ΕΛΑΧΙΣΤΟ - ΜΕΓΙΣΤΟ ΜΙΓΑΔΙΚΟΥ
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Μέγιστη και ελάχιστη τι μ ή μ έτρου Μιγαδικοί αριθ μ οί Λυμένες ασκήσεις 1 - 25
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ΕΛΑΧΙΣΤΟ - ΜΕΓΙΣΤΟ ΜΙΓΑΔΙΚΟΥ
Transcript of ΕΛΑΧΙΣΤΟ - ΜΕΓΙΣΤΟ ΜΙΓΑΔΙΚΟΥ
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1 - 25
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7 .
((((
(
1(-(25
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( 1 - 9) pipi * http://blogs.sch.gr/gkaraferis/4les/2012/08/Migad4koi_arithmoi_Karaferis.pdf* http://lisari.blogspot.gr/p/blog-page_20.html
( 10 - 11) pipi * http://bouboulis.mysch.gr/education/C_Lyk_MK/C_MK-01A.pdf
( 12 - 23) pipi * http://blogs.sch.gr/smichailog/4les/2012/11/1..pdf* http://blogs.sch.gr/smichailog/
( 24 - 25) ' 1. . , .4 pi, 2010.
7. .
-
1. z = + 2 + (33)i , . ) pi z.
) pi w, pi w = z + (1+ i) ) z, pi pi .
) z = x +yi (x,y ) , pi
x = + 2y = 33
, pi' pi ' pi .
3x =36y = 33
(+)
3x +y =9 3x y9 = 0 , pi pi , pi
pi z.
) w = z + (1+ i) = + 2 + (33)i +1+ i = (+ 3)+ (33 +1)i w = (+ 3)+ (32)i .
w = x +yi (x,y ) , pi
x = + 3y = 32
, pi' pi ' pi .
3x =39y = 32
(+)
3x +y =11 3x y11 = 0 , pi pi ,
pi pi w.
) 1 , () - pi pi . : 3x y9 = 0 pi z.
() pi pi , , pi pi . pi pi- () .
, . , pi pi ,
:y = x .
, pi
=1
31
=1
=
13
.
, () , .
:y =
13x , pi pi pi
7. .
-
:y =13x
: 3x y9 = 0
3x +
x39 = 0 9x + x 27 = 0 10x = 27 x =
2710
= x
.
pi pipi y =
132710
y =910
= y
, pi
2710
, 910
, -
pi
zmin
=2710
910
i .
2. () , pi z1 ,z2 pipi, pi z1 + z2 = 12i z1z2 = 3 4i . w, pi () .
, pi pi , ! , , , pi pi (), pi - .
pi
z1
+ z2
= 12i
z1z
2= 3 4i
, pi, 2z1 = 46i z1 = 23i , pi
, , (2,3) .
pi z1 + z2 = 12i z2 = 12iz1 = 12i2 + 3i z2 =1+ i , pi
, , (1,1) .
, () :yy = (x x) :y1 = (x +1) .
=yy
xx
=1+ 312
=43
=43
, pi
:y1 =
43(x +1) : 3y3 =4x 4
: 4x + 3y +1 = 0 .
() pi' pi . , -. pi pi (), pi . , , pi pi - () .
:y = x , pi .
, pi
=1
=1
43
=1
=34
.
7. .
-
:y =
34
x , pi pi
:y =34
x
: 4x + 3y +1 = 0
4x + 3
3x4
+1 = 0 16x + 9x + 4 = 0 25x =4 x =425
= x
.
pi y =
34425
y =325
= y
, pi
425
, 325
, pi -
wmin
=425
325
i .
3. z, w , pi | (1 i)z 2 | = 2 , |w + 2i | = |w2 + 4i | . ) pi z.
) pi w.
) | z w | .
) | (1 i)z 2 | = 2 (1 i) z
21 i
= 2 | 1 i | z 2(1+ i)
(1 i)(1+ i)= 2
12 +12 z
2(1+ i)12 +12
= 2 2 | z (1+ i) | = 22
| z 1 i | = 2 ,
pi pi (1, 1) 2 .
|z z0 | = > 0 , pi pi ( -) z pipi , ( pi pi- ).
) |w + 2i | = |w2 + 4i | pi , pi (0,2) , (2, 4) .
w = x +yi (x,y ) , | x +yi + 2i | = | x +yi2 + 4i |
| x + (y + 2)i |2= | (x 2)+ (y + 4)i |2 x 2 + (y + 2)2 = (x 2)2 + (y + 4)2
x2 +y 2 + 4 + 4y = x 2 + 4 4x +y 2 +16 + 8y 4y =4x +16 + 8y
4x 4y16 = 0 x y 4 = 0 , pi pi pi.
7. .
-
) | z w | pi pi pi (), pi pi .
pi pi : x y 4 = 0 , - , pi
| z w |min= () = ()() = d(,) =
pi pi .
=| 11 4 |
12 + (1)2 2 =
4
2 2 =
4 22 2 =
= 2 2 2 | z w |
min= 2 .
) , () !) , ' pi pipi, , ! , pi , pi. pipi, ' pi pi ! pi ! pi!
4. z, w , pi | z + 2 | = 1 , |w 4i | = 2 . - | z w | .
| z + 2 | = 1 pi (2,0) 1 = 1 , |w 4i | = 2 pi- (0,4) 2 = 2 .
| z w | pi pi pi . , , pi - pi , pi :
) | z w |min= () = ()()() =
= ()12 = ()12 | z w |min= ()3 .
() = (x
x
)2 + (y
y
)2 =
= (0 + 2)2 + (40)2 = 4 +16 = 20 = 2 5 ,
pi
| z w |min
= 2 53 .
) | z w |max= () = ()+ ()+ () =
= 21 + | z w |min +22 = 2 1+ 2 53 + 2 2
| z w |
max= 2 5 + 3 .
pi pi .
7. .
-
5. pi z, pipi, pi
1| z 3i |
+1
| z + 3i |=
10| z 2 + 9 |
.
z1 ,z2 , pi pipi pi - pi , | z1z2 | .
z2 + 9 = (z 3i)(z + 3i) .
pi | z2 + 9 | 0 z 2 + 9 0 (z 3i)(z + 3i) 0 z 3i 0 z + 3i 0
z 3i z 3i .
pi
1| z 3i |
+1
| z + 3i |=
10| (z 3i)(z + 3i) |
1| z 3i |
+1
| z + 3i |=
10| z 3i | | z + 3i |
| z + 3i | + | z 3i | = 10 .
, , , , z ,3,3 , pi () + () = 10, pi z, pi ' , 10, pi E 2 = 10.
2 = 10 = 5 .
E 2 = 2 + 2 , pi' pi
52 = 2 + 32 2 = 259 = 16 = 4 .
| z1z2 | pi pi E. pi
pi pipi | z1z2 |max= ( ') = 2 = 10
| z1 z2 |min= ( ') = 2 = 2 4 = 8 .
pi pi 7. pi ? pi pipi pi .
7. .
-
6. z = 2x +yi , x ,y , | z | = 2 . pi M(x, y) - pipi. 1 ,2 , pi , pipi -
pi w1 ,w2 1 ,2 ,
|w1w2 | .
pi | z | = 2 | z |2= 4 (2x)2 +y 2 = 4
4x 2 +y 2 = 4
4x 2
4+
y 2
4= 1 x 2 +
y 2
4= 1 , pi pi-
@, y'y, pi pi pi .
pipi @ 2 = 1 = 1 ,
2 = 4 = 2 .
1 ,2 @, |w1w2 | pi pi-
. pi , :
) |w1 w2 |min= ( ') = 2 = 2 1 = 2 .
) |w1 w2 |max= ( ') = 2 = 2 2 = 4 .
7. pi z, pi | z 3 | + | z + 3 | = 10 . , | z1z2 | , pi z1 ,z2 , pi , pi , pi-
pi pi.
, , ' , ,
z ,3,3 , - () + () = 10, pi pi 10, pi @ 2 = 10.
pipi @ 2 = 10 = 5 ,
= 3 ( (3, 0)), pi
2 = 2 + 2 5
2 = 2 + 32
2 = 259 = 16 = 4 .
| z1z2 | , z1 ,z2 @ pi , pi-
pi , pi :
) | z1 z2 |min= ( ') = 2 = 2 4 = 8 .
) | z1z2 |max= ( ') = 2 = 10 .
7. .
-
8. f (z) =
z +1z
, z = x +yi (x,y ) , z 0 .
) f(z) + i ( ).
) pi : f (z) z .
) f (z) f (z ) = 2 , pi pi z ,
(1, 0) = 2 .
) (), | f (z)1 | .
) f (z) =
z +1z
=z (z +1)
zz=
zz + z| z |2
=| z |2 +z
| z |2=
x 2 +y 2 + x yix 2 +y 2
f (z) =
x 2 +y 2 + xx 2 +y 2
y
x 2 +y 2i .
) : f (z) f (z) = f (z)
z +1z
=z +1
z z(z +1) = z (z +1) zz + z = zz + z
z = z z .
) pi f (z) f (z ) = 2 z +1
zz +1
z= 2
(z +1)(z +1)zz
= 2
zz + z + z +1 = 2zz zz (z + z )1 = 0 | z |2 2Re(z)1 = 0
x 2 +y 22x 1 = 0 (1)
(2)2 + 02 4 (1) = 4 + 4 = 8 > 0 , pi (1) pi
22
,02
(1,0)
=
82
=2 22 = 2 .
z = 0 (0, 0), pi .
x = 0 , y = 0 (1) 02 + 02 2 01 =1 0 , pi
.
pi z (1, 0) 2 .
) | f (z)1 | =
z +1z
1 =z +1z
z=
=
1z | f (z)1 | =
1| z |
, pi:
| f (z)1 | , |z| .
| f (z)1 | , |z| .
7. .
-
. pi pipi pi;
, | f (z)1 | | z | , , pi - , pi pipi pi pi . , pipi
| z |min= () , | z |max= () .
y = 0 (1), x22x 1 = 0
x = 1+ 2 = x x = 1 2 = x , pi pipi pi . :
, , pi , pi pi. , , pi .
) | z |min= () = | 1 2 | = 21 , 1 2 < 0 2 >1 2 >1 .
pi
| f (z)1 |max
= 21 .
) | z |max= () = 1+ 2 , pi | f (z)1 |
min= 1+ 2 .
9. w (0,) , w2 2 w +1 = 0 . ) .
) w1 ,w2 , :
. |w1 | , |w2 | .
. pi w1 ,w2 .
. |w1w2 | .
) , pi
= (2)2 4 1 1 = 42 4 =4(12)=42 .
(0,) , > 0 2 > 0 , pi < 0 , pi -
, w
1,2=
2 42 i2
=22 i
2=
2( i)2
w
1= + i , w
2= i .
) . |w
1| = | + i | = 2 + 2 = 1 |w
1| = 1 .
pi, |w
2| = |w
1| = |w
1| |w
2| = 1 .
7. .
-
. w1 = + i , w1 = x +yi (x,y ) , pi
x = y =
.
pi 2 + 2 = 1 , x
2 +y 2 = 1 , pi pi - , pi pi w1 .
pi w2 w1 , , pi x'x, w1 .
pi x'x, pi w2 .
. w1 ,w2 , |w1w2 | pi pi ., ,
|w
1w
2|max
= 2 1 |w1w
2|max
= 2 .
pi , pi , pi pi , pi !
10. z, pipi, pi x 2 +y 2 = 9 . - w = 512i . | z +w | .
x2 +y 2 = 9 pi (0, 0)
3, | z | = 3 .
| z +w | = | z + 512i | pi pi pi (5,12) . , , (), pi :
) | z +w |min= () = ()() = ()3 , pi
() = x
2 +y
2 = (5)2 +122 = 25 +144 =
= 169 = 13 ,
pi | z +w |
min= 133 | z +w |
min= 10 .
) | z +w |max= () = ()+ () = | z +w |min +2 3 =
= 10 + 2 3 | z +w |
max= 16 .
7. .
-
11. pi | z 3 | , z | z + 4i | 2 . | z + 4 | 2 pi
(0, 4) 2 ( , pi-).
(3, 0) 3, | z 3 | pi pi pi , pi , , pi' pi :
) | z 3 |min= () = ()() = ()2 ,
() = (x
x
)2 + (y
y
)2 =
= (03)2 + (40)2 = 9 +16 = 25 = 5 ,
pi | z 3 |
min= 52 | z 3 |
min= 3 .
) | z 3 |
max= () = ()+ () = 3 + 2 3 | z 3 |
max= 9 .
12. z, w, pi zw = 1 i . w (0, 1) 1, :
) , pi , , z. ) |z|.
) w ,
|w i | = 1 (1)
z = 0 , pi zw = 1 i 0 w = 1 i , pi pi.
z 0 , pi pi zw = 1 i w =
1 iz
, pi (1)
1 iz i = 1
1 i izz
= 1| 1 i iz |
| z |= 1 | 1 i iz | = | z |
i
1i1z
= | z | | i | ii2
1z = | z | |z 1 i | = | z |
| z +1+ i | = | z | , pi pi , pi
(1,1) , (0,0) .
z = x +yi (x,y ) , | x +yi +1+ i | = | x +yi |
| (x +1)+ (y +1)i |2= | x +yi |2 (x +1)2 + (y +1)2 = x 2 +y 2
x2 +1+ 2x +y 2 +1+ 2y = x 2 +y 2 2x + 2y + 2 = 0 x +y +1 = 0 ,
pi pi .
7. .
-
) 1 , pi pi. : x +y +1 = 0 ( pipi ), pi .
2 , pi pi . pi pi , |z|. pi pi (), pi .
| z |
min= () = d(,) =
| 0 + 0 +1 |
12 +12=
1
2
| z |
min=
22
.
13. z, w, pi zz + 2z + 2z = 5 ,w = (+ 2)i , . ) pi pi z , pi - .
) pi w, pipi, (), pi - .
) pi () .
) z, pi .
) w, pi .
) pi | z |2 +2(z + z )5 = 0 | z |2 +2 2Re(z)5 = 0
| z |2 +4Re(z)5 = 0 .
z = x +yi (x,y ) , x2 +y 2 + 4x 5 = 0 .
42 + 02 4 (5) = 16 + 20 = 36 > 0 , pi pi
42
,02
(2,0)
=
362
=62 = 3 , pi pi z.
) w = x +yi (x,y ) ,
x = y = (+ 2)
, pi' pi ' pi .
x = y =2
(+)
x +y =2 x +y + 2 = 0 , pi pi , pi
pi w.
7. .
-
) o, pipi , pi ! , () , .
, d(,) < d(,) < 3 .
d(,) =
|2 + 0 + 2 |
12 +12=
0
2= 0 < 3 , pi pi .
, d(,) = 0 , , () , , () pi , , . () , pi .
pi , , ' ' pi= pi! pi pi , .pi, () pi : , ' , , - . pi, pi ., ' pi pi pi pi (pi , pi ' , pi pi, pi !).
) (), -pi , pi' pi pipi , - zmax =5i ,
(5,0) .
) : x +y + 2 = 0 ( - ) pi pi -, .
pi pi () = () = 2, pi . , ( ) , ,
x
+ x
2,y
+y
2
022
,2 + 0
2
(1,1) ,
pi wmin =1 i .
7. .
-
14. ) pi , (), z, pi pi | z | = 3 Im(z) 0 .
) z (), pi - w = z + 43i .
) w, pi .
) pi | z | = 3 pipi pi (0, 0) 3. z = x +yi (x,y ) , pi Im(z) 0 y 0 , pi , (), - pipi , pi pi x'x, pi x'x.
) pi w = z + 43i z = w 4 + 3i .pi | z | = 3 |w 4 + 3i | = 3 , pi pi (4,3) 3.
z = + i (, ) , pi Im(z) 0 0 , z ().
w = x +yi (x,y ) , pi w = z + 43i
x +yi = + i + 43i x +yi = (+ 4)+ (3)i ,
pi
x = + 4y = 3
.
pi = y + 3 .
, pi 0 , y + 3 0
y 3 , pi pi w - (4,3) 3, y 3 , pi - . pi , , pi pi y =3 .
) , pipi , pi -, pi pi pi (-, ) . (x 4)
2 + (y + 3)2 = 9 , y 3 .
:y = x , pi .
' , x = 4 , y =3 3 =
4
=
34
,
pi :y =
34
x .
7. .
-
pi
:y =34
x
(x 4)2 + (y + 3)2 = 9 , y 3
(x 4)2 +
34
x + 3
2
= 9 (x 4)2 +3x + 3 4
4
2
= 9 (x 4)2 +3(x 4)
4
2
= 9
(x 4)2 +
9(x 4)2
16= 9 16(x 4)2 + 9(x 4)2 = 9 16 25(x 4)2 = 9 16
(x 4)2 =
9 1625
x 4 =9 1625
x 4 =3 45
x = 4 +125
x = 4
125
.
pi pi! , pi ,
-
15. z, w, pi w =
5 z| z |
, w2 = 3 +i , > 0 .
) pi = 4.
) pi 5z (3 + 4i)z = 0 . ) pi , , z.
) | z 3 + 4i | .
) pi w =
5 z| z |
|w | =5 z
| z | |w | =
5| z || z |
|w | = 5 |w |2= 5
32 +2 = 5 9 +2 = 25 2 = 16
>0
= 4 .
) = 4 w2 = 3 + 4i .
pi w =
5 z| z |
w2 =
5z 2
| z |2 3 + 4i =
5z 2
zz 3 + 4i =
5zz
(3 + 4i)z = 5z 5z (3 + 4i)z = 0 (1)
) z = x +yi (x,y ) , pi (1) 5(x +yi)(3 + 4i)(x yi) = 0
5x + 5yi3x + 3yi 4xi + 4yi2 = 0 (2x 4y)+ (4x + 8y)i = 0 ,
pi
2x 4y = 04x + 8y = 0
x 2y = 0x 2y = 0
x 2y = 0 , pi pi ,
pi pi z.
) | z 3 + 4i | pi pi : x 2y = 0 pi pi, pi
(3, 4) .
pi pi (),
| z 3 + 4i |min= () = d(,) =
=| 32 (4) |
12 + (2)2=
11
5
| z 3 + 4i |
min=
11 55
.
7. .
-
16. z, w, pi | z 2 | = | z + 23i | |w + i | = |w 4 + 4i | .
) pi z w.
) | z w | .
) pi z.pi | z 2 | = | z + 23i | pipi pi z -
, pi (2,0) , (2,3) .
z = x +yi (x,y ) , | x +yi2 | = | x +yi + 23i |
| (x 2)+yi |2= | (x + 2)+ (y3)i |2 (x 2)2 +y 2 = (x + 2)2 + (y3)2
x2 + 4 4x +y 2 = x 2 + 4 + 4x +y 2 + 96y 8x 6y + 9 = 0 ,
pi pi .
pi w.
pi |w + i | = |w 4 + 4i | pipi pi w -
, pi (0,1) , (4, 4) .
w = x +yi (x,y ) , | x +yi + i | = | x +yi 4 + 4i |
| x + (y +1)i |2= | (x 4)+ (y + 4)i |2 x 2 + (y +1)2 = (x 4)2 + (y + 4)2
x2 +y 2 +1+ 2y = x 2 +168x +y 2 +16 + 8y 1+ 2y = 328x + 8y 8x 6y31 = 0 ,
pi pi .
) 1 : 8x 6y + 9 = 0 , 2 : 8x 6y31 = 0 , pi
1
=86
=43
, 2
=86
=43
,
(1),(2) piE.
pi ,
| z w |min= d(1 ,2) .
1: 8x 6y + 9 = 0
1: 6y = 8x + 9
1:y =
86
x +96
2: 8x 6y31 = 0
2: 6y = 8x 31
2:y =
86
x 316
.
| z w |min
=
96
316
1+43
2=
406
1+169
=
203259
=
20353
| z w |min
= 4 .
7. .
-
17. z, Re
1z
=14
, z 0 .
) Im(z) = 2 , Re(z) .
) pi , , z.
) |z|.
) z1 ,z2 , Re
1z
1
= Re1z
2
=14
, | z1z2 | .
) pi Re
1z
=14
2Re
1z
= 2 14
1z
+1z
=12
z + zzz
=12
2Re(z)| z |2
=12 2Re(z) =
| z |2
2 Re(z) =
| z |2
4.
z = x +yi (x,y ) Im(z) = 2 , z = x + 2i , pi pi pi
x =
| x + 2i |2
4 x =
x 2 + 22
4 4x = x 2 + 4 x 2 + 4 4x = 0 (x 2)2 = 0
x 2 = 0 x = 2 , Re(z) = 2 .
) pi 2Re(z) =
| z |2
2 4Re(z) = | z |
2 | z |2 4Re(z) = 0 .
z = x +yi (x,y ) , x2 +y 2 4x = 0 .
(4)2 + 02 4 0 = 16 > 0 , pi pi
42
,02
(2,0)
=
162
=42 = 2 , pi pi.
) pi pipi pi pi , pi- pi , (), pi
| z |max
= 4 .
) pi Re
1z
1
= Re1z
2
=14
pi
z1 ,z2 pi , pi | z1z2 | pi pi .pi , pi
| z
1z
2|max
= 2 2 = 4 .
7. .
-
18. z 2 8 z + 72 + 9 = 0 , , z . ) .
) pi =. ) z3 ,z4 , pi pipi =, pi
| z3z4 | 8 .
) , pi
= (8)2 4 1 (72 + 9) = 64228236 = 36236
=36(1 2) =362 0 .
pipi:
. = 0, = 0, pi pi ,
z
0=
82 z
0= 4 .
pi 2 + 2 = 1 pi
2 + 02 = 1 2 = 1 = 1 =1 .
i) = 1 , pi z0 = 4 1 = 4 .
ii) =1 , pi z0 = 4 (1) = 4 .
. < 0, 2 > 0 0 , ,
z
1,2=
86 i2
= 43 i z1
= 4 + 3 i , z2
= 43 i .
) z1 = 4 + 3 i , z1 = x +yi (x,y ) ,
x = 4y = 3
=x4
=y3
.
pi 2 + 2 = 1
x4
2
+y3
2
= 1x 2
16+
y 2
9= 1 , pi pi-
= ( x'x).
pi z2 z1 , z1 pi x'x.
= pi x'x , pi z2
=.
7. .
-
) z3 ,z4 3, | z3z4 | pi pi . 3, pi - , | z3z4 | 2 .
pi pipi 3 2 = 16 = 4 , pi
| z
3z
4| 8 .
19. z, w, pi z = (+1)+
2234
i , , |w | = 1 .
) pi , , , z, w . ) z, pi .
) z = 3 + 4i , :
. w, pi | z +w | = 6 .
. | z +w | .
) z = x +yi (x,y ) ,
x = +1
y =223
4
, pi' pi
= x 14y = 223
4y = 223(x 1) 4y = 223x + 3 3x + 4y25 = 0 ,
pi pi , pi pi z.
pi |w | = 1 pipi pi w .
) : 3x + 4y25 = 0 , -pi . pi pi (), , pi . , -, pi pi pi ().
:y = x , pi .
, =1
34
=1
=43
.
:y =
43
x , pi pi
:y =43
x
: 3x + 4y25 = 0
7. .
-
3x + 4
4x325 = 0 9x +16x 3 25 = 0 25x = 3 25 x = 3 = x
.
pi y =
433 y = 4 = y
, pi (3, 4) -
zmin
= 3 + 4i .
) . z = 3 + 4i , | z +w | = 6 |w + 3 + 4i | = 6 . w0 = x0 +y0i (x0 ,y0 ) , pi |w | = 1 ,
|w
0| = 1 |w
0|2= 1 x
02 +y
02 = 1 (1)
pi, pi |w + 3 + 4i | = 6 | x0 +y0i + 3 + 4i | = 6
| (x0 + 3)+ (y0 + 4)i |2= 62 (x
0+ 3)2 + (y
0+ 4)2 = 36
x02 + 9 + 6x
0+y
02 +16 + 8y
036 = 0
(1)
1+ 9 + 6x0
+16 + 8y036 = 0
6x
0+ 8y
010 = 0 3x
0+ 4y
05 = 0 4y
0= 53x
0 y
0=
53x0
4 (2)
(2) (1) x
02 +
53x0
4
2
= 1 x02 +
(53x0)2
16= 1
16x02 + (3x
05)2 = 16 16x
02 + 9x
02 + 2530x
016 = 0 25x
02 + 930x
0= 0
25x
0230x
0+ 9 = 0 (5x
03)2 = 0 5x
03 = 0 5x
0= 3 x
0=
35
.
pi (2) y
0=
53 35
4=
595
4=
1654 y
0=
45
.
w0
=35
+45
i .
. | z +w | = | z (w) | , pi pi- pi z pi w .
pi |w | = |w | |w | = 1 , pi- w , pi pi- pi .
() , pi .
7. .
-
pi pi (), pi , , pi . :
| z +w |
min= () = ()() = d(,) =
| 3 0 + 4 025 |
32 + 421 =
2551
| z +w |
min= 4 .
| z +w |
max= () = ()+ () = 2 1+ | z +w |
min= 2 + 4 | z +w |
max= 6 .
20. z w | (i + 2 2)z | = 6 , |w(1 i) | = |w(33i) | , :
) pi z.
) pi w.
) |w|.
) | z w | .
) | (i + 2 2)z | = 6 | 2 2 + i | | z | = 6 (2 2)2 +12 | z | = 6
4 2 +1 | z | = 6 3 | z | = 6 | z | = 2 ,
pi pi (0, 0) 2, pi pi z.
) pi w |w1+ i | = |w3 + 3i | pi , pi (1,1) , (3,3) .
w = x +yi (x,y ) , | x +yi1+ i | = | x +yi3 + 3i |
| (x 1)+ (y +1)i |2= | (x 3)+ (y + 3)i |2 (x 1)2 + (y +1)2 = (x 3)2 + (y + 3)2
x2 +12x +y 2 +1+ 2y = x 2 + 96x +y 2 + 9 + 6y 22x + 2y = 186x + 6y
4x 4y16 = 0 x y 4 = 0 , pi pipi .
) : x y 4 = 0 , pi . pi pi (), pi .
|w |min
= () = d(,) =| 00 4 |
12 + (1)2=
4
2=
=
4 22 |w |
min= 2 2 .
) | z w | pi pi pi (). ,
| z w |
min= () = ()() = |w |
min | z w |
min= 2 22 .
7. .
-
21. f (z) =
z + 3iz 3i
, z 3i , z1 = 3i33f (i) .
) pi | f (z) | = 1 , z , z 3i .
) f (z) = f (z ) , pi z .
) z1 .
) pi w, pi
|w + z1| =
i f (z1) z
1
2 + 5 i.
) pi w pi pi, pi pi .
) f (z) =
z + 3iz 3i
=z 3iz 3i
, pi | f (z) | =
z 3iz 3i
=z 3iz 3i
| f (z) | = 1 .
) pi f (z) = f (z )
z 3iz + 3i
=z + 3iz 3i
(z 3i)(z 3i) = (z + 3i)(z + 3i)
zz 3i z 3iz + 9i2 = zz + 3iz + 3i z + 9i2 6i z =6iz z =z z .
) f (i) =
i + 3ii 3i
=4i
i3i=
4i4i
=1 , pi z
1= 3 (i)3 (1) z
1= 33i .
) f (z
1) = f (33i) =
33i + 3i
33i3i=
33 + 3i3i
=33 f (z
1) = 1 , pi
i f (z1) z
1
2 + 5 i=
i 1 (33i)
2 + 5 i=
| i | | 33i |
| 2 + 5 i |=
32 + (3)2
22 + 52
=3 23
= 2 .
|w + 33i | = 2 , pi pi (3,3) 2 .
) pipi , pi . , pi , pi , , , pi . ' pi, .
:y = x , pi .
, y = 3 , x =3
3 = (3) =1 .
:y =x .
(x + 3)2 + (y3)2 = 2 .
7. .
-
y =x (x + 3)2 + (x 3)2 = 2 (x + 3)2 + (x + 3)2 = 2 2(x + 3)2 = 2
(x + 3)2 = 1 x + 3 = 1 x + 3 =1 x =2 x =4 .
pi x =2 , x =4 .
pi y = 2 , y = 4 , pi (2,2) , (4,4) .
wmin
=2 + 2i , wmax
=4 + 4i .
22. z1 ,z2 , z1 z2 | z1 + z2 | = | z1z2 | . ) pi
z1
z2
.
) pi
z1
z1
+ z2
+iz
2
z1
+ z2
1 .
) z2 = 4 + 3i , :
. pi z1 . . | z1z2 | .
) | z1 + z2 | = | z1z2 | | 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 2z1z2 =2z1z2 z1z2 =z1z2 .
z2 0 , z2 0 , pi pi pi
z1z
2
z2z
2
= z
1z
2
z2z
2
z
1
z2
= z
1
z2
z
1
z2
= z
1
z2
z
1
z2
.
)
| z1|
| z1
+ z2|
+| i | | z
2|
| z1
+ z2|1
| z1| + | z
2|
| z1
+ z2|1 .
z1 z2 , z1 + z2 0 | z1 + z2 | > 0 , pi
| z1 | + | z2 | | z1 + z2 | | z1 + z2 | | z1 | + | z2 | ,
pi , .
) . z2 = 4 + 3i , | z1 + 4 + 3i | = | z1 43i | pi , pi (4,3) , (4,3) .
z1 = x +yi (x,y ) , | x +yi + 4 + 3i | = | x +yi 43i |
| (x + 4)+ (y + 3)i |2= | (x 4)+ (y3)i |2 (x + 4)2 + (y + 3)2 = (x 4)2 + (y3)2
x2 +16 + 8x +y 2 + 9 + 6y = x 2 +168x +y 2 + 96y 8x + 6y =8x 6y
16x +12y = 0 4x + 3y = 0 , pi pipi .
7. .
-
. | z1z2 | = | z1 43i | pi pi pi : 4x + 3y = 0 .
pi pi (), -
| z
1z
2|min
= () = d(,) =| 4 4 + 3 3 |
42 + 32=
255
| z
1z
2|min
= 5 .
23. f (z) =
iz 1z +1
, z {1} .
) pi f12(i) =64 .
) z, pi f (z) =
12
12i .
) | f (z1) | = 1 | f (z2) | = 2 , z1 ,z2 {1} , | z1z2 | .
) f (i) =
i i1i +1
=111+ i
=2
1+ i=
2(1 i)(1+ i)(1 i)
=2(1 i)12 +12
f (i) =(1 i)
[f (i)]12 = [(1 i)]12 = (1 i)12 = [(1 i)2 ]6 = (1+ i2 2i)6 = (2i)6 = 26i6 = 64i 41+2
f 12(i) = 64i2 f 12(i) =64 .
) f (z) =
12
12i
iz 1z +1
=1 i
2 2(iz 1) = (z +1)(1 i)
2iz 2 = z iz +1 i 3iz z = 3 i (1+ 3i)z = 3 i z =
3 i1+ 3i
z =
(3 i)(13i)(1+ 3i)(13i)
=39i + i + 3i2
(1)2 + 32=68i
10=
610
810
i z =35
45
i .
) pi | f (z1) | = 1
iz11
z1
+1= 1
| iz11 |
| z1
+1 |= 1 | iz
11 | = | z
1+1 |
i z
1
1i
= | z1
+1 | | i | z1
ii2
= | z1
+1 | | z1
+ i | = | z1
+1 | ,
pi pi , pi (0,1) , (1,0) .
7. .
-
, pipi , pi pi pi () = () = 1 pi -. , , pi, , y = x x y = 0 . pipi .
pi | f (z2) | = 2
iz21
z2
+1= 2
| iz21 |
| z2
+1 |= 2 | iz
21 | = 2 | z
2+1 | | z
2+ i | = 2 | z
2+1 |
| z2 + i |2= 4 | z
2+1 |2 (z
2+ i)(z
2 i) = 4(z
2+1)(z
2+1)
z2z2 iz2 + i z2 i2 = 4z
2z
2+ 4z
2+ 4z
2+ 4 3z
2z
2+ iz
2 i z
2+ 4(z
2+ z
2)+ 3 = 0
3 | z2 |2 +i(z
2z
2)+ 4 2Re(z
2)+ 3 = 0 3 | z
2|2 +i 2 i Im(z
2)+ 8Re(z
2)+ 3 = 0
3 | z2 |2 +8Re(z
2)2Im(z
2)+ 3 = 0 .
z2 = x +yi (x,y ) , 3(x 2 +y 2)+ 8x 2y + 3 = 0 x 2 +y 2 +
83x
23y +1 = 0 .
83
2
+ 23
2
4 1 =649
+49
4 =689
4 =6836
9=
329
> 0 , pi pi -
83
2,
232
43
,13
, =
12
329
=124 23
=2 23
.
, z1 :yx = 0
z2 , | z1z2 | pi pi
pi , pi -,
| z1z2 |min= () = ()() = d(,) =
=
43
13
12 + (1)2
2 23
=
532
2 23
=5
3 2
2 23
=
=
5 26
2 23
=5 2 4 2
6 | z
1z
2|min
=26
.
7. .
-
24. z 0 , pi Re
1z
=14
. :
) pi z.
) | z 6 + 3i | .
) pi 2Re
1z
= 2 14
1z
+1z
=12 2z + 2z = zz
2(z + z ) = | z |2 | z |2 2 2Re(z) = 0 | z |2 4Re(z) = 0 .
z = x +yi (x,y ) , x2 +y 2 4x = 0 .
(4)2 + 02 4 0 = 16 > 0 , pi pi
42
,02
(2,0) -
=
162
=42 = 2 .
z = 0 (0, 0), pi .
x = 0 , y = 0 02 + 02 4 0 = 0 , pi , pi
, pipi pi' , pi z 0 .
pi (2, 0) 2, (0, 0).
) | z 6 + 3i | pi pi - (6,3) pi .
, , pi :
| z 6 + 3i |min= () = ()() =
= (x
x
)2 + (y
y
)2 =
= (26)2 + (0 + 3)2 2 = 16 + 92
| z 6 + 3i |
min= 3 .
| z 6 + 3i |max= () = ()+ () = | z 6 + 3i |min +2 = 3 + 2 2
| z 6 + 3i |
max= 7 .
7. .
-
25. z w, pi | z 1 | = | z 3i | w +w3i
35(ww) = 10 + i 40 i50 . :
) pi z. ) pi w.
) | z w | .
) | z 1 | = | z 3i | pi , pi (1, 0) (0, 3).
z = x +yi (x,y ) , | x +yi1 | = | x +yi1 |
| (x 1)+yi |2= | x + (y3)i |2 (x 1)2 +y 2 = x 2 + (y3)2
x2 +12x +y 2 = x 2 +y 2 + 96y 12x = 96y 2x 6y + 8 = 0 x 3y + 4 = 0 ,
pi pi .
) i35 = i 48+3 = i3 = i , i 40 = i 410+0 = i0 = 1 , i50 = i 412+2 = i2 = 1 , pi pi pipi w +w + 3i(ww) = 10 +1(1) 2Re(w)+ 3i 2 i Im(w) = 12
Re(w)3Im(w) = 6 .
w = x +yi (x,y ) , x 3y = 6 x 3y6 = 0 , pi .
) 1 : x 3y + 4 = 0 , 1
=13
=13
, 2 : x 3y6 = 0 , 2
=13
=13
.
piG .
| z w | pi pi () pi (), pi- pi .
1: x 3y + 4 = 0
1: 3y = x + 4
1:y =
13x +
43
2: x 3y6 = 0
2: 3y = x 6
2:y =
13x 2 , pi
| z w |min
= d(1,
2) =
43
+ 2
1+13
2=
103
1+19
=
103109
=
103103
=10
10=
102
10
| z w |
min= 10 .
7. .
