Μιγαδικοί-Λύσεις Σχολικού Βιβλίου
28
Μαθηματικά Γ Λυκείου θετικής-τεχνολογικής κατεύθυνσης Νέα Μουδανιά , Ιούλιος 2014 Μιγαδικοί αριθ μ οί Δη μ ήτρης Αντ . Μοσχόπουλος Καθηγητ'ς Μαθηματικ-ν Πτυχιο3χος Αριστοτελε9ου Πανεπιστημ9ου Θεσσαλον9κης Λύσεις ασκήσεων σχολικού βιβλίου
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Transcript of Μιγαδικοί-Λύσεις Σχολικού Βιβλίου
-
-
, 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
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