Post on 28-Dec-2015
description
1
() i2 = 1i2 = 1i2 = 1
1.1 + i, , R :) i(1 2i) + 2(i 1) ) (i2 2i+ 3)(i2 + i 1)) (i2 + i)(1+ 2i) ) (i2 i)(1 i)(2+ i2)(2 i2)) i(2 i)+2i(1 i)3i(i22) ) (i1)(3i2+2i+5)) (3i2 7i+ 5)(4i2 2i+ 7) + (1 i2)(2+ 3i 5i2)) ix(i x 2) i(2x 3) + i(x2 3), x R
1.2 + i, , R :
)5
1 2i) i
1i
)1+ 2i2 i
)3 7i1+ i
+7 3i1 i
)3+ i2i
)1+ i
2
2 i)
3i2 + i2i 1
) i + i
)1 i
3
3+ i)
(2+ 3i)(1 i)1 2i
)1 2i3+ i
1+ 2i1+ 3i
)(2 5i)(1+ i) (2+ 3i)
3+ 4i)
1+ 2i2 i
+2i
1.3 + i, , R :) (i 3)2 + (2+ i)2 ) (3+ 4i)2 + (4 3i)2 + i2) (1
2i)2 ) (3+ 4i)2(1 i) + (3 4i)2(1+ i)
) (1 i)(1 2i)(1+ i)(1+ 2i) + i2 ) (1 2i)3) (1+ i)3 (1 i)3 ) (2+ 3i)3 + (2 3i)3) (2 i)5(2+ i)6
1.4 pi :) (i 1)i1 2(1 i)2 ) (1 2i)2 + (1+ 2i)2) (6 4i)(1 i)1 + (17+ 19i)(1 5i)1
1.5 + i, , R :
)(2+ i)2
3+ i)
(1 i1+ i
)2)
(1 5i3 2i
)2+
(4 2i1 3i
)2)
(1 i)3 + (1+ i)3
(1 i)3 (1+ i)3)
(i 32
)3
(i+ 32
)3)
(3+ 3ii 1
)2)
22 +22i4 22
22i4
1.6 x R, pi :
(x 1 i)(x 1+ i)(x+ 1+ i)(x+ 1 i) = x4 + 4
1.7 z = 3+4i w = 23i, + i, , R, :
z+ w, zw,26w
25wz
1.8 z = 2 3i, + i,
, R, w = z+ 2iz 2i
1.9 :
z =(1 i)2
1+ i w =
z2 + 1z
pi w
1.10 z =1
1 i+
11+ i
+i2.
pi 4z2 8z+ 5 = 0
1.11 z = 9+ xi, x > 0. x z2 z3
1.12 a, b, c C :a
7+ 4i=
b1+ 2i
=c
1+ 3i
pi a
a 6b+ 2c= 1+ 2i
1.13 pi :
)1+ i1
1 i+
1 i1
1+ i+
1+ i1 i1
+1 i1+ i1
= 4
)1i
(1i 1
)
(1i i
) (1i+ 1
)= 1+ 3i
)(1+ i)n
(1 i)n2= 2in1, n N
) (1+ i)4n = (1 i)4n, n N
1.14 z C, : Re(z) > 0 Re ( 1z
)> 0
1.15 , > 0 n N (+i)n = (i)n) pi n 3) n = 3, pi
=
33
1.16 :
z =1
1+ + i, pi , 2pi+ pi, Z
pi (Re(z))2+(Im(z))2 =1
2(1+ )
1.17 z =12+
32
i w = 12+
32
i. pi- pi:
1
A =1z3+
1w3
+3z2w
+3zw2
1.18 z1 = 1 + i z2 = 1 i pi
E(z) =z2 + z+ 1z3 + z2 + 1
pi E(z1) E(z2)
1.19 (1, 1), pi - :
) z =1+ + i
1
1+ i1
1 + i
1+
1 i1+
) w =1 i
1+
1 + i1+
1+ i
1
1+ + i1
1.20 z = 12+ i
32
w =12+ i
32.
pi :) 1+ z+ z2 = 0 z3 = 1) 1+ z+ z2 + z3 + z4 + z5 + z6 = 1) w2n = zn, n N) w300 = 1 w333 = 1) z2016 +
1z2016
= z2013 +1
z2013= 2
1.21 z =1+ i
3
2
pi:
E = (+ z+ z2)(+ z2 + z), , , R
) pi z3 = 1) pi pi E
1.22 a, b, c pi c = (a+ bi)3 107i
()
1.23 z = (2x + y + 5) + 2i w = (4x 2y + 4) + (3x 4y)i. z = w, pi pi x, y
1.24 :
z = 6i (3 4i)x 3yi (3i 2)x+ (4 2yi)
x, y R, pi z = 0
1.25 pi x, y pi :) x 2+ (y 3)i = (1+ 3i)(y+ i)) (x+ y) + (x2 y2)i = 5+ 5i) x(1+ xi) + 2(1+ xi) = 2(3y 1)i y(1+ yi)) x(x+ 2y+ i) + 1+ yi = 6x+ 3i) (x2 + y2i)i+ x2 = 9(1+ i) + y
) (x+ yi)2 =12+ 5i
i
)x
2+ i+
y1 2i
=6 2i4 3i
) (x+ i)2 +4i= (i x)2 + y+ 1
)x yi1+ i
+1i=
x2+
11 i
) (x+ yi)2 =55+ 10i1 2i
)x
1+ 2i+
y3+ 2i
=5+ 6i1+ 8i
1.26 f (z) = z3 + 2z2 + z+ 4 i. , R, pi f (1+ i) = 0
() z R Im(z) = 0z R Im(z) = 0z R Im(z) = 0 z I Re(z) = 0z I Re(z) = 0z I Re(z) = 01.27 pi pi
pi:
) (1 i)24 ) (1+ i3)3 )
(1
1 i
11+ i
)2) (3+ i)6
1.28 z =1+ i1 i
, pi R. pi :
) w = z+1z
pi
) u = z1z
1.29 :
z =(1+ 3i)(2 3i) (4+ i)
2x+ i, x R
) pi - z) x, pi :
) z R ) z I1.30 pi pi pipi,
x R pi :) z R ) z I
2
) z = 6+2x2i3(x+6i) ) z = (2 x+3xi)(i x)) z = (2x+ i)(1+ i)2 + (x2 + 4)i+ x
) z =x+ 2x+ i
) z =1 i
3
x+ (x+ 1)i) z =
x+ ix xi
) z =x+ ix
3+ i
1.31 z = x+ yi, x, y R, pi :
(z 1)(iz 1) R x = y x y = 11.32 , R :
z =+ i i
w =+ i i
pi :) z+ w I ) zw R ) z = w || = ||
() z 0 Re(z) 0z 0 Re(z) 0z 0 Re(z) 0 Im(z) = 0Im(z) = 0Im(z) = 01.33 z C, z2 0, pi z R1.34 z , z , i, -
pi z3 iz+ i
< 0
1.35 z, pi - pi :
) z2 + z 0 ) z2 4z+ 3 < 0 ) z+ 1z< 0
() in = iin = iin = i
1.36 + i, , R pi :
)1+ ii5
+i2 ii7
+i4 ii9
+i 1i3
)(1i3+ i
) (1i7 1
)+
(1i5 i
) (1i9 1
)1.37 pi pi pi:
) i31 + i46 + i265 ) i2012 + i2013 + i2014 + i2015
) 5i2 + 7i78 4i31 + 7i 3i44 + 15
)1i+
2i2+
3i3+
4i4+
5i5+
6i6+
7i7
)14i3
1
5i13+
12i23
+1
3i37+
16i49
) i10 +1i15
+ i20 +1i25
+ i30 +1i35
1.38 + i, , R pi :) (1+ i)20 ) (1+ i)21 ) (2+ 2i)20 + (2 2i)20
1.39 , , , 4 pipi, pi i+++ = 1
1.40 pi pi pi:) i57 + i58 + i59 + + i213 ) i i2 i3 i2015) i+ i2 + i3 + + i4, N
1.41 + i, , R :
z =(1 9i5 4i
)28+
(2+ 3i3+ 2i
)171.42 :
z =(13+ 7i7 13i
)36+
(21 4i4+ 21i
)53 pi [z Re(z)]2015 = i
1.43 z =12
12i, pi z127
1.44 n Z, pi (3+ i)6n = (3 i)6n
1.45 n Z, pi :) i3n+31 = in5 ) i3n13 = i3n ) i3n2 = i32n
1.46 n N, pi :) 2in + (1)n = n 1 ) 2in 3(1)n = n 3) 8in + 5(1)n = n+ 5 ) 2i2n+4 + 5(1)n = 4n 1
1.47 pi , . pi- : (+ i)2014 + ( i)2014 = 0
1.48 f (n) = inz, z C n N. pi f (3) + f (8) + f (13) + f (18) = 0
1.49 2in = 3n 20, n N, pi :) n = 6 ) (3+ 5i)n + (5 3i)n = 0
1.50 in+2 i = 0, n N, pi- (1+ i)n+1 pi
1.51 f (n) = in(1 i), n N. pi :) f (4n) + f (4n+ 1) + f (4n+ 2) + f (4n+ 3) = 0) f (1) + f (2) + f (3) + + f (101) = 1+ i
1.52 f (z) = i(z2 + 1) + z, z C. pi:) f (i) f (i)) pi ( f (i))n + ( f (i))n, n N
3
1.53 z =3+ i w = 1
3i.
) pi :
)zw= i )
z wz+ w
=zw, R
) z22 + w22 = 0) n N, pi zn + wn = 0
1.54 n N, pi :) in + in+1 + in+2 + in+3 =
1in+
1in+1
+1
in+2+
1in+3
) i2n+i2n+1+i2n+2+i2n+3 =1i2n
1
i2n+1+
1i2n+2
1
i2n+3
1.55 n N, pi pi-:) i5n+7 ) i3n+1 + i2n + i5n+3 ) (1+ in)(1+ i2n)
) in+ i3n+2+ i2n+1+ in+3 ) in+1+ i2n+3+ i3n+ in+2
1.56 z =2(1 i)2
. pi:) z8 = 1 ) 1+ z+ z2 + z3 + + z79 = 0
1.57 z C z2 iz 1 = 0. pi pi:
A = z63 z42 + z33 + z120
1.58 pi 3+ 1
22
+
3 122
i72 = 1
1.59 n N, pi :
)(1+ i1 i
)2n+
(1 i1+ i
)2n= 2(1)n
)(4 2i1+ 2i
)n+
(6+ 2i1 3i
)n=
{0, n pi2n+1in, n
)( x+ i1 xi
)2n+
( i x1+ xi
)2n= 2(1)n, x, y R
)(xi yx+ yi
)4n
(x+ yixi y
)4n+1= 1+ i, x, y R
1.60 n N, :) (2 3i)n + (3+ 2i)n = 0 )
(2 i1+ 2i
)3n+1= 1
)(2 3i3+ 2i
)2n+
(4+ 5i5 4i
)2n= 2n
27n+13
1.61 f (z) = z3n, n N) n = 10, pi pi- f (1+ i
3) f (
3 i)
) n pi f (3+ 4i) + f (4 3i) = 0
1.62 pi 99=0
(1)(100 )i+1
() z = w, zn = wn z = w, zn = wn z = w, zn = wn
1.63 z,w z2 + w2 = 0. pi- z2014 + w2014 = 0
1.64 z,w pi 2z = (1+ i)w. pi z4 + w4 = 0.
1.65 z C, z2 + z+ 10 = 0. pi (2z+ 1)10 < 0
()
1.66 a, b, c a, b , 0, pi- :
b+ ca R c+ a
b R a
b< R
pi a+ b+ c = 0
1.67 z,w C z + w = 1 z3 + w3 = 2. pi z100 + w100 = 1
1.68 z C z4 + z2 + 1 = 0 , , N, pi z6 + z6+2 + z6+4 = 0
1.69 a, b, c :
a+ b+ c = a2 + b2 + c2 = 0
pi :) ab+ bc+ ca = 0 ) a4 + b4 + c4 = 0) a4b4 + b4c4 + c4a4 = 0
1.70 z C z2 + z + 1 = 0. , , C, pi :) (+ )(z+ z2)(z2 + z) = 3 + 3) (z2 + z+ )3 = (z2 + z+ )3
() pi
1.71 f : C C, pi 2 f (z) + f (iz) = z2, z C. pi f (z) = z2, z C
1.72 f : C C, pi - f (z) + f (iz) + f (iz) = z, z C. pi f (z) = z, z C
1.73 f : C C, pi f (z) + f (z) = z, z C, 3 = 1. pi f (z) =
z + 1
4
2
()
2.74 z =1+ i
3
2. pi
z2 = z
2.75 z1 = 2 i z2 = 3+ 2i. w = z1 + z1 z2 u =
z1z2
2.76 x + yi, x, y R :
) w = (1+ i)3 (1 i)3 ) z =(2+ i)2 + (2 i)2
1+ i
2.77 z C, pi :(1+ zz
)
(z+
1z
)= 1 z
2.78 x, y R, z,w , :) z = x2 + (y+ 3)i w = xy 2(x+ i)) z = 3+ (3x+ y)i z = 2y x 5i) z = (x2 + i)i w = x(5i x) + 4i
2.79 z, x+ yi, x, y R, :
) z =(1+ 2i)2(3 i)
i(1+ i)5) z =
1 i1+ i
(1+ i)20
(1 i)10
2.80 z =(3+ i)2
1+ i.
pi -
w =1+ zz
2.81 z C I, pi Re( zz+ z
)=
12
2.82 z,w C, pi (zw zw)2 0
2.83 f (z) =2z+ iz 2i
, z , 2i. f (9 5i) = w, :
) pi w =32
32i
) pi 23 w
20142.84 z,w Re(z) Im(w) , 0.
(z + w) I (z2 + w2) R, pi z+ w = 0
2.85 z,w C z+ w R zw R, pi z = w
2.86 z , 0 w =zzzz.
pi : ) w I ) 2 Im(w) 22.87 a, b, c C, pi :
) a Im(bc) + b Im(ca) + c Im(ab) = 0) Im(ab) = Im(bc) = Im(ca) , 0, a+ b+ c = 0
() f (z, z) = 0f (z, z) = 0f (z, z) = 0
2.88 z pi pi pi pi :) Im(z+ 1) + iRe(2 z) = 2 i) Re(2z+ 1) + iIm(3 z) = 5 2i) [Re(z) + 3i][Im(z) i] = 3 4i z) 3(1+ i) z 2z = [Re(z)+ 5i]i+ [Im(z)+ 2i](1 i)
2.89 z pi pi pi pi :) z(1+ i) = 1+ i z ) z+ z+ i(z+ i) + 2 = 0) 2z+ z = 3+ i ) (3+ 2i)z (1 2i)z = 4 4i) z+ i(z z) + 3 = 0 ) zz+ 2i(z+ z) = 4(5+ 2i)) z i(z+ z) 1 = 0 ) i(z+ z) + i(z z) = 6i 4) (i 1)(2z i) = z 1 ) zz+ 5(z z) = 61+ 50i) (1+ i)2z+ 5 = z+ 4i
2.90 pi :) z2 + z = 0 ) z2 + 4z+ 4 = 0) z2 + 2z2 + z z+ 9 = 0 ) zz+ z z z2 = 18) z2 + 2z+ 1 = 0 ) z2 2z+ 1 2i = 0) iz2 2z i = 0
2.91 1+ z+ z+ z2 + z2 + z3 = 0
() z R z = zz R z = zz R z = z z I z = zz I z = zz I z = z2.92 a, b, c C, pi pi -
pi:
) z = ab+ ab ) z =a+ ba
+a+ ba
) z = (ab+ c)(ab+ c) ) z = iab+ ba
ab ba
) z = (a b)15 + (a b)15
2.93 z C, pi pi pi:
) w =z iz+ i
+z+ iz i
) w =z2 + z2 z z
zz+ 1) w = z2 zz+ z2 ) w = (z i)2 + (z+ i)2
5
) w =z+ 1z
+z+ 1z
) w =zz(z+ z)(z z)
2zz
) w =z i zz i z
2.94 pi pi -:) (3i 1)9 + (3i+ 1)9 ) (2 i
3)5 (2+ i
3)5
2.95 z,w C, pi pi - :) u = zw zw ) u = (z w)5 (z w)5
) u =z+ w1 zw
z+ w1 zw
) u =(z+ i)2 (z i)2
(z+ i)3 + (z i)3
) u =z2 z2
1+ zz2.96 z :
f (z) = (1 i)z+ (1+ i)z+ 1
pi f (z) = f (z) z R2.97 , , , pi
w = ( )( ) ( )( )
2.98 z , 1 w =z
z+ 1.
pi : w R z R2.99 z , i w =
z+ iz i
. pi- : w R z I
2.100 n N, pi pi pi:) (1+ i)n + (1 i)n ) (3+ 4i)4n + (4+ 3i)4n
)(+ i)n
i+( i)n
+ i, , R
)(1 i)n
(2+ i)(3 i)+
(1+ i)n
(2 i)(3+ i)+ (5 2i)n(5+ 2i)n
)(19+ 7i9 i
)n+
(20+ 5i7+ 6i
)n2.101 z w z+w , 0.
pi u = i 1 zwz+ w
2.102 n N z = (1+i)n+(1i)n+in pi, pi n
2.103 f (z) = z7 2z8 + 1, z C. pi f (z) + f (z) R
2.104 R Im(+ 2z 2z
)= 0, z ,
2,
pi z R
2.105 z C, pi :(z+
1z
) R z = 1
z z R
2.106 z , i, R. pi- :
z+ iiz+
I z I2.107 z, z , 0
w =1 zz2
. pi :
w R z R z z = 2Re(z)2.108 z,w :
z7 = 4+ 3i w7 = 3+ 4i
u =zw, pi : ) u < R ) u < I
2.109 z = x+yi w = y+xi, x, y R. z5 R, pi w5 I
2.110 z C - :
w =zz+
zz
pi :) w R 2 w 2 ) w = 2, z R) w = 2, z I
2.111 a, b c. pi- :) z = a(b+ c) + b(c+ a) + c(a+ b) pi) w = a(b c) + b(c a) + c(a b)
2.112 pi x, (1+ xi)7 R. pi (x+ i)7 I
2.113 z,w C, (z + w) R (z w) I. pi :) zw 0) n N, : ) zn + wn R ) zn wn I
2.114 z C (z i)(z + i) = 1. pi w =
(z 2i)3
(z i)3 i3 pi
2.115 z,w C u = (zw)(zw). pi- : u R z R (z w) I
6
2.116 f (z) =(1+ z)n
1+ zn, z C
n N) pi f
(1z
)= f (z), z , 0
) zz = 1, pi f (z) R) n, pi f (i) pi f (i) R
2.117 a, b, c pi :
a+ b = c, b+ c = a c+ a = b
pi : ) a+ b+ c = 0 ) a, b, c I2.118 a, b, c pi
:
a = b+ c, b = c+ a c = a+ b
pi : ) a + b + c = 0 ) a, b, c R) a = b = c = 0
2.119 , , R , w C, pi w+ w+ = 0. pi w R
2.120 pi , n (+ i)n + (+ i)n R. pi n = 4, N
2.121 z =2+ 2
3i
1+ i. zn R,
n N, pi n pipi 12.
2.122 z,w z , w :
a =1 zwz w
, b =1+ zwz w
i c =z+ wz w
pi :) a2 + b2 + c2 = 1) a, b, c R, z 1
w
3
() pi
3.123 pi :) (1+ i)z = 2 ) z (1 i)(1+ i) = i(z 2)) 2iz (1+ i)z = 2 ) (3+ i)z+ (2+ i)2 = 7+ 2i
) (2+3i)z2 = (1+2i)+2z+ i ) (1 i)2z = 4i2) (i z)(1+ 2i) + (1 iz)(3+ 4i) = 14+ 6i) (1 i)z+ z = 2(1+ i)2 + 7 ) iz+ z = i2 + i3) 3iz (1 i)(z+ 1) = (5+ i)2 + 23i2
3.124 pi :
)z+ 31 i
z 11+ i
= 2i ) 2ziz+ 3i1 2i
= 9+ 2i
) z+iz
1 i= (1+ i)3 )
z 1i+ 1
=z+ 1i 1
i
3.125 z1 = 3 i z2 = 2+ i, :
zz1+ z2 = (z1 z2)2
3.126 f (z) = (1+ i)z+3 i, z C.) :
) f (z) = 0 ) f (z) = z ) f (z) = f (1 i)) x, y R, f (x 2yi) = f (2 4i)
3.127 (1 i)30z = (1+ i)33 216i
() pi
3.128 z,w pi -pi :) z+ w = 4+ i 2z w = 1+ 5i) z+ iw = i 3z+ 2w = 3+ 5i) (1+ 2i)z iw = 7 (3 i)z w = 2 10i) w iz = 3(1+ i) z 2iw = 3(1 i)) (1+ i)z+ (1 i)w = 2+ i 3z 2iw = 4 i) (1 i)z+ iw = i(4+3i) iz(2+ i)w = 23i) 3z+ (1+ i)w = 2+ 3i (2+ i)z+ (1+ i)w = 1
3.129 z1, z2, z3 pi- : z1 = z2+iz3, z2 = z3+iz1 z3 = z1+iz2
()
3.130 pi :) z2+1 = 0 ) z22z+4 = 0 ) z26z+25 = 0) z2 + 4z+ 8 = 0 ) z2 = 4z 13
3.131 pi :
)(z+ 7iz 3i
)2+ 9 = 0 )
(z iz+ i
)2 3
(z iz+ i
)+ 2 = 0
3.132 pi :) z2 = 2i ) z2 = 5 12i ) z2 2iz 1+ 2i = 0) z2 (5+ i)z+ 12+ 5i = 0 ) z2 iz+ 6 = 0) z2 5z+ 7 i = 0 ) z(z i) = 2(iz+ 1)) (z2 4z+ 5) + i(z+ 1) = 0) (1+ i)z2 2
2iz (1 i) = 0
7
3.133 , R, z2 + z+ = 0 z1 = 2 i.
3.134 f (z) =z 1+ i
z2 2z+ 2, z C
) pi z -
) f (z) =z 1+ i
2
3.135 z1, z2 z2 2z+ 10 = 0. pi :
)1z1+
1z2=
15
) z21 z2+z1z22 = 20 ) z
21+z
22 = 16
3.136 w1 = 4 + 7i w2 = 2i2020 + 3i2019.
x2 (4+ )x+ 2 = 0, , R
w1w2, :
) , )
z+ z i
= i(1+ i)
3.137 z1, z2 z +3z= 2.
pi :
(z1 z222
)2016+
(z1 + z22
)2017+
(z1z23
)2018= 3
3.138 pi :
)(z 1z i
)2+ 4 = 0 )
(z 3i2+ i
)2+
(z+ 3i2 i
)2= 0
3.139 z1, z2 z2+4z+8 = 0, pi :
z1 + z2 + 4iz1z2 + 8i
I
3.140 z1, z2 z2 + 2z+ 2 = 0.
w =(z1 + z2)4 + 2i(z1z2)2 2i
+ i, , R.
3.141 z1z= i3
) pipi ) z1 Re(z1) < 0, pi z31 = 1 pi pi- z20151 + z
20151
3.142 , z2+ 5z+ 13 = 0 M(0, 1), pi (MA) (MB) = 13
3.143 z1, z2 z2+z+ = 0, , , R , 0. 2 = 2++2, pi z21 (1+ z2) + z
22(1+ z1) = 1
3.144 z2 + 2z+ 2 = 0) z1, z2) pi z41 = z
42 = 4
) n N, zn1 + zn2 = 03.145 z2 + z+ = 0, , C.
z1, z2, pi :) , R) z2 z = 0 pi
3.146 z24 z+4 = 0, 0 pi) z1, z2 pipi ) pi pi:
S 1 =1z1+
1z2
S 2 = z41 + z42
) , S 2 = 0
() pi
3.147 pi z2016 + z + i = 0, , > 0, pi
3.148 , R pi z3 + z + = 0 1 + i. ;
3.149 pi P(z) = z3 + 7iz2 z 7i, z C) pi P(1) = 0) pi Q(z) P(z) = (z 1)Q(z) z C) P(z) = 0
3.150 pi :) z31 = 0 ) z3+z22 = 0 ) z33z2+2 = 0) z3 + 3z2 + 3z+ 2 = 0 ) z3 3z2 + z+ 2 = 0) z4 = 1 ) z4 + 2z3 + 2z2 + 2z+ 1 = 0) z6 + 2z5 + 2z4 + 2z3 + z2 + (z+ 1)2 = 0
3.151 R, pi :) z3 2(1+ )z2 + (1+ 4)z 2 = 0
) z2 +1z2= 2
3.152 z1, z2, z3
8
z3 z2 + z 1 = 0
pi z20161 + z20162 + z
20163 = 3
3.153 pi :
)(z+ iz i
)3+
(z+ iz i
)2+
z+ iz i
+ 1 = 0
)(z+ 1i
)3 7
(z+ 1i
)2+ 16
z+ 1i
10 = 0
3.154 pi :) (z2 + 1)2 + (z2 z 1)2 = 0) (z2 4z+ 5)2 + (z+ 1)2 = 0) (3z 1)4 + (z 2)4 = 0
3.155 z1, z2 - z3 = 1+
3, pi
z1z2< R
3.156 pi:
P(z) = z3 (5+ 2i)z2 + (11+ 5i)z 12i
) pi ) P(z)
3.157 pi - z4+z2(z1)2+(z1)4 = 0
3.158 n N n , pi - (z + 1)n + (z 1)n = 0 pi
3.159 z z4 + z2 + 1 = 0 (1)) pi : ) z6 = 1 ) z2018 = z2) (1)
3.160 z0
nxn + n1xn1 + + 1x+ 0 = 0
0, 1, ..., n R n , 0, pi z0 pi
3.161 zn = , n N n 2, , pi R
3.162 z C z5 = 1, pi :
)z
1+ z2+
z2
1+ z4+
z3
1+ z+
z4
1+ z3= 2
)z
1 z2+
z2
1 z4+
z3
1 z+
z4
1 z3= 0, z , 1
)z4
z 1+
z3
z2 1+
z2
z3 1+
z4
z 1= 1, z , 1
3.163 z5 + z3 + z + 3 = 0. z1, z2, z3, z4, z5 , -pi :
(1+ z21 )(1+ z22)(1+ z
23)(1+ z
24)(1+ z
25) = 10
()
3.164 z,w pi pi pipi:) z2 = w w2 = z ) z3+w5 = 0 z2 w4 = 1
3.165 z,w pi- z2 = w w2 = z.
) pi : ) z3 = 1 ) w = 1z
) z,w
3.166 pi pi pipi, :) z16 + 2z15 + 1 = 0 (z+ 1)2 + z2 + z = 0) z16 + 2z14 + 1 = 0 (z2 + 1)2 + z3 + z = 0) z3 + 4z = 3z2 + 2 z10 32z+ 32 = 0
4 pi
()
4.167 pi z pi pi pipi:) Re(z) = 2 ) Im(z) = 3 ) Im(z) = 3 Re(z) > 3 ) Im(z) = 1 2 Re(z) 2
4.168 pi z, :) z+ z = 2 ) z z = 2i ) 1+ Im(z) = 2Re(z)) Im(z2) = 0 ) z2 = z2 ) z2 + 4z = z2 + 4z) (2+ 5i)z+ (2 5i)z = 2 ) Re[(2 i)z] = 4) Re(z+ 2) = Im(z) ) Re(2+ iz) = Im(zi) + 2Re(z)) Re(z+ 5+ 7i) = Im(z 2+ 6i) ) z3 = z3
) (1+ i3)z = 2z
4.169 R, pi z, pi pi- pipi:) z = 5+ + (12+ 3)i ) z = 4 (1 )i) z = 2+ (1 2)i ) z = 2 + 1 (2 2)i
) z = (4+ 3i) + (13i
14
)) z =
2 i1+ 2i
4.170 z = 1++ i w = 1 + i, R. , z w, pi .
9
4.171 z = (1 + t) 2ti, t R w = iz + 2. pi z w
4.172 z = (1 + i) + 1 i, R. :) pi z
) , R z = + i1+ i
4.173 pi z pi w ,:) w = z(1 2i) ) w =
iz+ 1
) w =2
z+ 1+ i
4.174 pi z pi w pi,:) w = iz+2z ) w = (zi)(z1) ) w = z2+z+1) w =
zz+ i
) w =z+ 1z 1
) w =z+ i
z 1+ 2i
) w =z+ 2zz2
) w =z+ 1z+ 3
) w =2 2iz+ 1
4.175 z w = (z i)(z+zi). w y = x, pi z
4.176 z w = zz z2. pi z, Re(w) + Im(w) = 0
4.177 z C w = (1z)(1 iz). M(z) pi Re(w) + Im(z2) = 0
4.178
z =x+ yi1 i
+2 4i1+ i
, x, y R
pi M(x, y), z y = x.
4.179 z = 2x(y+1)i w =z 2z+ 2i
.
w R, pi M(x, y) - pi .
4.180 z = + 1 + (2 1)i, pi R) pi pi - z () : y = 2x3) () pi- pi A(0, 2) ()
) z pi pi-
4.181 z 2x + 3y + 7 = 0, pi w, pi (2+ i)w = z 1+ 2i
4.182 z pipi () : y = x + 1, pi w = (2 + i)z + z + i pi pi .
4.183 z , pi w, pi pi pipi:
) w =z 1z+ i
) w =z+ i1+ iz
4.184 :
z2 + (2 2)z+ (52 2+ 1) = 0, R
:) z1, z2 pipi ) pi z1, z2
()
4.185 pi z, :) zz = z+ z ) 2zz+ 3(z+ z) + 2i(z z) + 2 = 0) zz+ i(z z) = 3(2 z z) ) zz+ i(z z) = 3) zz 3z 5z = 2iIm(z) ) zz+ z+ z = i(z z)) (z )(z+ ) = (z+ )( z), > 0) (z (1+ i))(z (1 i)) = 9
4.186 R, pi - z, pi pipi, pi -:) z = (1 ) + (2+ )i) z = 3+ 2+ (4+ 2)i) z = 1+ 2i+ 5(+ i)
4.187 z,w z , 1, w =z+ 1z 1
.
Re(w) = Im(w), pi z
4.188 z , 2 w =z iz 2
. pi :) w R, z pi pi A(2, 0)) w I, z pi pi A(2, 0)
10
4.189 pi z, :
)z
iz 2 R ) z
2
z+ i I ) z
2
z 2 R
)(z
1z
) I ) z i
z 1 I ) 1 z
1 iz R
)z
z+ 4 I )
(z+
4z
) R ) z+ 2i
z i I
)z+ 1z 2i
I ) z2 z+ 1z2 + 1
R ) z 1 iiz+ 1
R
)i(z2 + 1)z+ 1
I ) z(z )1 z
R, (0, 1)
4.190 pi z, :
) Re(z2 1) = 2Im2(z) ) Re(z) + Im(1iz
)= 0
) Re(z+ 1z 1
)= 2 ) Re
(z+ 3z 1
)= Im
(z+ 3z 1
)) Im
(1
z+ 1
)=
16
) Im(z
1z
)= 2Re(i z)
) Re(z+
2
z
)= (+ 1)Re(z), > 0
) Im(
z z
)= (1 )Im(z), R
) Im(
1 z
)=
12, R
4.191 z1 = 3+ i z2 = 1+ 3i. pi z, pi pi pipi:
)z z1z z2
I ) (z z1)(z z1) = z2z2
4.192 R, pi pi pi-pi, pi - z pi :
) z =+ 1+ i
) z = 2i+ 2i
) z =+ 2i1+ i
) z =1+
1+ 2+ i
4.193 z = x 2+ yi w = x+ yi. pi M(x, y) pi
zw
.
4.194 z,w w =z+ 1z 1
. z x + y = 1, pi w.
4.195 , z,w - pi zw = 2i. () : x+y+ 1 = 0, pi- pi .
4.196 z x2 + y2 = 4, pi -
w = z+4iz
pi .
4.197 z,w C, w , 0, pi :
2Im(w) + wRe(z) + iw[1+ Im(z)] = 0
pi z - (0,0) 1.
4.198 z =1
2+ + i, R.
pi z - pi
() pi
4.199 pi z, :) 4(z+ z) = (z z)2 ) [Im(z)]2 = 2(z+ z)) (z z)2+ 4(z+ z) = 0 ) 2zz = z2+ z2+ 8(z+ z)
4.200 R, pi - z, pi pipi:) z = (+ i)2 + 1 ) z = ( 1)2 + 2(1 )i
4.201 z pi- y = x2, > 0, pi w = iz
4.202 pi - z = + i2, R pi.
() -pi
4.203 pi z, :) z2 + z2 = 14zz 48 ) 10zz 3(z2 + z2) = 16) z2 + z2 = zz+ 1 ) Re(z2 + z) = 1+ Re(z)) z2 z2 = 8i ) 2(z+ z)2 + 3(z z)2 = 24) (z2 1) I ) z2 = 1+ 2iRe(z) Im(z)) z2 + z2 + 2(1 i)zz = 2i, R
4.204 z w = 2z z. Re(w2) = 1, pi - z
11
4.205 R, pi z pi pi pipi:) z = 3 + 4i) z = (3+ 4) + (4+ )i
4.206 :
z = x+ yi w = 2x+ 3+ (3y 5)i, x, y R
pi w (3,-5) 6, - pi z
4.207 pi z, pi :
)1
z2 + 9 I ) i z+ 1
z i R ) z 3z+ 2
z 2 I
)i
z2 + 1 R ) 1
2(z+z)2(z2+z2)+(zz)+16 I
4.208 > 0 (0, 1). z , pi w pi 2w = (1+ )z+ (1 )z
4.209 pi z pi pi pi-pi:) z = + i
1
, (pi
2,pi
2
)) z = + i
1
, (0, pi)
4.210 z = x + yi, x, y R
w = x 11(y+ 2)i u = x 1+ (y 2)i
R w = i u, pi z pi-.
4.211 z = x + yi, x, y R
w = ( i)z+ 3Re(z) u =2(1+ 2)1 i
R w = u, pi z .
4.212 z,w pi 3w (Im(w)) i = z+ 2i. z 4x2 + 9y2 = 36, pi w .
4.213 z,w w = z+22
z,
> 0. z O(0, 0) , pi- w pi
4.214 w = +i, , R, pi :
x2
25+
y2
9= 1
z = x+ yi, x, y R, w :
3z+ 2Re(z) = 5+ 6i+ w
pi pi - z K(1, 2) = 1.
4.215 0 < < pi pi pi :) z2 10 z 92 + 25 = 0) 2 z2 2 z+ 5 42 = 0) (2)z2 (2)z+ 2 2 = 0) (2)z2 2 z+ 5 42 = 0
4.216 z , 0 y = x, pi
w = z+1z
5
()
(i) - |zw| = |z||w||zw| = |z||w||zw| = |z||w|
5.217 :
) z = (2 3i) + (6 3i) ) z = (12i)(3+
3i)
) z = (1 i)5(5+3i) ) z =
3+ 2i(1 i)2 2 i
) z =43 4i
1+3i
) z =(1 2i)2 + 2+ 3i(3 i)2 7+ 5i
) z = (3 4i)(1 xix i
)5, x R
5.218 z = 2 i w =3 i.
zw zw
12
5.219 z,w C Re(z) Im(w) = Im(z) Re(w), pi |zw| = |Re(zw)|
5.220 z C, pi :
|Im(z)| =
|z|2 Re(z2)
2
5.221 z = (1+ xi)(x+ i)+ (1 xi)2, x R.) z R, pi x = 1) x pi |z| = 8
x2 + 1
5.222 z C, |z| = 1. w =
z2(z i)1+ zi
5.223 z,w C w = z1+ |z|
, pi
z =w
1 |w|
5.224 z = 1 i w =z 2z4
, pi
4|w| = |z|
5.225 z, pi :) z = |z| 1+ (|z| 2)i ) z = |z| 1+ |1 2i|i) z+ |z| = 2+ 8i
5.226 z,w pi |z| = |w| = 1, zw R z
w I
5.227 z C w = z+ |z|2(Re(z) + |z|)
. pi-
w2 = z
5.228 z = x + yi, x, y R, pi 2 (x+ iy2)3 I
5.229 z,w C, pi :
|z|2 |w|2 = (Re(zw))2 + (Im(zw))2
5.230 z1, z2 C w = z1 + z2+
, , > 0.
pi |w z1|+ |w z2| = |z1 z2|
5.231 zn =(1+ i
3)3
(3 i)n
, n N. n pi :) |zn| = 641 ) zn I
5.232 z C {1}, |z| = 1. pi pi R, z = 1+ i
1 i
5.233 z,w, u C, :
zw+ wu+ uz = 1 |(z+ w)(w+ u)(u+ z)| = 10
pi |(1+ z2)(1+ w2)(1+ u2)|
5.234 n N, :
zn =(1+
i1
) (1+
i2
)
(1+
in
) pi |zn+1 zn| = 1
(ii)
5.235 z2+2z+ = 0, > > 0.) z1, z2 pipi ) pi |z1| = |z2| = 1
5.236 z2 (2)z + 1 = 0, pi
(0,pi
4
). z1, z2
pi
|z1 + z2| =|z1|+ |z2|
3
z1, z2.
5.237 [2, 2] z,w z2 zw+ w2 = 0. pi |z| = |w|
(iii)
5.238 :) |z+ i2| = 2iz ) |z2| = z ) |z 1| = z i) z+2z = |z+2| ) |z|2 = z+1 i ) |z|+z = 2+ i) |z|2 2z+ 2i = 0 ) |z i| = 2z ) |z 2| = z) z+ |z+ i| = 8+ 3i ) 3z |z+ 1+ 3i| = 1+ 3i) z2 2iz+ |z| = 6 ) |z|2 2(z z) = 2(5 6i)
5.239 :) 4z2 + 8|z|2 = 8 ) z2 + z+ |z+ 1| = 0) |z| z = 1+ 2i ) |z|2 3z+ 2z = 21+ 15i) z+ |z 9 6i| = 17+ 2i ) 2z |z| = 1 8i) z2 + 2|z| 8z+ 15 = 0 ) z2 + z+ |z+ 1| = 0) z2 + |z| = 0 ) 3(z z) + 2|z|2 = 26+ 12i
5.240 z pi - :
|z 1 2i|+ |z 2|i = |z+ 1 3i|i+ 22
5.241 R, pi :) |z|2 z 1+ i = 0 ) z2 3|z|+ 2 = 0) 4|z 1|2 4iz+ 1+ 4i = 0
13
5.242 z |z| = 1 - w = (z+ z)(1 i)n + i(z z)(1+ i)n, n Z pi
5.243 z, pi pi pipi:) Re(iz) = 0 |z+ i| = 2) z = i z |z+ 1| = |z|) |z|+ |i z| = 2 Re(z2) = 1) |z+ 2z| =
10 |z|2 + Re(z2) = 2Im(z)
) |z| = 1 |z2 + z2| = 1) |z| = 1 |z2 + z2 + z2z+ zz2 4| = 6) (i|z|3i)(z2 z) = 3i+ z3 z2+ z i|z| piz2012 + z1000 + 1 = 0
5.244 pi :
) z2 = 4z+ |z|2 +16|z|3
)zz+
zz=
z|z|+
|z|z
) z2 + 2+ i = (1+ i)|z| ) |z2 + 4|2 + 6|z+ 2|2 = 30) 8z3 + 12z|z|2 + 6z|z|2 + z3 8 = 0) 2014(z i)2(z+ i) 2013(z+ i)2(z i) = 1
5.245 z,w C, pi -: (
z+1z
) (|w|+
1|w|
)= 4 |w| < 1
pi |z| = 1
(iv) |z|2 = zz|z|2 = zz|z|2 = zz
5.246 z C, pi :)
z|z|+
|z|z R ) z
|z|
|z|z I
5.247 z1, z2, z3 C |z1| = |z2| = |z3| = 1 z1 + z2 + z3 = 1, pi
1z1+
1z2+
1z3= 1
5.248 z. pi :) |zi| = |z+i| z R ) |1iz| = |1+iz| z R
5.249 , R , , pi :
|z i| = |z i| Im(z) = + 2
5.250 z C, |z| = 2, pi pi- |iz+ 2|2 + |2i z|2
5.251 z C. pi :
z2 2iz |z|2 = 0 Im(z) = 1
5.252 z C, pi :) |z+ i z|2 = 2|z|2 + 2Im(z2)) |z 3i|2 |z+ i|2 = 8 Re(8i z)
5.253 z,w C, pi :
|zw| Re(zw) =|z w|2 (|z| |w|)2
2
5.254 z,w w , 0. pi:) |z+ w|2 = |z|2 + |w|2,
zw I
) |z+ w| = |z w|, zw I
) |z+ w| = |z|+ |w|, zw 0
) |z+ iw| = |z|+ |w|, ( zw
)2 0
5.255 z,w C |z| = 3, |w| = 4 |z w| = 5, pi 16z2 + 9w2 = 0
5.256 z C |z| = 1, pi :
|1+ z2| = |2 |1 z|2|
5.257 z,w) pi :
|z+ w|2 + |z w|2 = 2(|z|2 + |w|2)
) pi pi pi:
) |z+w| =3 |z| = |w| = 1, |zw| = 1
) a = |z|+|w|+i(|z||w|) b = |z+w|+i|zw|,
|a| = |b|
) |z 2i| = 1 |z| = 2, |z+ 2i| =15
v) |z w| = |z| = |w|, |z+ w| =3|z|
v) |z+ w| =2|z| |z w| = 2|w|
v) |z 2| = |w 2| = 2 |z w| = 4,
|z+ w| = 4
v) R w , 0, :
4 z2 + w
2 + |z 2w|2 = 2|w|2 ( zw2 + 4)
v) a = 3
|z3 + 3z2i i 3z|2
b = 3
|z3 3z2i i3 3z|2, a+b = 2(|z|2+1)
5.258 z,w C , R, pi :
|z+ w|2 + |z w|2 = (2 + 2)(|z|2 + |w|2)
14
5.259 pi pi pi:) z C, (1+ z)(1 z) I |z| = 1) z , i,
z iz+ i
I |z| = 1) z C R, (1+ z)(1 z)
z z R |z| = 1
) z , 1, z iz 1
I |2z 1 i| = 25.260 z,w C, pi :
) z , w, i(z+ w)z w
R |z| = |w|) z , w,
z+ wz w
I |z| = |w|) w , 1,
z zw1 w
R z R |w| = 15.261 z,w C, |z| = |w| = 1, pi :
)(z w)2
zw 0 ) u+ zwu (z+ w)
z w I, u C
5.262 z C, pi :) |z+ 8| = 2|z+ 2|, |z| = 4) |z 2| = |2z 1|, |z| = 1) |z 8i| = 2|2z i|, |z| = 2) |2z i| = |iz+ 2|, |z| = 1) |z 1| = |z |, R {1, 1}, |z| = 1) |z+ 9+ 3i| = |3z+ 3+ i|, |z| =
10
5.263 z C, pi :) |z 10| = 3|z 2|, |z 1| = 3) |z+ 3| = 2|z 3|, |z 5| = 4) |z i| = 2|z+ i|, |3z+ 5i| = 4) |z+ 1 2i| = |2z+ 2+ i|, |z+ 1| = 1
5.264 z,w C, pi z|w| w|z|zw+ |zw|
I
5.265 z,w C. pi :) |z+ w|2 = |z|2 + |w|2 + 2Re(zw)) |z+ w|2 |z w|2 = 4Re(zw)) |zw 1|2 |z w|2 = (|z|2 1)(|w|2 1)) (|z|+ |w|)2 |z+ w|2 = 2(|zw| Re(zw))
5.266 z,w C, z , w, pi :
z+ wz w
=|z|2 |w|2
|z w|2+ i
zw zw|z w|2
5.267 z C |z22iz1|+|z2+2iz1| = 2|z2+1|. pi : ) |z i| = |z+ i| ) z R
5.268 z,w |w| , 1
u = ( + i)z w1 zw
, R
pi |u| = 1 |z| = 15.269 z1, z2, z3,
pi pi :
z1 + z3 , 0 |z1 + z2 + z3| = |z2 + z3| = |z1|
z1
z2 + z3
5.270 z,w C, |z| = |w| = > 0 pi :) (z+ w)
(1z+
1w
) R ) z+ w
2 + zw R
)(z+ wz w
)2 R, z , w ) z
3 + w3
(z w)3 I, z , w
5.271 z,w C z + w , 0, zw < R |z| =|w| = . R, z+ w+ zw
R
5.272 z,w, u C :
|z+ u w|2 + |z u+ w|2 = 2|z|2
pi u = w
5.273 z w, w , 0. pi- :
) Re( zw
)=
Re(zw)|w|2
) Im( zw
)=
Im(zw)|w|2
5.274 z, pi :) |z 1| = |z i| = |z 2| ) |z| = |z|2 = |1 z|) |z 1| = |z+ i| |z 2| = 2|z+ 3|) |z+ i| = |z+ 5 4i| |z 6i| = |z|
) |z| =2|z|= |z 2| ) |z| = 1
z 1+ i2 = 2
) |z| =1+ i
3
iz
= |2 z|5.275 z,w C, pi pi pi-
:) |z+ w| = |z w| zw I) |1 zw| = |z w| |z| = 1 |w| = 1) |z 2|2 + |z+ 2|2 = 10 |z| = 1) |z+ w|2 = 2zw |z w|2 = 2zw) |z|2 + |w|2 = |z w|2 |z+ w| = |z w|) |z+ w| = |z|+ |w| |z w| = ||z| |w||)
zw+
wz I |z+ w| = |z w|, R
5.276 z,w, u C, pi :
|z w|2 + |z u|2 = 2z u+ w2
2 + |u w|2215
5.277 z,w C, |z| = |w| = 1. pi: z+ zw
2 + z zw2 = 4
5.278 z,w C, pi : |w||z| z+ |z||w|w = |z+ w|
5.279 z,w. pi :) |1+ zw|2 + |z w|2 = (1+ |z|2)(1+ |w|2)
) |z w| =(1+ |z|2)(1+ |w|2), zw = 1
5.280 z,w z , w u =
z+ wz w
. pi :) zw R, u R ) |z| = |w|, u I
5.281 z,w C z , iw w , 0. |z2+w2| = |z2 2izww2|, pi
zw R
5.282 z C, pi :
|z+ |z||+ |z |z|| = 2|z| z R5.283 z C, |z+ i|+ |z i| = 2(|z|+ 1).
pi z+ 1z
= 25.284 z pi -
|z 3|+ |z+ 3| = 10 |z| = 4
5.285 z C, |z 1| = 3 |z| = 1. pi : ) z2 + z+ 1 = 0 ) z2013 = 1
5.286 z C, |z+ i|+ |z i| = |z+ 1|+ |z 1|, pi |Re(z)| = |Im(z)|
5.287 z C z < R w = 1+ z+ z2
1 z+ z2 pi, pi |z| = 1.
5.288 z w =1 iz+ z2
1+ iz+ z2.
pi : |w| = 1 |z| = 1 z R5.289 f (z) = z+
1z, pi z C.
pi :
) f (z) I z I ) f ( z1z2
) I |z1+z2| = |z1z2|
5.290 a, b, c C |a| = |b| = |c| = 1. pi- :
) u =a+ bc
+b+ ca
+c+ ab
pi-
) v =a bc
+b ca
+c ab
-) pipi a + b + c = 0, pi
Re(ab+
bc+
ca
)=
32
5.291 z1, z2, z3 C, pi :
|z1| = |z2| = |z3| = 1 1+ z1z2z3z1 + z2 + z3
R
pi z1z2z3 = 1 z1z2 + z2z3 + z3z1 =z1 + z2 + z3
5.292 z,w R, zw = 1. pi :z w2 + i+ z w2 i
= |z|+ |w|5.293 z,w, u C, z2 = wu. pi
:
|w+ u+ 2z|+ |w+ u 2z| = 2|w|+ 2|u|
5.294 z,w, u C, w2 = z2 u2. pi- :
|z w|+ |z+ w| = |z u|+ |z+ u|
5.295 z1, z2, ..., zn :
|z1| = |z2| = = |zn| = > 0
pi
w = (z1 + z2)(z2 + z3) (zn1 + zn)(zn + z1)
pi
5.296 z1, z2, z3, ..., zn, pi . pi (
1+z2z1
) (1+
z3z2
)
(1+
znzn1
) (1+
z1zn
) R
5.297 z,w C, |z| = |w| |z + w| = |z| + |w|. pi z = w.
5.298 z,w pi :) z+ w+ 2zw = 1 |z| = |w| = 1) z+ w+ 1 = zw |z| = |w| = 1
16
5.299 z,w, u C, pi :) |z w|2 + |w u|2 + |u z|2 + |z+ w+ u|2 == 3(|z|2 + |w|2 + |u|2)) |z+w|2+|w+u|2+|u+z|2 = |z|2+|w|2+|u|2+|z+w+u|2) |z+w+u|2+|z+w+u|2+|zw+u|2+|z+wu|2 == 4(|z|2 + |w|2 + |u|2)
5.300 z,w C, |z2 w2| = 2|zw|. pi- :
||z+ w| |z w|| = 22||z| |w||
5.301 z,w C, |z| = |w| = |z+w|) pi z3 w3 = 0) pi
zw
5.302 1, 2, ..., n > 0 z C :1
z+ 1+
2z+ 2
+ + nz+ n
= 0
pi z R5.303 z,w pi
: |z| = |w| = 1 |z+ w+ 2| = |zw 1|
5.304 , z,w |z| = |w| AOB = 450, pi z2 + w2 =
2zw
(v) |z| = |z||z| = |z||z| = |z|
5.305 z C |i z| + |z + i| = 2, pi |z i| = 1
5.306 z C, pi :) |z2 + i| = |i z2| ) |z+ |z|2| = |z2 + z|
5.307 z1, z2, z3 C |z1| = |z2| = |z3| = 1 z1 + z2 + z3 = 4i, pi
w =1z1+
1z2+
1z3
5.308 z,w C, |z| = |w| = 1 R, pi- |zw+ | = |zw+ 1|
5.309 z,w C, |z| = |w| = 1, pi |z+ w+ zw 1| = |z+ w zw+ 1|
5.310 |z1| = |z2| = |z3| = 2 |z1 + z2 + z3| = 4, pi |z1z2 + z2z3 + z3z1| = 8.
5.311 z1, z2, z3 C, |z1| = |z2| = |z3| = 1, pi |z1 + z2 + z3| = |z1z2 + z2z3 + z3z1|
5.312 z1, z2, z3 C, :
z1 + z2 + z3 =32
|z1 1| = |z2 1| = |z3 1| = 3
pi |z1z2+z2z3+z3z1| = 3|z1+z2+z33|
(vi) z = w,z = w,z = w, |z| = |w||z| = |w||z| = |w|
5.313 z C, 8 + z2 = (3z2 6)i, z
5.314 z C z4 + (z 1)4 = 0, pi Re(z) =
12
5.315 z C, (z 2)n = (z + 2)n, n N, pi z I
5.316 n N, pi pi pipi:) (1+ i)n = 16 ) 8i(1+ i)n = (
3 i)n
) (3 i)n = 8i ) (2i)n + (2i)n = 2048
) (1+ 2i)n = 3+ 4i ) + in = 0, , > 0) (+ i)2n = (+ i)n+1, pi , R) n(2+ 3i)n + (3 2i)n + (3+ 2i)n = 0
5.317 n N, (1+ i)n = 2n
5.318 z C {i} n N (z 3iiz+ 1
)n=
1 i3
2,
pi Im(z) = 2
5.319 z,w C zw
wz= i, pi
|z| = |w|
5.320 z,w C (zw)2+w2 = 0. pi:
) Re(zw) =|w|2
2) |w| =
2|z|
5.321 z,w C z20+w20 = 0. pi:) |z| = |w| ) w , 0,
( zw
)30 I
5.322 pi pi , pi :) (1+ iz)7 = (z+ i)5
) (1+ iz)n =1+ i
3
3+ i, n N
) (1+ iz)2+1 = (i+ z)2+1, , N , ,
5.323 z C, 5(z + i)n = (4 + 3i)(1 + iz)n, n N, pi z R
5.324 z1, z2 z2+z+ = 0, , , R > 2. pi :) z1, z2 < R) (w z1)n = (w z2)n, n N n > 1, w R
17
5.325 z = x + yi, x, y R, pi (z 1)22 = (z i)22. pi :) x = y ) z22 I
5.326 z C (1+z)2n+(1z)2n = (1z2)n, pin N n 2. pi |1 z| = |1+ z|
5.327 z , i pi (z+ i)3 + 2i(z i)2 = 0. pi :
) |z+ i| = 2 )(z+ i)2 + 4
z+ i R
5.328 pi , , 0 z :(z i
z+ i
)n=+ i i
, pi n N
pi : ) |z i| = |z+ i| ) z R5.329 pi :
) z3 = |z| ) z3 |z| 6 = 0 ) 2z3 = |z|+ 1) z3 = z ) z3 = z|z| ) 2z6 = |z|+ 1
5.330 z n pi :
zn = 1 (z+ 1)n = 1
pi :) |z| = |z+ 1| = 1 ) z2 + z+ 1 = 0
5.331 z C, z14 = 27z11, pi z25 R5.332 z C, [(1 2i)z]7 = (z + 2)7, pi
: )5|z| = |z+ 2| ) |2z 1| =
5
5.333 z C, z100 + iz99 iz + 1 = 0. pi- |z| = 1
5.334 z,w C, z2+zw+w2 = 0. pi |z+ w| = |z| = |w|
5.335 z C, 1+ 2z+ 22z2+ + 2nzn = 0, n N. pi (1+ 2z)(1 2z) I
(vii)
5.336 z,w pi :
z+20iw
= 5+ i w+12iz= 4+ 10i
|zw|
5.337 z1, z2 C |z1| = |z2|. x, y pi xy = z1 xy = z2
()
(i) pi
5.338 :
z = 8i
(3+ 4i)4 w =
+ 17i(4 3i)4
, R
pi - pipi
(ii)
5.339 , , z1 = 1 + 3i, z2 = 3 3i z3 = 1 9i . pi .
5.340 z , 0 w = t(3 1)z,
t R. t pi , =
5.341 pi - z25(1+ i)z+ 13i = 0
5.342 z,w C, 2zw+
wz= 2.
pi :) z,w
) |z| =22
|w|
5.343 a, b, c :
c = za+ (1 z)b, a , b z = 12+ i
32
pi a, b, c - .
5.344 R -:
z1 = + i(1+ ) z2 = (1+ ) + i
, z1, z2 pi .
5.345 z C z < R z < I, pi z, z,
z2
z -
18
5.346 a, b, c - pi a + b + c = 0 |b| = |c|. pi a, b, c -
5.347 ,, , , -. ,, - , pi Re[(+ 2)( )] = 0
5.348 ,, , , -. ,, , pi pi C R || = 1 (1 )+ = 0
(iii)
5.349 pi pi pipi, pi z1, z2, z3 - :) z1 = 3+ 4i, z2 = 5+ 5i z3 = 2+ 6i) z1 =
4 2i1 i
, z2 =6+ 2i1 3i
z3 =2 6i2 i
5.350 z,w z , 0 pi w = z(1 + i), > 0. pi z,w - , pi .
5.351 , z,w , 0 z , w. pi pi pi :) AOB = 900 ) |z w| = |z+ w|) |z|2 + |w|2 = |z+ w|2 ) zw+ zw = 0
5.352 pi pi pipi, pi z,w, u, pi , - :) 2u = iz+ (2 i)w ) (1 2i)u = w 2iz
5.353 z,w C, pi -, pi pipi, :) z,w u = iz(1 i)w, z , w) z i, 1+ iz (1+ i)(z i), z , i
5.354 , , , , -, pi :
||2=
||
2=
||
22
, A = 900, pi :
Re()Re()
=12
5.355 z1, z2 z21 + z
22 = 0. pi :
) |z1| = |z2| ) |z1 + z2| = |z1 z2|) , z1, z2
5.356 z1, z2, z3 , :
|z1 z2|2 + |z1 z3|2 = |z2 z3|2
pi z2 + z3 = 0
5.357 , , z1, z1, z3 , pi - . pi :
A = 900 |z2 + z3 z1|2 = |z2|2 + |z3|2 |z1|2(iv) pi
5.358 pi pi- , pi -:) z2 2i z = 0, z , 0 ) z3 + 3z2 + 3z+ 9 = 0) z3 = 27
5.359 z,w C 2z = (1+ i3)w. w x2 + y2 = 4, pi- :) z ) pi
5.360 z,w C, pi z + w, z w z + i
3w
pi .
5.361 z, pi pi pipi, pi :) i, z, iz ) 1, z, z2
5.362 z,w C z2 + w2 = zw, pi pi z,w pi
5.363 :
19
(2x)z2 (6x)z+ 42x+ 9 = 0, x (0,pi
2
)) z1, z2 pipi -.) , z1, z2 , -:) pi x, -pi ,) pi x pi;
5.364 :
z2 (2+1)z+ 22 = 0, [0, 2pi)
) z1, z2 pipi ) , z1, z2 , pi pi pi.
5.365 , , z1, z2, z3 pi :
z1 z2z3 z1
= 12 i
32
pi pi
5.366 a, b, c C, |a| = |b| = |c| a, b, c pi , pi ab, bc, ca pi pi .
5.367 3 = 1, , 1, pi z,w z 2w z , w pi
5.368 a, b, c pi :
|a| = |b| = |c| = Re(ab) = Re(bc) = 2
2
, , a, b, c, pi :) pi)
) Re(ca) = 2
2
(v)
5.369 z,w |z| = 1 |w| = 2. z,w - 600, :) pi pi z w) pi : ) |z+ w| ) |2z+ w|
5.370 , z,w , pi :
|z w|2 +3|z| |w| = |z|2 + |w|2
pi (OAB) =|zw|4
5.371 z,w C zw+
wz=3, pi-
O(0, 0), A(z) B(w).
5.372 , , z1, z2, z3. ,
= A =z2 z3z1 z2
2 , pi 5(z22 + z
23) = 4z1(z3 + z2 z1) + 6z2z3
(vi) pi
5.373 pi pi pipi, pi z1, z2, z3, z4 pi:) z1 = 4+ i, z2 = 6+ 5i, z3 = 4+ 11i z4 = 2+ 7i) z1 = 1+ 2i, z2 = 5+ 8i, z3 = 7+ 12i z4 = 3+ 6i
5.374 pi
z2 z+ 1 = 0 z2 +3z+ 1 = 0
pi.
5.375 , a , 2 + 2 = 0. pi - , , +
5.376 z,w z , w. pi z,w, z + (z w)i w+ (z w)i
5.377 , , C pi - . , , , , , pi pi- pi, Im[( )] = 0 | | = ||
20
() pi
(i)
5.378 pi z, pi :) |z+ 2| = |z+ 1 i| ) |z 1 2i| = |z 3+ 4i|) |z+ i| = |z 3+ i| ) |iz 3+ 2i| = |iz+ 2 i|) |z+ z|2 = 2|z|2 ) |2iz 1+ i| = |2iz+ 1+ i|
5.379 z C n N, z2n = |z|n(z + 1)n. pi z 2x+ 1 = 0
5.380 f (z) =z 3iz+ i
, z , i) f (z) = 1, pi z) f (z) < 1, pi Im(z) > 1
5.381 f (z) = iz+ 1, pi z C) f (z) = z) | f (z)| = |z|, pi z
5.382 z,w. pi z, pi pi- pipi:) |w 1| = 1 zw = i) |w| = 1 zw = z+ 1+ wi
5.383 z,w pi :
2|z| = z+ z
wiw 2
R
) pi - z,w) z = w
5.384 z,w C, |z+w| = |zw|. pi u = zw.
5.385 z,w w , 1 pi :
|2zw 2z+ w 1| = |2zw 2z+ 2w 2|
pi z
(ii)
5.386 pi z, pi :) |2z 1| = 4 ) |z+ 1+ 2i| = 3 ) |z i| = 2) |(1 i)z+ 2i| = 2 ) |3 2iz+ 4z| = 7
) 4z2+ i
+ 5 = |z5|
5.387 pi z pi :) |z 3 4i| 5 ) 1 |z 3 4i| 2) 2 < |2z 2+ 6i| < 4 ) |1 z|2 1 |z|2
5.388 pi z, pi :) |4z i| = 2|z+ i| ) |z 1|2 + |z+ 1|2 = 10) |z+ 1| = 2|z 2| ) |z 1 i| = 2|z 1+ i|) 2|z| = |z 3| ) |z 2i|2 + |z 3+ i|2 = 13) 2|z 3 2i|2 + 2|z+ 1 3i|2 = 21
5.389 z, pi :
|z 4| 52
|z 3i| 52
5.390 z,w C, pi w, pi pi pipi:) |z i| = 2 w = 4z+ i) |z| =
5 w = (2 i)z+ 3i
) |z| = 4 2iw = z 1 i) |z 1| = 2 (w+ 2i)z = 2w) |z 3+ 4i| = 5 zw = 1) |z 2i| = 3 w = 4z i
) |z| = 1 w =z 1z
5.391 z pi |z2+ i| = 3. pi :) z ) z+ 3 2i ) iz+ 1
5.392 z, pi |z3+ i| = 2. pi w, pi pipi:) w = z+ 1 ) w = z 3+ 5i ) w = 2z 4+ 3i
5.393 z (0,0) 1, pi
w =z iiz+
, R
5.394 z,w C |z| = 1 z2 = 2+ izw) pi z iw) pi w
5.395 z,w zw iw = z (2 + 3i). z (0,1) (2,3), pi w -
21
5.396 pi z pi (z 2i) = 4, N 4 .
5.397 z , 0, pi -
Re(1z
)=
14
) pi z |z 2| = 2) Im(z) = 1, pi pi- z
5.398 z1, z2, z3 - , > 0 z1+z2+z3 , 0. pi
z =z1z2 + z2z3 + z3z1
z1 + z2 + z3
.
5.399 z pi - 2|z| = |z 3|) pi z) pi
w =z 3z+ 1
pi .
5.400 z, pi |z+ 3| = 2|z 3|. pi:
) z ) w =z 9z 5
5.401 z , i w =z+ 2iz i
.
pi z, pi- pipi:) |w| = 2 ) w I Im(w) > 0
5.402 z0 , 0 - z pi |z z0| = |z|. pi w pi zw = 1
5.403 z pi pi , pi w, pi zw = 1,
5.404 z, O(0, 0) 4, pi w = z2 + 3i pi pi .
5.405 z pi :
|z i| = 1 Im(z) 1
) pi z) pi
w =12
(z i
1z i
) y y
5.406 pi z, pi |z2 + 1| = 2|z|.
5.407 pi z , 0, pi :z+ 1z + 2
+ z+ 1z 2 = 4
(iii) pi
5.408 pi z, pi- pipi:) |z+ 2| = |Re(z) 2| ) |4z i| = 1+ 4Im(z)
) |z|2 (Im(z) +
12
)2+
54= 0
) (|z| 2i)(|z|+ 2i) = (Re(z) + 2)2
5.409 p R z C, pi :) |2z p| = |z + z + p|, z pi y2 = 2px) |2z pi| = |z z + pi|, z pi x2 = 2py
5.410 z,w C w = z2, R |z 1| = 1,
pi w
5.411 z pi :
(2z+ z+ i)10 = 3 i2
50 (z i)10 pi :) |2z+ z+ i| = |z i|) z pi pi
5.412 z C, |z|2 (Im(z) + 1)2 + 1 = 0. :) pi z) z pi
8
22
(iv) -pi
5.413 pi z, pi- pipi:) |z 1|+ |z+ 1| = 4 ) |z 3i|+ |z+ 3i| = 10) ||z 4i| |z+ 4i|| = 6 ) |z 2| = |z+ 2|+ 1
) |z10i||z+10i| = 12 )1
|z i|+
1|z+ i|
=4
|z2 + 1|
) |z |+ |z+ | = 4, > 0 )z+ 1z
= |z|5.414 z pi
|z+ 6|+ |z 6| = 20
) pi z) pi pi |9z z|
5.415 > > 0, pi z pi :
|(+ )z ( )z| = 2
()
(i)
5.416 z , pi pi :) |z i| < |z+ i| Im(z) > 0) |z| < |1 z| 2Re(z) < 1) |iz+ 1| > |z| 2Im(z) < 1) |6z i| |2+ 3iz| |z| 1
3
5.417 z C {1}, pi :
Re(
2z+ 1
)> 1 |z| < 1
5.418 z C Re(z) > 1, pi :z 12 < 12
5.419 z,w C, pi pi -:) |z w| > |z+ w| Re(z) Re(w) < 0) |z w| |z w| Im(z) Im(w) 0
5.420 z C, |z| 1, pi :
|2z z|2 + |2z+ z|2 10
5.421 z,w , pi (z2 3z+ 5)(w2 3w+ 5) 0
5.422 z,w C |z| < |w| |z w| > |1 zw|, pi |z| < 1 |w| > 1
5.423 a, b, z,w C |a| 1, |b| = 1 w = b(a z)az 1
.
pi |w| 1 |z| 15.424 z C, pi pi -
:
)z 2iz
< |z| Im(z2) < 1)
z+ 1z < |z| Re(z2) < 12
5.425 a, b, c C |a| 1, |b| 1 |c| 1, pi |ab|+ |bc|+ |ca| 3|abc|
5.426 z,w C, pi :) |z|2 + |w|2 zw+ zw) 2|zw| zw+ zw 2|zw|) 2(|z|2 + |w|2) (z+ w)2) (1+ |z|2)(1+ |w|2) (|z|+ |w|)2) 2|z+ w|2 6|z|2 + 3|w|2
5.427 z,w C, |z| 1 |w| 1, pi: ) |z|+ |w| 1+ |zw| ) |z w| |1 zw|
5.428 z,w, x, y C, pi :
(|z|2 + |w|2)(|x|2 + |y|2) |zx+ wy|2
5.429 z C |z| = 2 w = z2 z 4. pi :
) |w| = 21+ 4(Im(z))2 ) |w| 217
) (z2 z 4)(16z2
4z 4
) 68
5.430 z,w C |z| = |w| = 2 x R |z+ xw| > 1, pi |Re(zw)| < 2
3
5.431 z,w C pi - zw R 8z|w| w = z2|w|w. pi 0 < |w| 16
5.432 z,w C, pi pi pi-:) |z| 1, |w| 1 |z w| = 3, |z+ w| 1) |z| 5, |w| 5 |z + w| = 6, |z w| 8
5.433 z,w C, |z| = |w| = 1, pi :
|z+ w|+ |z w| 22
5.434 z C, Re(z) 0, pi
23
2|1+ z| 1+ |z|
5.435 z,w C, Re(z) 0 Re(w) 0, pi- |z|2 + |w|2 + zw+ z w 0
5.436 z,w C |w| = 1, pi :
|z w| 1 12(zw+ zw)
5.437 z,w C, |z + w| = 1. pi
Re(zw) 14
5.438 z,w C ww+1 = |zwiw|. pi |z i| 2
5.439 z C, pi |z + 1| 12
|z2 + 1| 1
5.440 , R ( + ) , 0 1+
1< 0,
pi :
|z+ w|2
+ |z|
2
+
|w|2
5.441 z,w C, |w| = 1, pi
|z w| 12(zw+ zw) 1
5.442 z = x + yi, x, y R. pi :
||x| |y|| |z| |x|+ |y| 2|z|
5.443 (,1) (1,+) z C.) pi :
|z+ i| > |z+ i| |z| < 1) z4 + iz3 + z+ i = 0, pi |z| = 1
5.444 z |z| = 1. pi |z2 + z 1| 5
5.445 a, b, c pi |a| = |b| = |c| = 1 a + b + c = 3. z C [Im(z)]2 1, pi :
|z a|2 + |z b|2 + |z c|2 3
5.446 :
A = {z C : z = + i, > 0, |z| < 1}
pi z A pi x A z =
1 x1+ x
5.447 z1, z2, ..., zn C {i}, n N pi :z1 iz1 + i
+ z2 iz2 + i+ + zn izn + i
< 1 pi :
)(z1 + z2 + + zn) i(z1 + z2 + + zn) + i
< 1) pi pi pi z1, z2, ..., zn pi pi
5.448 , > 0, + = z,w C, pi- |z|2 + |w|2 |z+ w|2
5.449 z,w C > 0, pi :
|z+ w|2 (1+ )|z|2 +(1+
1
)|w|2
5.450 z C, pi : 1z iz+ 1z zi
12 iz zi
5.451 x, y R, x2 + y2 = 1. z,w C,
pi : zx2 + wy
2 |z+ w|25.452 z,w C, |z| = |w| z + w > 0, pi-
zw > 0
(ii)
5.453 z,w C, pi :
||z| |w|| |z w| |z|+ |w|
5.454 z C. pi :) |z| = 3 w = 6+ 8i, 7 |z+ w| 13) |z| = 4 w = 3 4i, 1 |z w| 9) |z 1| = 3 5 |2z 3| 7) |z| = 3 w = 8+6i, 24 |2z+3w| 36) |z 3i| = 3, 2 |z 4| 8) |z 6i| 4, 6 |z+ 8| 14) |z 2+ 3i| = 2, 3 |z 5+ 7i| 7
5.455 a = 3+ 5i, b = 7+ 5i z C, pi |z a|+ |z b| 4
24
5.456 z,w C |z| = 1 |w| = 2, pi |2z+ (3+ 4i)w| 12
5.457 z,w C, pi :) |z| = |w| = 1, |zw+ zw| 2) |z| = 1, |zw |z|| 1+ |w|
5.458 z C |z| 3 1 R, pi |2z + z2| 2
5.459 z C, pi pi pi:) |z| 0. pi 310 m 1
25.477 a, b, c a + b + c = 0
|a| = |b| = |c| = 1. z C, pi :
3|z| |z a|+ |z b|+ |z c| 3(|z|+ 1)
5.478 z,w C u = |z|w + |w|z pi |u| = |z|2 + |w|2. pi |z| = |w|
5.479 z,w C, |z| = 1 |w| = 2. pi- :
(z+ w)(1z+
4w
) 9
5.480 a, b, c |a| = 1, |b| = 2 |c| = 3. pi :
) (a+ b+ c) (1a+
4b+
9c
) 36
) |a b c|+ 1a + 4b + 9c
25.481 ) z C, pi :
|z 2i|+ |z+ 3i| |z|+ |z+ 5i|
) w C |w| = 1, pi :
25
|2w 1|+ |3w 1| 1+ |5w 1|
5.482 z C z+ 2z
= 3, pi :|z|+
2|z| 17
5.483 z C, |z| = 1, pi :) |3z 2| = |2z 3| ) |3z 2|+ |5z2 2z 1| 1
5.484 z C |z| = 1, pi :
|2013z 1|+ |2014 z+ 2013z2| 2014
5.485 z ,12
w , 12, pi :
|3z+ w||2z 1|+ |2w+ 1|
12+
|z||z+ w|
5.486 z1, z2, z3 C |z1| = |z2| = |z3| = 1, pi : 1z1 + 1z2 + 1z3
35.487 z1, z2, ..., zn
(0,0) 1, pi : 1z1 + 1z2 + + 1zn = |z1 + z2 + + zn| n
5.488 z C R, pi zz+ |z|+
zz |z| 1
5.489 z C |z| = > 2, pi : 1z2 + z+ 1 12 1
5.490 z C :
|z 1|+ |z|+ |z+ 2|+ |z+ 3| = 8
pi |z+ 1| 25.491 , , C :
K = {z C/|z |+ |z | ||}
z1, z2 K [0, 1], pi z = z1 + (1 )z2
5.492 z C n N :
|z 1|+ |z+ 1|+ |z2|+ |z+2|+ + |zn|+ |z+n| =
= 2n2 + 4n
pi |z| n+ 25.493 z,w C, pi :
|z+ w| z|z| + w|w|
2(|z|+ |w|)5.494 , R, + = 1 z,w C, |z| =
|w| = 1, pi :
|z+ w| |z+ w|2
5.495 z,w C, |z| = |w| x 1, pi:
(x+ 1)|z+ w| 2|xz+ w|
5.496 z,w C, |z + w| = |zw|, pi |z| 2 |w| 2
5.497 z C, |z| = 2, pi :
|z4 5z2 + 6| 2
5.498 z1, z2, z3, z4 C, pi :
|z1 z2| |z3 z4|+ |z2 z3| |z1 z4| |z1 z3| |z2 z4|
5.499 z,w C, pi :) |z|+ |w| |z|+ |w|+ ||z| |w|| |z+ w|+ |z w|)
|z|+ |w|+
||z| |w||
|z+ w|+
|z w|
5.500 , , , z C, pi :
|z |+ |z |+ |z | | |+ | |+ | |2
5.501 z,w u = |z|w + |w|z, |u| = |z|2 + |w|2. pi |z| = |w|
5.502 z C, pi :) |1+ z|+ |1 z| 2(1+ |z|)) |1+ z| |z| |1+ z|2) |z2 + 2z+ 2|+ |z 1|+ |z2 + z| 3) |z2 + z|+ |z2 + 1|+ 3|z+ 1| 2) |z+ 1|+ |z+ 2| |z|+ |z+ 3|) |z+ 2|+ |z+ 3| |z|+ |z+ 1|+ 4) |z+ 3|+ |z+ 4| |z+ 1| |z 2| 8) 2|z+ 5| |z|+ |z+ 1|+ 9) |z|+ |z+ 1| |z 1|+ |z+ 2|) |z+ 2|+ |z 1| |z+ 1|+ |z|+ 2
26
5.503 z C, |z2 + 1| 1 |z + 1| 1. pi :) |z2 + 2z+ 1| 1 ) |z| 1
5.504 z,w C x (0,pi
2
), pi :
|z|x
+|w|x
|z+ w|
5.505 z,w C = pi3+ i
pi
3, pi
|z w| |z|+ |z w|
5.506 z C, z4 4z+ 6 = 0, pi |z| > 1.
5.507 z,w C [0, 1], pi :
|z+ w| |z+ (1 )w|+ |(1 )z+ w| |z|+ |w|
5.508 z C, |z| = 1, pi :) |1+ z|+ |1+ z2|+ |1+ z3| 2) |1+ 3z2014|+ |1 z2014|+ |1+ 2z2014|+ |1+ z2014| 5) 2|1+ z|+ |1+ z2|+ |1+ z3|+ |1+ z4|+ |1+ z5| 4
5.509 z,w C, |z| = |w| = 1, pi :
|z+ 1|+ |w+ 1|+ |zw+ 1| 2
5.510 z C, |z2+ 1| = 2|z+ 1|. pi |z| 3
5.511 z C, pi :
) z2 + 1z2
2, z+ 1z 2
) z3 + 1z3
2, z+ 1z 2
5.512 z,w C, z3 + z2w = z w. |w| 2, pi |z| 2.
5.513 a, b, c C, pi :
|(1 a)(1 b)(1 c)| 1 |a| |b| |c|
5.514 z1, z2, ..., zn C, pi :
|1+ z1|+ |z1+ 2z2|+ + |(n 1)zn1+ nzn| 1 n|zn|
() -
(i)
5.515 pi pi pipi, |z| z pi :) Re(z) + 1 = Im(z) ) |z 5+ 3i| = |z 1 i|) |iz+ 3| = |z 4| ) |z+ 4i|2 |z+ 3|2 = 17) z = (+ 2) + (2+ 1)i, R
5.516 z C Re(z) = Im(z) + 3, pi |z|2 8
5.517 () pi pi z1, z2 pi z1+ z2 = 12i z1 z2 = 34i. w pi () .
5.518 z pi :
2|z+ 1| = |(1+ i3)z 3
3+ 3i|
:) pi z) |z|
5.519 z C, |z 4|+ |z 3i| = 5.) pi z) |z|
(ii) -
5.520 z 3x + 4y 60 = 0. |z 3 4i|
5.521 z = (2)+(1+2)i, R. :) pi z) z pi ) |z 1 i|
5.522 z, pi - |z+ 2 3i| = |z+ 1 i|. :) pi z) z, pi pi -pi pi A(1, 1)
5.523 z C |2iz 2 6i| = 2|z 5+ 3i|) pi z) |z|) pi |z 2+ 3i| 25
27
5.524 z, pi |z 1| = |z i| |z 2i|+ |z+ 3 4i|
(iii)
5.525 pi - z w |z w|, pi pipipi:) z(43i)+z(4+3i) = 2 4(w+w)3i(ww) =22) |z 2| = |z+ 2 3i| |w+ i| = |w 4+ 4i|
5.526 z |z2|+ Im(z2) = 1) pi z (1) (2)) z1, z2 (1) (2), pi |z1 z2|
2
5.527 z,w C, w = 3z i z+4. z y = x 2, :) pi w) pi |z w| 52
5.528 z1 = + ( 1)i z2 = ( 3) i, R.) pi z1 z2 pi .) |z1 z2|) = 2, pi pi
K = 1+(z1z2
)20145.529 z,w pi
:
|z+ 2| = |iz+ 2| w = (1+ i) +2+ ii, R
) pi - z w
) pi |z w| 22
(iv)
5.530 pi pi pipi, - z, pi :) |z 2| = 1 ) 2|z|2 = (1 i)z+ (1+ i)z) |z 4 3i| = 2 ) |z|2 + 2i(z z) + 3 = 0
5.531 z1, z2 C, |z|2 = 2Re(z), pi |z1 z2| 2
5.532 z,w C, |w| = 1 z5 = 22w5i. pi :) z ) 32 |z| 72
5.533 z pi - :
|z 1 3i|2 + |z 3 3i|2 = 4
) pi z) pi 4 |z z| 8
5.534 z,w pi |w| = 2 z = 2w + 2i. |z w|
5.535 z pi - :
Re[(z 1)(z+ 1)] = z+ z
) pi z) z1, z2 pi pipi - z, pi |z1 z2| 2
2
5.536 z, :
(z 3 4i)(z 3+ 4i) = 25
:) pi z) z -
5.537 z pi - :
|z 1+ 2i| =22
:) pi w = 2z+ 1+ i) w pi
5.538 z C pi |z i| = 2|z 1|) pi z) pi
5 2 |z| 5+ 2
5.539 z pi :
|(3+ 4i)z+ 25|+ |z+ 3+ 4i| = 6
28
) pi z) pi z pi ) z1, z2 pi pi , |z1 z2|
5.540 z1, z2 |z1 z2| = 2 w pi :
(w z1)(w z1) + (w z2)(w z2) = 4
) pi w) pi |wz2|
5.541 :
z = (3+2) + (3+
2)i, pi [0, 2pi)
) pi pi - z) pi pipi z, pi pi
(v) -
5.542 z C, pi , pi pipipi:) A = |z 4 3i|, |z| = 2) A = |z 1 i|, |z+ 2+ 3i| = 3) A = |z 2+ 5i|, |z+ 1+ i| = 2) A = |z 6+ 2i|, |z+ 2i| 1
5.543 z C. pi :) |z 2 i| 5, 8 |z 14 6i| 18) |z 1 i| < 5, |z 10 13i| > 10) |z 1+ 2i| < 3, 2 < |z+ 2 2i| < 8
5.544 z C |z| = 1, pi |z2 + 2z|
5.545 z pi - |z i|2 + |z + i|2 = 4. pi 4 |z 3+ 4i| 6
5.546 z = x + (y 1)i, x, y R pi |z 1 i| 1 = 0) pi w = x+ yi) pi
2 1 |w 2 i| 2+ 1
5.547 z pi (2z 3)n = (z 3i)n, n N.) pi z pi -) z1, z2 pi pi , pi |z1 z2| 2
5
) w = 2+5i, z
pi |z w|
5.548 z C, |(3 + 4i)z + 4 3i| = 10. :) pi z) pi u = |iz2+3i|) z pi pi u
5.549 z C |z + 1 i| = 1. pi:
|z 1 2i|2 + |z 5+ 4i|2
(vi) -
5.550 z,w C, pi- |z w| pi pipipi:) |z 1| = |z i| |w 2| = 1) (1+ i)z+ (1 i)z+ 4 = 0 (w 2)(w 2) = 1) |z 1 i| =
2 zw = 6
5.551 z,w pi :
|z+ 1| 1 w = 1 t2
+ i1+ t
2, t R
- |z w|
5.552 z,w C, zw = 2w + 1. z (1,0) = 1, :) pi w) |z w|
5.553 z,w pi - :
|(1+ i)z 1| =2 |w+ 3i| = |w 1+ 2i|
:) pi z,w) |z w|
29
5.554 z,w pi :
zz+ 2|z| = 3 w = ( 3)i, R
) pi z w
) pi |z w| 32 22
) pi |2z w| 32 42
5.555 z,w pi - :
|z 2 4i| = 2 |w 2+ 4i| = |w 6|
pi:) |z w| ) |z w 1+ 3i|
(vii)
5.556 pi pi pipi, |z w|) |z+ 2| = 1 |w 4i| = 2) |2z 4i| = 2 |w 3+ 3i| = 2) |z+ 1 2i| = 2 |w 6 2i| = 3) |z 1 i| = 1 w = 2z+ 3
5.557 z , pi :
) w =z 2i2iz+ 1
) |z w| 2
5.558 z,w pi - :
|z 3+ 2i| 3 |(1+ i)w+ 2| 22
pi |z w| 105.559 z,w C :
1+ izz+ i
R |iw+ 1| = |2w+ i|
:) pi z,w) pi z,w
5.560 z,w pi :
|z 1 4i| 22 |iw+ 11 2i| 32
) |z w|) pi z = w
5.561 z C, |z 1| 1 |z 2| = 1, pi 1 |z| 3
5.562 z,w C |z| = 2 |w 3 + 4i| = 5. pi |2z+ w|
(viii) pi-
5.563 z = 2+ 2i w = t 1+ 2ti, , t R. :) pi z,w) |z w|
5.564 z,w pi :
|4z 1| = 1+ Re(z) Re(w) + Im(2w) = 6
) pi - z w) pi |z w| 5
5.565 z,w pi :
z2 + z2 + 4Re(z) = 2|z|2 |w+ 1|2 |w 2i|2 = 15
) pi - z w) pi |z w| 5
(ix) pi-
5.566 z,w pi :
|z+ 1| = 1 Re(z) w 2 =(1+ 3i6+ 2i
)7) pi z) pi |z w| 1
30
(x)
5.567 z C |z 4|+ |z+ 4| = 10) pi (C) z) z1, z2 (C) pi , pi 6 |z1z2| 10
5.568 z pi - :
1|z 3i|
+1
|z+ 3i|=
10z2 + 9
) pi z) z1, z2 pi pipi - z |z1 z2| = 6, pi |z1+ z2| 8
5.569 z,w pi |z3|+ |z+3| = 10 |w| = 4. pi|z w|
5.570 z C, |iz 3| + |iz + 3| = 10. pi :) z , pi , ) 4 |z| 5 ) 16 |z2 9| 25
(xi) pi
5.571 z C |z+ 5| |z 5| = 8, pi |z| 4
5.572 z pi :
|z+ 2i| |z 2i| = 23
) pi z) pi |z 1| 6
6
()
6.573 pi pi pipi, pi z,w :) zw = zw ) |z| = |w| = |Re(zw)| = 1
6.574 , z,w -. pi pi pi:)
zw R OM//ON ) z
w I OMON
) |z+ w| = |z|+ |w| OM ON6.575 pi pi pipi,
pi - z, pi pi , :) 1+ i, z, z+ i ) z, z+ 1, z2 ) 1, iz, 1 z2) 1, z, 1+ z2
6.576 A1, A2, A3, A4 - z1, z2, z3, z4 . pi :
A1A2//
A3A4 z2 z1z4 z3 R
6.577 , C f (z) = z+ , z C. z1, z2, z3 , pi - f (z1), f (z2), f (z3) -
6.578 z C, pi - z,z
1z
.
6.579 z,w C zw = 1, pi z,w
6.580 a, b, c pi - pi ab+ bc+ ca = 1. pi :
)b ac a
R) a, b, c
6.581 z C R a, b R a , b a + b + 1 > 0. pi z2, az+b, bz+a |z 1| =
a+ b+ 1
6.582 z C :
w =(z|z|+ 1
)2n u =
(z|z| 1
)2n, n N
pi w, u
6.583 , z,w. pi :
(z+ w)2
zw R , , |OA| = |OB|
31
6.584 z , 0 w =
z, > 0.
, z,w , pi :)OM2 ON = OM
) , ,
6.585 a, b, c C |a| = |b| = |c| = 1 |a+b+c| = 3, pi a = b = c
()
6.586 z = 2 + 3i w = 1 + 4i , . () : 3x + y 5 = 0, MAB = 900
6.587 z,w , .) pi
OA OB = Re(zw)
) pi x x pi pi - z 1 z i) pi z, z1 z i
6.588 , z2 + 2z + = 0, R. w = 1 i, pi |
MA| |MB| = 4
6.589 z , z 1 2z + i .
AMAB
6.590 z w = 1 + i. , z+w zw , pi z,
OA OB = 2
6.591 z , 0 , z z2 1 . pi z,
OAOB
6.592 A1, A2, A3, A4 - z1, z2, z3, z4 . pi :
A1A2A3A4 z2 z1z4 z3 I
6.593 [0, pi] - z2 2 z+ 1 = 0 (1)) z1, z2 (1)) M1,M2 z1, z2 -, ,
OM1OM2
) pi , .
6.594 , , , || = ||. z = ( + )n w = ( )n n N, pi :) n , , ) n pi,
() pi
6.595 , , , z1, z2, z3, z4 . pi pi z1 z2 =z4 z3
6.596 , , z, z+ w,w
. z+ iwz iw
-, pi pi .
7 pipi
()
7.597 z , 1 w =1 zz 1
) z = 2+ i, |w|) pi |w| = 1
) pi u =w 1z 1
pi) w = 1, pi z
7.598 pipi, OA -
z , 0 OB -
w = z, =1+ i
3
2) pi 3 = 1) pi -pi) pi z3 + w3 = 0 z2 + w2 = zw) z , pi w
7.599 z , 1 w =z+ iz+ 1
) |w| = 1, pi Re(z) = Im(z)) w I, pi - z) |w| = 2, pi z - , pi
32
7.600 z pi :
|z 1| = 10 z+ 13
= 12 w 3wz + 6 = 6z w, :) w) pi u -pi :(
2u32
) (u
34
)= 6ww
7.601 z |z 3 6i| = 2|z|) pi |z+ 1+ 2i| = 2
5
) z1, z2 pi pipi - z, pi |z1 z2| 4
5
) z1, z2 pi pipi - z, |z1 z2| = 4
5, pi :
|z1 + z2|n = 2n 5 n2 , pi n N
7.602 z,w, u pi- :
w = z+ i z u = z+ iz
) pi :
) |w|2 = 2|z|2 + 2Im(z2)
) |w+ u| 4|z|) |w u| 22|z|
) |w| =2, pi
z
7.603 f (z) =1z, z C
) pi ( f (i))2015
) z 1, pi f (z)
) pi | f (z) f (z)| 2|z|
) | f (z) z| = 3|z|, pi z
7.604 z pi |z i|2 + |z+ i|2 = 4 w = (4+ 4i)z) , , i,i z -, pi AMB = 900) pi z) pi w) pi |z w| = 5
7.605 z = + i, [0, 2pi) :
w = 3z+ 10+4iz
) pi - z w) pi 5 Re(w) 15 5 Im(w) 5) |z w|
7.606 w pi - 3(w+ w) + 2i75(w w) = 26i90) pi w) w0 pi - ) z C |z+ i10|2+ |zw0|2 = 6, |z+ 1 5i|
7.607 z,w pi - :
|w| = 1 z =1+ 2ww 2
) pi z
K(43, 0
)
=53
) z1, z2 pi pipi - z, z1 , z2, pi
u =3z1 + 3z2 + 83z1 3z2
) pi 1z
) pi |12z2 + 7z 12| = 25|z|
7.608 R z , i w =
z+ iiz+
. pi :) w , z ) |w| = 1, z R) w R, pi z
7.609 z1, z2 - w = |z1z2|2 + (z1z2 + z1z2)i pi y = x) pi z1z2 K(1, 0) = 1) z = 2+ 2i Im(z1z2) > 0, :
33
) z1z2 z1z2) pi (z1z2)20 + (z1z2)20 = 211
) z , z1z2 z1z2, pi (MAB) = 1
7.610 z,w pi :
w =(4 3i3+ 4i
)13+ (2 i)2 + 1+ 2i
|z+ 3i|+ |i z+ 3| = 2|1+ i|
) pi Re(w) = 4 Im(w) = 3) pi z - K(0,3) = 1) pi pi z pi ) |z w|
7.611 f (z) = z+ i z, z C) z , 0, pi
f (z)z 2
) pi f (i), f (1 + i) -
) f (z) , 0, pi f (z)f (z)
2014 = 1) f (z) + f (z) = 4, :
) pi z
) |z|
7.612 f : C C pi:
2 f (z) + f (1 z) = 3z2 7z+ 14, z C
) pi f (z) = z2 5z + 6, z C) f (z) = 5z+ 9+ 4i) f (z) = 54z pi- ) f (z) > 0) pi f (z) f (z) 0, z C
7.613 f (z) =2iz+ 9 12i
z 2i, z , 2i
w1 = z 2i w2 = f (z) 2i.) pi w1w2 = 5 12i
) w1 = w2, w1,w2) :
) f (z) = 2 2i ) f (z2) =8 7i6+ 5i
) f (z) R, pi z
7.614 f (z) =iz 1z+ 1
, z C {1}) pi ( f (i))12 = 64) z pi :
f (z) =12
12i
) | f (z)| = 1 | f (w)| = 2, - |z w|
7.615 f (z) = |z+ 2i|, z C) pi :
) f (z) = |i z+ 2| ) f (z) = f (z)) f (1+i+z) = f (2+3i+z), pi z pi ) pi pi :
) f (z) = f (z) z R) f 2(z) + f 2(z) = 10 |z| = 1
) :
z+ f (z 2i) = f 2(1 i) + i f (i)
7.616 f (z) =(z i)(z+ i)
z z,
z C Im(z) , 0) pi f (z) I) pi f
(1z
)= f (z)
) z pi:
f(1z
)=
z z2
+ i
pi:
) z
) w = 2z2 + 1
7.617 z,w pi w+ 2+ i = (1+ i)|z|) z = w, z) pi z : ) w R ) w I) pi
2|z| |w|+ 5
7.618 z,w pi :
34
w =
5z
|z| w2 = 3+ i, > 0
) pi = 4) pi 5z (3+ 4i)z = 0) pi z) |z 3+ 4i|
7.619 z,w pi :
|z+ 2i| = |w+ 2i| = 2 |z w| = 4
pi :) z w ) |z+ w| = 4 ) |w| = 1, |z+ 3w| = 2
3
7.620 z,w pi :
z1+ i
+z
1 i= 2 Re(w2) + (Im(w))2 = Re(w)
) pi z = 2i w = 1+ i) pi (w z)30 + w30 = 0) n pi :
3( z2
)n= 4n 27
) pi u = w+ ( 1)z, R
7.621 z , 0 w = 1z
, z,w .) pi , , -) pi w+ 1 =
z 1z
) |z 1| = 1, pi |w+ 1| = |w|) |w|
7.622 z1, z2 z+1z= 0 -
w pi :
|wz1|+ |wz2| = |z1 z2|
) z1, z2 pi- z2n1 = z
2n2 , n N
) pi w) w1,w2 pi pipi - w, pi :
w1 w21+ w1w2
I
) pi 4 |w 3+ 4i| 67.623 z,w pi
:
z+ w = 2 z 2w = 1 i
) z w) pi z30 + w30 = 0) pi u pi :
|u z|2 + |u w|2 = 4
) :
v =(u 1
1u 1
)4 pi : ) v R ) v 16
7.624 z,w pi :
|z+ 2| = |iz+ 2| w = (1+ i) +2+ ii, R
) pi - z w
) pi |z w| 22
) u |4u+1+3i| =10
) pi u) v1, v2 pi
pi w u
) pi |v1|2 + |v2|2 =52
7.625 f (z) =12(z+ i z), z C
) f (z) = z, pi z) pi f ( f (z)) = f (z)
) pi f (z) z1 i
=12(Im(z) Re(z))
7.626 f (z) = iz (1+ i)z, z C) pi f (1i) f (2+i)) f (z) = f (1+ i)) pi z -pi f (z) = z) pi :
) f(1z
)=
f (z)zz
) f (z) I, z I) f (z) = f (z), z R
35
7.627 z pi :
4(Re(z))2 z2 z2 = 2
) pi |z| = 1) z , i u C, pi
w =u+ izu z i
z i
) pi m pi m(z 1) = 3(i z)
7.628 z pi :
Re( zz 1
)= Im
( zz 1
)) pi z) z0 pi -
) pi pi(z02
)20167.629 z = 2+
3 i
3,
R w < R pi- |w| = w+ w, R) pi z (2,0) =
3
) pi w) pi z = w ) |z z|
7.630 z pi :
(2z 3i)5 (z 3i)5 = 0
) pi :
) |z i| = 1 ) z , i, (z i
1z i
)5 I
) z1, z2 pi pipi - z |z1 z2| =
2, pi
|z1 + z2| 14
7.631 f (z) =z+ 2iz 2
, z , 2
) ( f (2 2i))2016) f (z) , :
) pi z , pi
) |z|
) pi
w = z 1+ i+2
z 1+ i
pi, 22 w 22
7.632 z pi |z 3i| = |z+ 1 2i|) pi z) z1, z2 Im(z1) > 0, pi pi
10 pi
2i) w =
1z1+iz22, pi
w,w2,w3
7.633 z1 = 1 2i z2 = 3+ 4i) pi -
z2z1
) , R pi x2 + x+ = 0
z2z1
) z pi |z z1| =
) pi z
) pi iz2z1
pipi pi
) w =iz 4 27i
z z2, pi :
w+ 5iw 5i
I
7.634 pi , 3 = 25
z,w w = +
4i
|iz 3i 4| = 5.) pi - z w) z pi -) |z w|) pi 1 |z 3| 9
36
7.635 z2 2z+ 2 + 3 = 0, R) pi pi- ) , z1, z2 (OMN) =
3, :
) z1, z2) pi z31 = z
32 = 8
) pi z20131 + z20132 = 2
2014
7.636 w pi :
(w+ w)(3 2i) + (w w)(5 i) = 8+ 6i
) w
) z1 =(w2
)50
z pi |z z1| = |z |z1||
) Re(z1) Im(z1)
) pi Re(z) = Im(z)
) z , 0, pi pi-
A =zz+
zz
7.637 z2 2z+ 22 = 0, > 0, pi z1, z2 Im(z1) > 0) z1, z2 pi z1 = iz2) n N, pi z4n+21 + z4n+22 = 0) z41 = 4, ) pi z1, z2
7.638 f (z) =3Re(z) + 4Im(z)
z,
z C) | f (z)| = 3, pi z) Re(z) = Im(z), R, pi :
) | f (z)| =|3+ 4|2 + 1
) | f (z)| 5
7.639 z = i + i
, R) pi z = i) n N, pi:
(1 z)n = (1+ z)n
) pi 1+ z+ z2 + + z99 = 07.640 z2 + 6z + 10 = 0, pi
z1, z2, Im(z1) < 0.) z1, z2 pipi ) w C, |w z1| = |w+ z26i|,:
) pi w
) w pi
) pi (w 2)2014 I7.641 2z2 + 2z+ 1 = 0
) pipi ) z1 Im(z1) > 0, pi- z21 z
31
) pi - z1, z21 , z
31 pi
pi .
7.642 z2 + z = 0, , R z1 z2 = 1+
2i2013
.
) pi < 0) , ) pi Im(z20151 + z
20152 ) = 0
Re(z20151 z20152 ) = 0
) pi|w z1|+ |w z2|, w C) (1 + iu)n =
z1z2, n N, pi
u < R
7.643 z =1
2+ i, R. pi-
:) z+ z = 4zz) pi
z (C) K(14, 0
)
=14
) z w =
2 4i
(C))
12+ i
,1
2 4i
12+ 2010i
7.644 z1 = + i z2 = i, R. pi :
37
) z1 = iz2 |z1| = |z2|) w =
z1z2, :
) 1+ w101 + w102 + w103 = 0) (1+ w)100 = (1 w)100
) z1, z2 ) 4 |z1 3 4i| 6) z3 + w2 = 0 pi ) z3 pi pi
, z903 = 1 z3 +1z3= 1
7.645 z,w pi |z2 i| = 1 |w+ 1+ i| = 2. :) pi - z w) |z+ 2w|) |z 1+ 2i|
7.646 z,w |z w| = |z| |w| = 1
) pi Re(zw) =12
) > 0 pi w = (1 2i)z
) =12, pi :
) |z| =22
) z,w
7.647 z,w C, w , 0 |z + w| = |z w|. pi :)
zw
) z , w, izz w
+ wz+ w 1
) w = 1+ i, z y = x
7.648 z,w C, |z| = |w| = 2.) pi z =
2z
) pi |z+w+zw2| = |z+wzw+2|) pi :
z+ w+ zw 2 = 0 z+ w zw+ 2 = 0) z+ w = 2, z,w
7.649 f (z) =z i|z|z+ i|z|
) z C pi f (z)) [ f (1)]2014) pi f (z) R z I
7.650 f (z) =z(z 2i)|z 2|
, z C{2}. :) f (1 i)) pi z, :
) f (z) R ) | f (z)| = |z|
7.651 z2+z+ = 0, , R, pi z1 =
5+ 33i
1 23i
) pi - z1) pi z31 = 8) , ) , (2,0), pi pi
7.652 f (z) =iz 2+ 4i
z i, z , i
u = z i w = f (z) i.) pi uw = 3+ 4i) z2 = uw) pi z, f (z) R
7.653 z,w :
|z 12 6i| = |z 4| 2|w+ i|2 = |w i|2 + 17
) pi z pi w) |z w|
7.654 z,w -pi :
zw+
wz= 1
pi :) z3 + w3 = 0 ) |z| = |w|) z,w , pi
)( zw
)20+
(wz
)20= 1
7.655 z = x + yi, x, y R :
f (z) =(2+
32i)z
52zi
38
) Re( f (z)) Im( f (z))) pi f (z)) pi | f (z)| = |x 2y|
5
) | f (z)| =5, pi
z
7.656 z,w C, pi -:
|zi 12| = |3z 4i| w =z 8z 2
) z w
) pi w2 + 4
w R
) pi |w 3+ 4i|
7.657 z , 1 -
w =2+ i z1 z
.
) z = 2, pi w2012 R) pi
w 2w+ i
= z
) z -, pi w) w pi, - pi z
7.658 f (z) =5 12i 2iz
z 2i, z , 2i
) f (z) = z f (z) = 5+2i) f (z) R, pi z
7.659 f (n) = 1+ i+ i2+ + in, n N) pi | f (n)| 2) n f (n) = 0
7.660 f (z) = (z i)(z+ 1), z C) pi
f (i)f (1)
2015) pi f (zi) = f (z), z C) pi z, Re( f (z)) = Re(z+ zi)
7.661 f (z) = |z + i| z pi f (z) = 2 f (z)) pi z) pi |3z 5i| = 4) z pi - ) pi w = z 3
7.662 z , 2i f (z) =z3 + 8iz 2i
) pi f (z) = z2 + 2iz 4) f (1+ i)) :
) f (z) = |z|2 + 2iz 4 ) f (z) = 2iz+ z 6) |z| = 1, pi - f (z) = 7
7.663 z C f (z) = 2z 1z2
) f(1z
)= 2
) pi f (z) R z R Re(z) = |z|2) pi z, pi f (z) R) pi - pi pi pi -.
7.664 f (n) = inz, n N z C) pi :
f (4n) + f (4n+ 1) + f (4n+ 2) + f (4n+ 3) = 0
) f (8) f (3) (1,1) = 4, pi z) z = 1+ i
3, pi pi
0, z, f (4n + 3) , pi pi
7.665 f (z) =z+ 2iz 2i
, z C {2i}) pi | f (z)| = 1, z , 2i) f (z) = f (z), pi z I) z0 = 2i3 2 f (i), :
) pi w,
|w+ z0| =
i f (z0)z01+ i3
) w pi
7.666 f (z) = z2 + 2z+ 3, z C) x, y R, f (x 2yi) = 2) f (z) = 0) , R, f (z) = z+ , 1 2i) z1 f (z) = 2z + 2, Im(z1) < 0, pi pi-
A = z531 +1z741
39
7.667 z2 2t z+ 5 4t = 0, t [0, pi]) z1, z2 pipi ) pi z1, z2) pi|z1 z2| |z1 + z2|
7.668 f (z) =2z i zz 1
, z , 1. pi :) f (1 i) = 3+ 3i ) ( f (1 i))2016 R) f (1 i), f (1 + i)
7.669 z C {1, 0} f (z) =
z 1z+ 1
) pi ( f (i))90
) pi f (z) + f(1z
)= 0
) f (z) = f (z), pi z R) pi z, pi | f (z)| = | f (2i)|
7.670 z1, z2,w :
|z1| = |z2| = 2, z21 + z22 , 0 w =
2z1z2z21 + z
22
) pi : ) w R ) z1z2 + z2z1
2) w = 2, pi pi z1, z2 pi
7.671 |2w+ 1| 4w = 1+ 8i) w pipi ) pi z, pi :
zz+ 4Re(zw) + 2 = i56 i72
) z1, z2 -pi pi , - |z1 z2|
7.672 3z2+ z+ = 0, , R, z1 = 1+ i z2. pi :) = 6 = 6) z21 + z
22 = 0 z
20161 + z
20162 = 2
1009
7.673 z , 0, pi z3 + |z| = 0 (1)) pi z) (1)
) z1, z2 pi (1), pi pi A =z20131 + z
20132
7.674 z,w |z| = 1 w = z2+ iz) |w| Im(z)) pi u = w iz) z |w|
7.675 z, pi |z|2 + z(1+ i) + z(1 i) + 1 = 0 (1)) pi z) pi
2 1 |z| 2+ 1
) z1, z2 pi pi (1), pi|z1 z2|
7.676 z = x + yi, x, y R z , 2i. w =
z 2z+ 2i
, :
) pi Re(w) =x2 + y2 2x+ 2yx2 + (y+ 2)2
Im(w) =2x+ 2y+ 4x2 + (y+ 2)2
) w pi, :
) pi z
) |z|
) z ) w , :
) pi z
) |z|
) z1, z2 -
7.677 z,w w =z 2iz 2
) z = 3 i, pi w100) w , - pi z) w Re(z) , Im(z), |z|) Re(z) = Im(z), w
7.678 z pi :
|z 2 3i|2 + |i z 3 6i|2 = 16
) pi z) |(4 3i)z 25|)
40
z) z0 pi pi - w1,w2 :
w2 + 2(|z0| 5)w+ 9 = 0
) pi w1,w2 < R
) w1,w2
) pi w1 3iw1 + 3i
I
7.679 z C, z , 1 :
f (z) =2+ i z1 z
) f (2)) pi w = ( f (2))2014 pi
) pi f (z) 2f (z) + i
= |z|) |z| = 1, pi M - f (z), pi .
7.680 z , i
f (z) =iz 1z i
) pi f ( f (z)) = z) f (z) = f (z), pi z R) | f (z)| = 2, pi z) | f (z1)| = | f (z2)| = 2, pi
|z1 z2| 837.681 z = (2+ 1)+ (3 2)i,
R) pi z) z pi ) pi z pi ) pi 3 |z 4+ i| 7
7.682 z pi |z+ 8| = 2|2z+ 1| (1)) pi |z| = 2) - 2z+ 1) z1, z2 pi pi (1), z1 , z2, pi
w =z1 + z
2
(z1 + z2) pi
7.683 pi , > > 0 z2 + 2z+ = 0, pi z1 z2.) z1 z2) z1 z2) pi w =
z1 + z2z1 z2
) u (u z1) = (u z2), N, pi u R
7.684 z,w pi :
|z i| = |2iz 1| 1+ 1+ iw
= 1) pi (C1) z) pi (C2) w) , (C1) (C2), pi |
OM|2 + |
ON|2 = 4
7.685 z pi |z 2i| = |z|) pi z) pi pipi z, z1, z2 pi 2) pi z41 + z
42 = 16
7.686 f (z) =z 3iz+ 1
, z , 1) z0 f (z) = 2 2i, pi pi A = (z20160 + i)
4032
) pi z, :
| f (z)| =1 3i2
300) pi pi | f (z) 1| |z+ 1|) z (-1,0)
10, -
pi f (z)
7.687 f (z) =z3 iz+ i
, z , i
) pi f (z) = z2 iz 1) f (z) = (2 i)z 21 3+ 4i, :
) , R) n N, n < 100,
pi (+ i)n = (5+ 3i)n
7.688 z,w pi :
41
|2w+ 1| 4w = 1+ 8i z = w( i), R
) w) pi z) |z|) z1, z2 pipi pi, z1 , z2 |z1| = |z2|, pi- z1 + z2
7.689 z1, z2 C z1 + z2 = 1 z31 + z32 = 2) z1, z2) pi - 0, z1 z2) pi z1001 + z
1002 = 1
7.690 z pi :
Re(2z+
8z
)= 4Re(z)
) pi z) Re(z) , 0, pi
w = 2z+8z
pi 8 w 8
7.691 z2 + z + = 0 (1) R, pi z1 z2) w =
z1 + z2 + i1+ z1z2i
, :
) pi |w| = 1
) pi w2012) (1) + i,
7.692 z , 0 w = z +
z,
R. z1 = 1 i3
w = , (1) R, :) , (1)) pi (1) z3 + 8 = 0) n pi (iz1)3n = i(z1)3n) w R, pi z
7.693 z,w pi :
z+ |z+ 2 i| = 6 3i zw = 6 8i
) pi z = 1 3i) pi |iz w|) pi z2014 + w2014 = 0) u =
w+ zw z
, R. pi u2n R, n N
7.694 f (z) =z3 iz+ i
, z , i.
) pi f (z) = z2 iz 1) , R pi f (z) = ( i)z+ i2018 3+ 4i) w =
iz+ 1z+ i
pi, :
) pi |z|
) pi | f (z)| 3
) pi f (iz)f (i z)
= 17.695 f (z) =
z+ 7iz 1
, z , 1) pi f (3 3i) ( f (3 3i))2011) w |w 13+ 3i| = 11+ i) Re( f (z)) = Im( f (z)), :
) pi z
) pi- |zw0|, pi w0 ()
7.696 z2 z + = 0, , R, pi z1 = 2+ i z2.) , ) pi z20141 + z
20142 R
) , ,
z1, z2, z3 =z1z2+
17+ i5
, pi ) |w z1| = |w z1|, pi w R) pi w, pi |w z2|+ |w z2| = 10
7.697 z(t) =1
1+ 2ti, t R.
pi :) z(t) + z(t) = 2z(t) z(t)) z(t) (C), pi ) t , 0, z(t)
z(
14t
) pipi
(C)
42
7.698 z() =
1+ i, R
) pi z() (C)
K(0,
12
) =
12
) z() pi - ) pi z()
z(1
) pi-
pi (C)) pi z(1), z(1) z(2012)
7.699 z1, z2 C z1 , z2 z1 , z2.
w =z1 + z2z1 z2
) pi : |w| = 1 z1z2 I
) |z1| = |z2|, pi w I (z1 z2z1 + z2
)2014< 0
7.700 f (z) = (z + i)(z + i), piz C) z1 = 1+ i, z2 = f (z1) z3 = f ( f (z1))) , , z1, z2, z3 -, pi -) pi z pi-pi: ) f (z) R ) f (z) I
7.701 z C, Re(1z
)=
14. pi :
) Im(z) = 1, Re(z)) pi z pipi) |z|
) z1, z2 C, Re(1z1
)= Re
(1z2
)=
14,
|z1 z2|
7.702 z , i w =z+ iz i
. w I,:) pi z - = 1
) z1, z2, z3 pi pipi - z. pi :
) |z1 + 2z2 + 3z3| = |3z1z2 + z2z3 + 2z3z1|
) |z1 z2|
2= |z3 z1| = 1, |z2 z3| =
3
7.703 z2 2z+ 2 + 3 = 0, R) pi pi ) , (OMN) =
3, :
) z1, z2) pi z31 = z
32 = 8
) pi z20131 + z20132 = 2
2014
7.704 ) pi (z+ 3+ i)(z 4+ 2i)) z2 (1 3i)z = 14 2i) z1, z2 pipi , Re(z1) > 0 , , z1, z2 z3 = 3+ i , :
) pi -
) pi M(z), pi :
MA2 + 2MB2 = 2M2 + 30
7.705 z2 z + 9 = 0, R pi z1, z2 < R.) pi :
) 6 < < 6 ) (z171 + z172 ) R
) z1, z2) ,
z1z2+
z2z1= 2
) = 0 Im(z1) > 0, pi w pi :
|w z1| = 4+ |w z2|
7.706 z , 0
w =1z
.) pi z ) pi z4 < 0
) z2 1z
= 2, :) pi |z| = 1
) pi Im((z+ i)5
z5 + i
)= 0
) u =3z
4zi
43
7.707 z1, z2, z3 :
z3 + (1 )z2 + (1 )z = 0, R
z1, z2, z3 , pi :
) = 12
) n N, (zn1 + zn2 + zn3) R
7.708 z,w pi :
|w| = 2 z = w+4iw
) pi z) z1, z2, z3 pi pipi - z, z1 + z2 + z3 , 0, pi :
)z18=
1z1,z28=
1z2
z38=
1z3
) (z1 + z2)(1z1+
1z2
) 4
)z1z2 + z2z3 + z3z1z1 + z2 + z3
= 22v)
z1 z2z3
+z2 z3
z1+
z3 z1z2
I
()
7.709 z,w pi - :
z 2i+ 4|z 2i|z+ 2i
=5
z+ 2i |iw 2 2i| = |i w|
) pi z w) pi z) |z w| |z w|) u v pi pi w z pi |u|2 + |v|2 = 10
7.710 w1,w2 |w1| = |w2| =2
Re(w1w2) > 0 z1, z2 z2 |w1 w2|z+ 1 = 0. pi :) z1, z2 < R) |z1| = |z2| = 1) (z1 z2)2 = |w1 w2|2 4
7.711 z,w pi :
|z 40| = |z 30i| |w 3 4i| = 1
) pi |z| , |w|) |z+ w|) |z| |w|) pi u = 6+ 8i z
7.712 z,w pi :
|z 1| = |z i| w =i 1z i
) pi - z w) z pi |w2 2i| = 2) n |z| 1+ in = (1463n2)i, pi zn3 R
7.713 z,w -pi :
2014|z| z+ 2014|w| w = 2014|z+w| (z+ w)
pi :) |z| = |w| = |z+ w|
zw R
) ,, - z,w, z+w , , ,
ABO
7.714 z,w z , 0 pi w2 2|z|w+ 2|z|2 = 0. pi:) w < R ) |w|2 = 2|z|2 ) w2 + w2 = 0) w2n + w2n = 0, n pi
7.715 f (z) =z iz
, z C) f (z) R, :
) pi z I) zn = 1, n N, pi z = i
) f (z) f (z) = 5, :) pi z
) | f (z) 1|) z2 + 4 f 2(z) = 0
7.716 z1, z2 z4=2z+ 1
) z1 z2
) pi (z1z2
)2012+
(z2z1
)2014= 0
) w C |w z1|2 + |w z2|2 = 16, :
44
) pi w pi |w 2| = 2
) w1,w2 pi pipi - w |w1 w2| = 4, pi |w1 + w2| = 4
) w1,w2,w3 pi pipi w, pi :w1 + w2 + w3 64
= 1w1 2 + 1w2 2 + 1w3 2
7.717 z (-2,1) = 1 w = 4z+ 8) pi iz1+2i) pi w) pi 0 |w| 8) pi w w =
|w|2
4 i
) |3w216|
7.718 z C {i} w = z2 2iz 1z2 + 1
. w , :) pi w =
z iz+ i
) pi |z| = 1) z1, z2 pi pipi - z, pi
v =(z1 + z2)7
z71 + z72
pi) w R :
wi+ w11 = + i
7.719 z C (z 4 + 2i)(z 4 6i) I w pi :
w 2z = 4 i2018 (1+ i)32
(8i)5
) pi w = 2z+ 4+ 2i) pi - z w) |z w|) , z w u , pi 1 |u 8+ 5i| 3
7.720 z pi -:
|z 2|2 + z(2+ 3i) + z(2 3i) + 1 = 0
) pi z) pi 1 |z| 5) w =
8z 3i
z
) pi w
) pi (z w)2 0) pi z = w Im(z) = 3iv) , z,w z , w,
pi x x
v) u , , pi :
|u z| |u w|+ |u 3i|2 = 4
7.721 u1, u2 pi- u1 + u2 = 3 + 9i y = x+7 y = 5 . - z1, z2 w z51 = u1, z
52 = u2 w =
u1u2
) u1 u2) pi w pi - ) pi w2 < R) w2 z2 + z + = 0, pi = 1
7.722 ) z,w ,. pi , zw R) pi z pi -
z i,1
z 2
) z0 pi- , - n zn0 = 5n 31) u C |u| = 1, z (), |u 2z|
7.723 z1, z2 z +1z= 1
w pi - Re(w2) = ww 8Re(w)
45
) z1, z2 pi z31 = z
32 = 1
) pi w pi y2 = 4x) pi z1, z2 pipi pi pi) u
2|u+ z1z2|+ z111 + z112 = 0
pi 2|w u| 17.724 z pi -
= 1
w =2z+ iiz 2
) pi : ) |w| = 1 ) |z+ w| 1) w < R, R w+
w R
) z1, z2 pi pipi - z |z1z2| =
3, pi |z1+z2| = 1
) pi |w+ 4|+ |3iw 1| 57.725 z,w, u pi-
:
w =z 2z 2i
, z , 2i |2u 3 3i| =2
) w = + i, , R, pi :
w2 R = 0 = 0) w2 R, pi z) pi u) pi z = u
7.726 z pi - :
|z+ 2i| = 1+ 2Im(z)
) pi z) pi |z+ 2i| |z 2i| = 2) pi |z2 + 4| = 2 + |z|2 - pi w = z2 + 2) z1, z2 pi pipi - z |z1z2| =
3, pi |z1+z2| 1
7.727 z pi zn0 = 3+4i z
n+10 = 2+11i, n N
z pi
w =zn
z z0 + 2
) z0 n) pi z) u = w z i
) z < I, pi u) u1, u2, u3 pi pipi
u, pi :
u1 + u2 + 3u3 , 0 3+ 4iu1 + u2 + 3u3
17.728 f (z) = z2+ z+ 3,
z C , R , 0, pi
i2. pi :
w =f (z)
2iz 1
) pi = 4 = 0) pi w = 2iz 1) |z 2| = 1, :
) pi z) pi 2 |z w| 8
7.729 z,w, u pi- :
2z+ |z+ 1+ 2i| = 1+ 2i w+z10
32u = 0
pi :) z = 1+ i) w, u pi pi ) |w u| =
2|w|
) w, u
7.730 z pi - :
(3z 2)5 (2z 3)5 = 0
pi :) |z| = 1
) w =(z i)9
z9 i pi
) |3z 1| = |z 3|) |3z 1|+ |2z2 z+ 6| 1) u = z
4z
pi 3 |u| 5
46
7.731 z,w pi :
z 2i|z 2i|
=
3
z+ 2i w =
5( i)3 4i
, R
) pi z) pi
u = z 2i+3
z 2i
x x) pi w |w|) pi 5|z w| 11 53
7.732 z3+z2+ 2 = 0, pi z0 = 1 + i w pi :
2Re(w) , 1 |w2 + w+ |w|2| = |i w|
) pi = 1 = 1) pi w R) u =
w 7w+ 7
, pi :
) |u| = 1 ) v =u2 + iu 1u2 iu 1
, v R
) |u2 + u 1| 57.733 z,w pi
:
z =2 i
, R w = z 1
) pi z) pi 2|z|2 3|z| 2 0) w1,w2 pi pipi w :
u =w1w2
+w2w1
pi : ) u R ) u = 2Re(w1w2)) z1, z2 pi pipi - z |z1 z2| = 2, |z1 + z2|
7.734 z |z i| = 1+ Im(z) :
f (w) =w2 + 1w+ i
, w C {i} | f (w)| = 1
) pi - z w) z0 pi :
|z0 i| = 1+ Im(z0) |iz0 + 1| = 1
) u pi (1+ i)u+ (1 i)u = 4
) pi u
) pi |z u| 22
7.735 z2 z + 2 = 0, R, pi z1, z2 < R) z1, z2 ) pi :
z1z2+
z2z1= 2
) = 0, :
) pi w pi :
|w z1| |w z2| = 2, Im(z1) < 0
) pi |w 1| 62
) pi w2
7.736 z pi :
(2z i)7 (iz+ 2)7 = 0
pi :) |z| = 1) w = 2z1 (-1,0) = 2
) u =(z
1z
)15, u I |Im(u)| 215
) |z2 2z+ 3| = |3z2 2z+ 1|) |2z 1|+ |z2 2z+ 3| 3
7.737 z,w pi :
|z| = 1, |w| = 3 |z+ w| = 4
47
) pi |z w| = 2
) pi (z+ w+ 1) (1z+
9w+ 1
) 25
) pi u pi 2Re(u2) = |z w|) pi u2 x = 1
7.738 z,w pi :
|iz+ 2| = |z+ 4| |4iw+ 1| 1 = 4Im(w)
) pi - z w) |z34i| z pi pipi
) pi |z w| 25
5
7.739 a, b, c pi pipi - |a| = |b| = |c| = > 0.) pi
w =(1
ac
) (1
ba
) (1
cb
) ) pi |ab+ bc+ ca| = |a+ b+ c|
) z = b+(1)c, R. pi :
) b, c, z
) |za| 12
|ab| |ac|
7.740 z,w C R z2 + w2 = zw.) pi |z| = |w| zw R) , z,w , pi :
) 2 < < 2
)
) AOB = 450, =2
7.741 z,w pi - , 0 < , 1, pi :
|z 2i| = |z i| z(w 2i) = 22 + wi
) pi pi - z (C1) ) pi w (C2) pi (C1), pi ) pi 2|z+ w+ 2i| = |w+ 4z zwi|
) pi 2z w2z+ w
I) , pi (C1) w pi (C2), pi -
13 MA
MB 3
7.742 z pi 2z4+z = z4+2z.) pi : ) z4 + z4 = z+ z ) z4 = z) 2z4 + z = z4 + 2z) z1, z2, z3 pi- , pi pi z1, z2, z3
7.743 z pi - :
32 + i22013 z6
= 64) pi |z| = 2) z1, z2 pi pi , pi :
)z1 + z2i4 z1z2i
I ) (z1 + z2)(1z1+
1z2
) 4
) w C, 2w = z 6 + 8i, pi:
) w , pi
) Re(w) Im(w) < 0) u C, z2 = 3+ 2iuz, pi :
) u , E(
3, 0) E (
3, 0)
)14 |u| 7
47.744 z pi
2z i z = 3) z
) pi (z+ izz 1
)2016= 1
) w pi
|w+ z| =
52 +
112
i z
48
) pi w
) pi 5 |w| 35
) pi pi - w1,w2 pi pipi pi
7.745 z |z 2i| = |2zi + 1|. pi :) z pi
) z5 + 1z5 1
) 1 Im(z) 1)
z 1z + 2i+ z 1z 2i
= 4) w C, zw , 1,
z+ w1+ zw = 1
7.746 w |z|2 + 2Im(wz) + 4 = 0, pi pi z, z0) z0 |w|) pi z20140 + w
2014 = 0) pi u pi |u + 2i| |u 2i| = |z0| u pi
7.747 z pi :
|z+ |z||2 + |z |z||2 + 2z2 = 0
) pi z I) z pi pipi :
1+ z+ z2 + + z9 = 1 i
) pi w pi :
w wi =iz z
) w pi -, pi x2 + |w| 2x, x R
7.748 a, b, c pi :
|a| = |b| = |c| = 1 a+ b+ c = 1
) z = (a+ b)(b+ c)(c+ a), pi :
) z R ) (iz+ 2013i3)2 0) pi pi:
A =1a+
1b+