.

) : z = z z z = 0 2 i Im(z) = 0 Im(z) = 0 z ! .

z =z z + z = 0 2Re(z) = 0 Re(z) = 0 z .

) (), : u , u = u .

v , v =v .

:

u =z

1+ z

2

1+ z1z

2

=

1z

1

+1z

2

1+1z

1

1z

2

=

z2

+ z1

z1z

2

z1z

2+1

z1z

2

=z

1+ z

2

1+ z1z

2

= u u = u .

v =z

1z

2

1+ z1z

2

=

1z

1

1z

2

1+1z

1

1z

2

=

z2z

1

z1z

2

z1z

2+1

z1z

2

=z

2z

1

1+ z1z

2

= z

1z

2

1+ z1z

2

=v v =v .

-

12

' , 9, 97.

) ' , z 0 . z = x +yi (x,y !) ,

1z

=1

x +yi=

x yi(x +yi)(x yi)

=x yix 2 +y 2

=x

x 2 +y 2

yx 2 +y 2

i ,

z +

1z

= x +yi +x

x 2 +y 2

yx 2 +y 2

i = x +x

x 2 +y 2

+ y y

x 2 +y 2

i ,

Re z +

1z

= 5Re(z)

x +

xx 2 +y 2

= 5x 4x =x

x 2 +y 2 4x

xx 2 +y 2

= 0 x 41

x 2 +y 2

= 0

x = 0 4

xx 2 +y 2

= 0 .

. x = 0 - z y'y.

z = 0 , y'y, , z 0 .

. 4

1x 2 +y 2

= 0

1x 2 +y 2

= 4 x 2 +y 2 =14

,

12

.

z = 0 , - .

, z y'y, -

(0,0) , (0,0) 12

.

) Im z +

1z

= 3Im(z)

y

yx 2 +y 2

= 3y 4y y

x 2 +y 2= 0 y 4

1x 2 +y 2

= 0

y = 0 4

1x 2 +y 2

= 0 .

. y = 0 - z x'x.

-

13

z = 0 , x'x, , z 0 .

. 4

1x 2 +y 2

= 0

1x 2 +y 2

= 4 x 2 +y 2 =14

,

12

.

z = 0 , - .

, z x'x, -

(0,0) , (0,0) 12

.

' , 1, 100.

| 1+ i | = 12 +12 = 2 .

| 1 i | = 12 + (1)2 = 2 .

| 3 + 4i | = 32 + 42 = 9 +16 = 25 = 5 .

| 3 4i | = 32 + (4)2 = 9 +16 = 25 = 5 .

|5i | = |5 | = 5 .

|4 | = 4 .

1+ i1 i

=| 1+ i || 1 i |

=2

2= 1 .

| (1 i)2(1+ i)4 | = | 1 i |2 | 1+ i |4= 2

2

24

= 2 22

2

= 2 4 = 8 .

| (2 i)(1+ 2i) | = | 2 i | | 1+ 2i | = 22 + (1)2 12 + 22 = 5 5 = 5 .

3 + i43i

=| 3 + i || 43i |

=32 +12

42 + (3)2=

10

25=

105

.

-

14

' , 2, 100, 101.

| (1+ i)2 | = | 1+ i |2= 12 +12

2

= 2 .

1+ i1 i

2

=1+ i1 i

2

=| 1+ i || 1 i |

2

=| 1+ i |

| 1+ i |

2

=| 1+ i || 1+ i |

2

= 12 = 1 .

.

1+ i1 i

2

=1+ i1 i

2

=i(1 i)1 i

2

= | i |2= 12 = 1 .

1 i1+ i

2

=1 i1+ i

2

=| 1 i || 1+ i |

2

=| 1 i |

| 1 i |

2

=| 1 i || 1 i |

2

= 12 = 1 .

.

1 i1+ i

2

=1 i1+ i

2

=i(1+ i)

1+ i= |i | = |1 | = 1 .

+ ii

2

=+ ii

2

=|+ i ||i |

2

=|+ i |

|+ i |

2

=|+ i ||+ i |

2

= 12 = 1 .

.

+ ii

2

=+ ii

2

=i(i)i

2

= | i |2= 12 = 1 .

' , 3, 101.

) | z 2 | = z 2 | z |2= z 2 zz = z 2 z 2zz = 0 z(z z ) = 0 z = 0 z z = 0 z = 0 z = z z = 0 z ! z ! .

) | z 1 | ! , z ! , z = x , x ! , | x 1 | = x .

| x 1 | 0 , x 0 ,

| x 1 | = x x 1 = x x 1 =x 1 = 0 () 2x = 1 x =

12

.

, z =

12

.

) | z + i | ! , 2z ! , '

(2z ) = 2z 2z = 2z z = z z ! z = x , x ! .

| x + i | = 2x x 2 +12 = 2x x 2 +1 = 2x (1)

-

15

x2 +1 > 0 , x ! , 2x > 0 x > 0 .

(1)

x 2 +12

= (2x)2 x 2 +1 = 4x 2 3x 2 = 1 x 2 =13x>0

x =13

x =1

3 x =

1 3

32 x =

33

.

, z =

33

.

' , 4, 101.

) (0,0) 1 ( ).

) (0,1) 1.

) (1,2) 3.

) - (0,0) 1 2 .

) (0,0) 2, .

' , 5, 101.

) z , (1,0) , (0,2) .

z = x +yi (x,y !) , | z +1 | = | z 2i |

| x +yi +1 | = | x +yi2i | | (x +1)+yi |2= | x + (y2)i |2

(x +1)2 +y 2 = x 2 + (y2)2 x 2 +1+ 2x +y 2 = x 2 +y 2 + 4 4y 1+ 2x = 4 4y

2x + 4y3 = 0 , .

) | z i | = | z +1 | , (0,1) , (1,0) , .

, , |y | = | x | , - , , :y =x . .

-

16

| z i | > | z +1 | () , - z.

' , 6, 101.

| z | = 1 .

| z | =

1+ xix + i

=| 1+ xi || x + i |

=12 + x 2

x 2 +12= 1 | z | = 1 .

' , 7, 101.

| z 4i | = 2 - z (0, 4) 2 (-).

(0,2) - , z1 = 2i - .

(0,6) - , z2 = 6i .

' , 8, 101.

w = 2z +1 w = 2z +1 2z = w1 z =

w12

.

| z | = 1

w12

= 1|w1 |

2= 1 |w1 | = 2 ,

(1,0) 2, w.

' , 9, 101.

| z1 + z2 |2 + | z

1z

2|2= (z

1+ z

2)(z

1+ z

2)+ (z

1z

2)(z

1z

2) =

= z1z1 + z1z2 + z1z2 + z2z2 + z1z1z1z2z1z2 + z2z2 = 2z1z1 + 2z2z2 = 2 | z1 |2 +2 | z

2|2 .

-

17

' , 1, 101.

z = x +yi (x,y !) ,

2 | z | | Re(z) | + | Im(z) | 2 x2 +y 2 | x | + |y | 2(x 2 +y 2) | x | + |y |

2(x2 +y 2)( | x | + |y | )2 2x 2 + 2y 2 | x |2 + |y |2 +2 | x | |y |

2 | x |2 +2 |y |2 | x |2 |y |2 2 | x | |y | 0 | x |2 + |y |2 2 | x | |y | 0

( | x | |y | )2 0 , .

' , 2, 101.

: | z | = 1 w =

z 1z +1

.

: w w =w

z 1z +1

=z 1z +1

(z 1)(z +1) =(z +1)(z 1)

zz + z z 1 =zz + z z +1 zz 1 =zz +1 2zz = 2 | z |2= 1 | z | = 1 .

' , 3, 101.

: w = z +

1z ! z ! | z | = 1 .

: w ! w = w z +

1z

= z +1z zz 2 + z = z 2z + z zz 2 + z z 2z z = 0

zz (z z)(z z) = 0 (z z)(zz 1) = 0 z z = 0 zz 1 = 0

z = z | z |2= 1 z ! | z | = 1 .

' , 4, 102.

: w =

z +iiz +

z .

: w w =w

z ii z +

=z +iiz +

(z i)(iz +) =(z +i)( i z )

izz +z i2z 2i =z + izz 2i +i2z z +z =z z

!*

!*

z + z =z z 2z =2z z =z z .

-

18

' , 5, 102.

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

w, |w | = 1 .

|w | = 1

2z iiz + 2

= 1| 2z i || iz + 2 |

= 1 | 2z i | = | iz + 2 | | 2z i |2= | iz + 2 |2

(2z i)(2z + i) = (iz + 2)(i z + 2) 4zz + 2iz 2i z i2 =i2zz + 2iz 2i z + 4

4 | z |2 +1 = | z |2 +4 3 | z |2= 3 | z |2= 1 | z | = 1 , .

' , 6, 102.

| z | = 1 .

| 2z 1 | = | z 2 |

| 2z 1 |2= | z 2 |2 (2z 1)(2z 1) = (z 2)(z 2) 4zz 2z 2z +1 = zz 2z 2z + 4

4 | z |2 +1 = | z |2 +4 3 | z |2= 3 | z |2= 1 | z | = 1 .

' , 7, 102.

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

= 1+ z + z + zz +1z z + zz = 2 + 2zz = 2 + 2 | z |2= 2 + 2 12 = 4 .

.

, : | 1+ z |2 + | 1z |2= 4 | z +1 |2 + | z 1 |2= 4 (1)

z, 1 1, (1)

()2 + ()2 = 4 (2)

, () = [1(1)]2 + (00)2 () = 2 ()2 = 4 .

(2) ()2 + ()2 = ()2 , -

, , , .

-

19

' , 8, 102.

| z +1 | = | z + 4i |

, (1,0) , (0, 4) .

z = x +yi (x,y !) ,

| x +yi +1 | = | x +yi + 4i | | (x +1)+yi |2= | x + (y + 4)i |2

(x +1)2 +y 2 = x 2 + (y + 4)2 x 2 +1+ 2x +y 2 = x 2 +y 2 +16 + 8y 1+ 2x = 16 + 8y

2x 8y15 = 0 , (

).

() -, (), - , () - .

- () .

, :y = x .

,

=1

28

=1

4=1

=4 .

:y =4x .

:y =4x : 2x 8y15 = 0

-

.

x =

1534

, y =3017

,

1534

, 3017

.

' , 9, 102.

1 (0,0) 4, | z1 | = 4 .

z = + i (, !) , 2 + 2 = 4 2 + 2 = 16 (1)

4z

1

=4

+ i=

4(i)(+ i)(i)

=4(i)2 + 2

=(1) 4(i)

16=i

4=44

i .

z2 = x +yi (x,y !) , z

2= z

1+

4z

1

-

20

x +yi = + i +

44

i x +yi =54

+34

i ,

x =54

y =34

4x = 54y = 3

=4x5

=4y3

.

, (1),

4x5

2

+4y3

2

= 16 16x 2

25+

16y 2

9= 16

x 2

25+

y 2

9= 1 ,

, 2 .

' , 10, 102.

) | z | = 1 | z |2= 1 zz = 1 z =

1z

.

) (),

1z

1

= z1,

1z

2

= z2,...,

1z

= z,

1z

1

+1z

2

+ ...+1z

= | z1+ z

2+ ...+ z

| = | z

1+ z

2+ ...+ z

| = | z

1+ z

2+ ...+ z

| .

, 1, 123.

)

f 1z

=

1z

1

1z

+1

1z

1z

=

1+ zz

1+ z

z

z + zzz

=(z +1)(z 1)

z + z= f (z) .

) f (z) =

(z 1)(z +1)z + z

=zz + z z 1

z + z=

| z |2 +2i Im(z)12Re(z)

.

z = x + yi , f (z) =

(x)2 + (y)2 + 2i y12x

=2x 2 + 2y 2 1+ 2yi

2x

f (z) =

2x 2 + 2y 2 12x

+2y2x

i f (z) =2x 2 + 2y 2 1

2x+yx

i .

Re(f (z)) = 0

2x 2 + 2y 212x

= 0 2x 2 + 2y 21 = 0 2x 2 + 2y 2 = 1 .

-

21

0 , 0 ,

x 2

12

+y 2

12

= 1 , -

, (x ,y) .

, 2, 123.

z = x +yi (x,y !) , w = w1

z zi =

1i x +yi i(x +yi) =

1i x +yixiyi2 =

1i

(x +y)(x y)i =

1i .

x +y =1

(x y) =

x +y =

1

x y =

.

,

(x +y)(x y) =

1 x 2 y 2 = 1 ,

.

, 3, 123.

) z = x +yi (x,y !) ,

x = + 2y = 31

.

= x 2 , y = 3(x 2)1 y = 3x 61 y = 3x 7 ,

, z.

) w = z + (1+ i)

w = + 2 + (31)i +1+ i = (+ 3)+ (31+1)iw = (+ 3)+ 3i .

w = x +yi (x,y !) ,

x = + 3y = 3

.

= x 3 , y = 3(x 3) y = 3x 9 , , -

w.

-

22

) :y = 3x 7 (), .

, z, , - zmin = x +y i .

() .

, - :y = x .

,

=1

3 =1

=

13

.

:y =

13x : x =3y ,

:y = 3x 7 : x =3y

,

x =

2110

, y =710

-

z

min=

2110

710

i .

, 4, 123.

) z = x +yi (x,y !) , | 2z +1 | < | z + i |

| 2(x +yi)+1 | < | x +yi + i | | 2x + 2yi +1 | < | x + (y +1)i |

| (2x +1)+ 2yi |2< | x + (y +1)i |2 (2x +1)2 + 4y 2 < x 2 + (y +1)2

4x2 +1+ 4x + 4y 2 < x 2 +y 2 +1+ 2y 3x 2 + 3y 2 + 4x 2y < 0

x 2 +y 2 +

43

x 23y < 0 (1)

x 2 +y 2 +

43

x 23y = 0

43

2

+ 23

2

4 0 =169

+49

=209

> 0 ,

43

2,

232

23

,13

=

12

209

=12

203

=122 53

=53

.

-

23

, (1) , .

) z = x +yi (x,y !) . | z 1 | 0 , | z 1 | = 1+ Re(z)

1+ Re(z) 0 1+ x 0 x 1 .

| z 1 | = 1+ Re(z)

| x +yi1 | = 1+ x | (x 1)+yi | = 1+ x (x 1)2 +y 2 = 1+ x

(x 1)2 +y 2 = (x +1)2 x 2 +12x +y 2 = x 2 +1+ 2x y 2 = 4x ,

(0,0)

x'x, - ( x 1 ).

, 5, 123.

z = x +y i , x ,y ! A 1, 2, ..., , ,

z1 + z2 + ...+ z = (x1 + x2 + ...+ x)+ (y1 +y2 + ...+y) i .

z1 + z2 + ...+ z = 0 ,

x

1+ x

2+ ...+ x

= 0 y

1+y

2+ ...+y

= 0 (1)

, , z1 ,z2 ,...,z ( , ' , y = x ) ' ,

y >x y (x1

+ x2

+ ...+ x)

y1

+y2

+ ...+y

0

0

, 6, 123.

(1z) = z

| (1z) | = | z | | 1z |= | z | | 1z | = | z | | z 1 | = | z | ,

' , (1,0) , (0,0) .

12

, 0

, -

x =

12 '

z.

, 7, 124.

) f (x) = x 2 + x + , 2 + + = f () 2 + + = f () , f () f () > 0 , , ! .

, , , , , f(x), , x ! , - .

) z1 ,z2 , z2 = z1 .

(z12 + z

1+ )(z

22 + z

2+ ) = (z

12 + z

1+ )(z

12 + z

1+ ) = |z

12 + z

1+ |2 ,

|z12 + z

1+ | > 0 z

12 + z

1+ 0 , -

, z1 .

8 124

1, 124.

i ) u2 + v 2 = 0 (u + vi)(uvi) = 0 u + vi = 0 uvi = 0 u =vi u = vi .

.

i i ) w = 3 + 5i , 35i = w , z = w10 +w10 = 2Re(w10) ! . .

-

25

2, 124.

( ).

3, 124, 125.

) (0,1) (0,1) , x'x ( -

), () ().

) (1,0) (1,0) ,

y'y ( ), () ().

) (1,0) (0,1) , -

y = x ( 1), () ().

) - (1,0) (0,1) ,

y = x ( 2), ()

().

4 125

5 125

6, 125.

z ,z ,z ,z . :

) , -, , 1.

) x'x 45o , , y = x .

(x ,x) z = x + xi .

-

26

) , , , (x ,x) , z =x xi .

) ( ). , y'y, (x ,x) , z =x + xi .

, , x'x, (x ,x) , z = x xi .

x.

| z | = 1 ,

| z

|2= 1 x 2 + x 2 = 1 2x 2 = 1 x 2 =12x>0

x =12

=1

2 x =

22

.

z

=22

+22

i , z

=22

+22

i , z

=22

22

i , z

=22

22

i .

7, 125.

:

) x'x, x'x.

, z, z .

) (0,0) , z .

) (0,0) , z .

) (0,0) 2, | z | = 2 .

12z =

12

| z | =12

2 = 1 ,

12z 1,

' .

12z .

)

1z

=1

| z |=

12

,

1z

-

(0,0) 12

,

1z

.

-

27