Migadikoi G Lykeiou
of 49
description
Complex Numbers
Transcript of Migadikoi G Lykeiou
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e-mail: [email protected] www.iliaskos.gr
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1
1. 1.1
CR
-R
i i 12 =- .z C -
z ia b= + R!a b .
I-
,i R!b b .-
R I C, ! R I C, 1 .
z ia b= + .-
Re z] g a Re z a=] g .-
Im z] g b Im z b=] g .
-, ,i R!a b a b+ .
Imz z 0R +! =] g .Rez z 0I +! =] g .
-
2
Im z 0!] g .Re z 0!] g .
,z i z i1 2a b c d= + = + .Re ReIm Imz z i i
z zz z1 21 2
1 2+ + +a b c d
a cb d
= + = +==
==
] ]] ]g gg g( (z ia b= + -
,M a b^ h ,M a b^ hz ia b= +
-'x x -
'y y
z ia b= + -OM ,M a b^ h.
M(, )
Im(z)
Re(z)O
1.2 ,z i z i1 2a b c d= + = +
-
-
-
3
z z i i i1 2 a b c d a c b d+ = + + + = + + +^ ^h hz z i i i1 2 a b c d a c b d- = + - + = - + -^ ^ ^h h h
z z i i i1 2$ a b c d ac bd ad bc= + + = - + +^ ^ ^h h h
zz
ii
i ii i i
2
12 2 2 2c d
a bc d c da b c d
c dac bd
c dad bc= +
+ = + -+ - = +
+ - +-^ ^^ ^h hh h
z i i212 2 2 2a b a b ab= + = - +^ hz i i3 313 3 3 2 2 3a b a ab a b b= + = - + -^ ^h h
i i i1 1 1 2 22+ = - + =] gi i i1 1 1 2 22- = - - =-] g
i 0!a b+ z x yi= + ,x y R!
z i x yi i x y xyi ix yxy2 2
2 2 2 22 2
+ + +a b a b a bab
= + + = + - + = +- =
=^ h (, , ,x y x y1 1 2 2^ ^h h ia b+ -
z x y i1 1 1= + z x y i2 2 2= + .
z ia b= +w ib a= -
z i i i i i iw2a b a b b a= + =- + = - =^ hz ia b= -
w ib a= + i-
z i i i i i iw2a b a b b a= - =- - =- + =-^ h
-
4
z ia b= + z ia b= -Re ReIm Im
z zz z
==-
] ]] ]g gg g( .
z z z z z zv v1 2 1 2+f f= = = = = =z z z z z zv v1 2 1 2f f+ + + = + + +z z z z z zv v1 2 1 2f f- - - = - - -z z z z z zv v1 2 1 2$ $ $ $ $ $f f=
zz
zz
2
1
2
1=b lz zv v=] ]g g
Re Imz z z z2 2 2 2$ a b= + = +] ]g gRez z z2 2a+ = = ] gImz z i z i2 2b- = = ] g
Imz z z z0R + +! = =] gRez z z z0I + +! = =-] g
-iy
,,,,
i i i i ii
i
1 01
1 23
v 4 4
yyyy
= = = =
==
- =- =
y y y+
Z
[
\
]]
]]
,,,,
i
vi vv
i v
1 44 1
1 4 24 3
v
llll
=
== +
- = +- = +
Z
[
\
]]
]]
z z 02a b c+ + = R*!a R!b c
-
5
42b acD = -
0>D
z 2,1 2!a
b D= - .
0D =z 2a
b=- .
0
-
6
f z 0=^ h f z^ h -
C--
C .
R I-
z x yi= + -ia b+
ImRe
z zz z
00
R
I
+
+
!
!
==
]]gg
,z wz z zz z zR
I
+
+
!
!
==-
z x yi= +,M x y^ h
w f z= ^ h-
x yi+,x y
x yi+
z x yi= +m z f g im m= +^ ^h h
,M x y^ h-
m
m x f m= ^ h y g m= ^ h
-
7
m-
12 2hn m vyo m+ = 1 12 2fz m vyo m+ =
w x yi= + z il m= +l mQ^ h
,M x y^ h
-
-
1.3
z ia b= +,M a b^ h
OM z 2 2a b= = +] g
M(, )
Im(z)
Re(z)O
z i1 a b= + z i2 c d= +,M a b^ h ,N c d^ h -
MN z z1 2 2 2a c b d= - = - + -] ^ ^g h h
-
8
M(, )
N(, )Im(z)
Re(z)O
-z .
,z z1 2
,z z1 2 z z1 2- .
-
-
z z z z= - = - =z zz2 =z z z z z zv v1 2 1 2f f=
zz
zz
2
1
2
1=
z zv v=
z z z z z z1 2 1 2 1 2!G G- +z z z R2 2 + !=
-
9
z z z I2 2 + !=-
z z z z1 2 1 2&= = z z z z1 2 1 2&= =Y
z z z z z z z z1 2 1 2 2 2 1 1 2 2+ += = =
z i i22 2 2 2a b a b ab= + = - +^ h z 2 2 2a b= +
z z z z z zz z z z z z z z
v v v v v v1 2 1 2 1 2
1 2 12
22
1 1 2 2
& + +
+ +
= = == = =
-
--
, ,f z z z 0=r^ h -z x yi= + ,x y R!
, ,f z z z 0=r^ h --
z
z z z 01 2 3+ + =
z z z 01 2 3+ + = .z z z z z z z z z z z z0 2 01 2 3 2 12 22 32 1 2 2 3 1 3++ + = + + + + + =^ ^h h .
-
Re
z z z z zz zz z
z z
z z z zz z z z
z z z z z z z zz z z2
1 2 3 1 2
1 2
1 1
1 2
3 1 22
33
1 2 3 3
1 2 1 2 2 2 3 3
32
12
22
& + +
+
+
+ =- ++
= - + =+ =
+ + + == - -
^ ^^h hh
z z z z zz z z z z
z z z z z z2
21 2 3 1 21 2 3
212
22
23
212
22
1 2 32& + ++ =- +
= - -= - + + =^ ^h h
1.4
,M x y^ h z x yi= + ,N x y0 0^ hz x y i0 0 0= + , , 0>1 2t t t
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11
N(x0, y0)
Im(z)
Re(z)Oz z >0 t-
z0 x x y y >0 2 0 2 2t- + -] ^g hH
N(x0, y0)
Im(z)
Re(z)O
z z<
-
12
,M x y^ h z x yi= + ,A x y1 1^ hz x y i1 1 1= + ,B x y2 2^ h
z x y i2 2 2= + .z z z z1 2- = -
z1 z2Im(z)
Re(z)O
B(x2, y2)
A(x1, y1)
-
, , ,x xy y x x x y y y1 2 2AB AB M M2 12 1 1 2 1 2$m m m= -- =- = + = +n
: y y x xM Mn m- = -n ] g-
z z z z
-
13
Im(z)
Re(z)O
B(x2, y2)
A(x1, y1)
z z z z>1 2- ---
z1 z2 z2 H -
Im(z)
Re(z)O
B(x2, y2)
A(x1, y1)
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14
,M x y^ h z x yi= + p R*! .z z p z z42- = +r r^ h
,E p2 0` j x p2=- .
2a,M x y^ h z x yi= + ,x yA 1 1^ h
z x y i1 1 1= + ,B x y2 2^ h -z x y i2 2 2= + c 0> .
z z z z c1 2- + - = z z c .
z z z z c1 2- - - = z z c>1 2-z1 z2
c2a =
x x y y 12 02
20
2
a b- - - =] ^g h ,x y0 0^ h -
-: x y 0f A B C+ + = -
,d O 0 02 2 2 2f A BA B C
A BC=
++ + =
+^ h
z0
-
-
15
z0 .
: , , : ,y x y xy xy x11 10 1 0
0 0f m b m m h m
m bm
= + =- == +=h h h
(
OK t-] g OK t+] g,z z1 2 -
,z z1 2 .: , , :c x x y y x
y OK y x
x x y yy x
x x x y
OKK
KOK
OKOK
02
02 2
02
02 2
02
02 2+
t m m
tm
m t
- + - = = =
- + - ==
- + - =
] ^] ^ ] ^
g hg h g h)
-z w- -
min z w d t- = - .
min z w 0- = .
: ,c K1 1 1t^ h : ,c K2 2 2t^ h -z w-K K1 2^ h
K K >1 2 1 2t t+^ hmin z w K K1 2 1 2t t- = - -^ hmax z w K K1 2 1 2t t- = + +^ h .
K K1 2 1 2 1 2G Gt t t t- +^ h -min z w 0- =
max z w K K1 2 1 2t t- = + +^ h .K K
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16
z w- -
1.5 C
z1 z2z1 z2
z1 z2 -
-z1 z2
,z z1 2 z z z z1 2 1 2+ = + .iv v N! .
Cz z 02a b c+ + = R!a b c !a .
z i0 a b= + R!a b -
z i0 a b= -,z z1 2 z z z z1 2 1 2= .z z z z z z1 2 1 2 1 2G G- + + z1 z2
1.6 i- .
,z z1 2 z z12 22 += z z1 2= z z1 2=- .
z 13 = z 1= .v N*! S i i iv v2 f= + + +
-
17
z- -y x=- .
z 1=
z1 z2 z z z z1 2 1 2G- - .z z2 2= .
z 1= iz
1.7 ,x y R!
x x y x y i3 3 2 02 - + + + + =^ h .x x i y yi1 2vyo - + = + .
ln x e i i2y+ = + .
xii yi1 2
1 hn+ = + .
x x yi y x i3 3 12 - + + = + -] g .,x y R!
x y xi x y i i6 5 52+ + + = + + +^ ^h h .x yi i xy xi3 6 7 22 + = + - - .
ix
iy
ii
1 2 2 3 1 86 7
- + - = -+ .
i x i x y i112 2- - + + + =^ ^h h .
x i x yi1+ = + .
z i x2 2hn= +z i2= + .
z i i5 4 4 5 5 6m n m n= + - + + +^ ^ ^ ^h h h hR!m n
z 0= .z i7 20= + .
R!a w ii
12a
a a= ++ + -
z i1 3 22 2m m m= - + - +] ]g g R!m
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18
z ii
2 1m mm= + +-] g R!m
z ii
ii
1 22 3
1 23 4= +
- + -+ .
lnz x x xi3 31 2= - - +] g z y i2 2= --
z x y i3 21 =- + -^ h z x y i5 32 = - - --
z i ii
11
11
2= - + + + z z2 45 02 - + = .
z i ii i
1 11 1
2 2
8 16
= - - ++ + -] ]] ]g gg g
z ii
ii
3 22 3
21 240 23= -
+ + -+b bl l .
x 81 04 + = -
z i23 2 1= -^ h.
z ii3
2= - Re z 2582 =-^ h .
z i23 24= +c m Re z 1=^ h .
z ii
21= -+
w z z1= + .
z i2 3= +, , ,z z z z- -z C! z z 1 02 + + =z 13 = .z z
1 22007 2007+ = .
z z 1 0v v3 2 6 1+ + =+ + v N! .f v i zv=] g v N*! -
f f f f3 8 13 18 0+ + + =] ] ] ]g g g g .f z i z z12= + +] ]g g .
f i] g f i-] g.f i f iv v+ - -]^ ]^gh gh .
-
19
A i i i i1 11015
2025= + + + .
B i i i i i5 7 4 7 3 152 18 31 44= + - + - + .i i i i57 58 59 213fC = + + + + .i i i i2 3 2006$ $ $ $fD = .
,E i i i v Nv v v3 1 2 5 3 != + ++ + .Z i i1 1v v2= + +] ]g g.
i i i i1 2 4 2 4G + - + -= ^ h666 @ @ @ .i i1 13 3H = + + -^ ^h h .
i i i i i i i i1 1 1 1v v v vv v v v
2 2 1 2 2 2 32 2 1 2 2 2 3+ + + = + + ++ + + + + + .
i i i i2 3 f+ + + + l
S i i i i1 1 v v2 3 f= - + - + + -] gv N*! .
S i i i i2 3 4 5 2 2 2 1f o o= + + + + + + - + -^ ^ ^ ^h h h h6 @.v N*!
i i 0v va b a b+ + - =^ ^h h i 0!a b- .z i
iii
2 32 3
2 32 32 2= -
+ + +-^ ^h h
-
, ,ii
ii v2 3
3 221 2 2 N
v4 4
!n-+ + +
- =nb bl l .
m l ii i
i11 5 2
2 5 02002 2004
-+ -- + =
l m` `j j .v N*! z i i3 2 2 3 Iv v2 2 != + + +] ]g g
z i i1 5 5 R132 132 != + + - -^ ^h h .x yi i3 2+ = + o^ h N*!o
x y 52 2+ = o .z i
i11 3= +- o` j N*!o Re Imz z 1+ =^ ^h h .
Re Rezz
zz
11 0 1
1 0++- = +
- =ooa `k j .
Im Imz iz i
z izi1 0 1 0++
+ = ++ =o
oa `k j .
-
z i iz i2 2 3 2
1- = - + .
iz i
iz i
2 3 2+- - = -
+ .
iz z z i
21
3 0-+ + - = .
i z zi1 2+ = +] g .zi i z4 1 2+ = +] g .
z i i1 23 3= - + -^ h .z i5 12=- - .
z i3 42 = + .z z2 2 02 - + = .z z z 1 03 2- + - = .z i z i2 3 02 - - + - =] g .z z2 = .
z z i3 2 0+ - =^ h .z z1 2 1+ = .z iz 1 02 + + = .z z z3 4 8 03 2+ + - = .z z i2 03 + + = .
z z z z1 9 0+ + =b bl l .
P z z z z7 16 103 2= - + -] g .z 1= P z] g.P z 0=] g .
P z z z z z4 8 4 74 3 2= + + + +] gP z] gP z 0=] g .
f z 0=^ h f z z i z i z i2 5 2 53 2= + - + - -^ ^ ^h h hii2 +
z z z3 10 7 10 03 2- + + =z z z z2 3 2 2 04 3 2+ + + + =
-
21
z i z i z i3 3 3 4 1 5 03 2- + + + + - =^ ^h h, ,2!a b
r rb l z i1 31 = +z z4 8 02 hna vyob- - = .
, 0>a b z i3 21 = -lnz z e2 3 02 b- + =a .z z2 2 02 + + = .
w z zz z
42
1 13
12
13
= -- z1
Im z 0
-
22
,z z C1 2 ! z z z z 11 1 2 2= = z z z z 1 01 2 1 2 !+ + -z z z z 1 01 2 1 2 !+ +- .,z z C1 2 ! z z z z 11 1 2 2= = w z z
z z1 R1 2
1 2 != ++ .
,z z C1 2 ! z z z z 11 1 2 2= = w z zz z1 I1 2
1 2 != +- .
w zz 12= +
zz
zz1 12 2+ = + zz 1=
,w zz
zzz z z z z2 0!= -+ -] ]g g
,w 1 1! -6 @.,w z
zzz z 0!= +
,w 2 2! -6 @.f z i z i z1 1 1= - + + +] ] ]g g g .
f z] g f z] g.f z f z=] ]g g -
f z z zz z1 1= +- +] ] ]g g g
z R!Y f z f z1- =b ]l g.
,z z1 2 w z z z z1 2 1 2 2= +] g,z z1 2 w z z z z1 2 1 2 2= -] g
x yi i3 7a b+ = +^ ^h h , , ,x y R!a bx y2 2 3 2 2 7a b+ = +^ ^h h .
w z iz i2
2= --
R I
w zz 1= +
w zz i
24= -
- -
-
23
w ziz z i1= +
-] g w w 1 R2 !+ + .
z i2 1 1l l= + + -] g R!l .z i2 3hni vyoi= + R!i .z ivyoi hni= + R!i .z i1 2 vyoi hni= + +^ h R!i .z i2 3vyoi hni= - + -^ h R!i .
z i i1 2 1 3a= + - -] g R!a .z i1 2 12m m= - + -] ]g g R!m .z i1 2vyoi= + +] g R!i .z i 12m= + +] g R!m .z i 1f{i
vyoi= + ,2 2
!i -b l.w z z i3 2= + -] ]g g R!
y x2 1= -
Im z 0!] g zw i1 2= + .w z z 12= + + R!
w z z1 1= +
Re w 1=] g .w z
1=- -
Re w 41=] g Im w 21=-] g
w zz
12= +
- .
Re Imw w=] ]g g -
Re Imz z iz4- =-b ]l g Re Rez z zi4 2+ =b ]l g
-
24
Re Imz z z i2 1 2 1 3+ + = + +] ]g g.z i2 41 = + z i1 22 = + -
Im Rew z w z 01 2+ =^ ^h h -OM
y x2= .z z R1 2 !+ z z R1 2 ! ,z z1 2
z z R1 2 !+ z z I1 2 !- ,z z1 2
w z zz
22
= + r -
M z^ h.
z zi z i1 2- +
z z 1- z2 -
z i zi
, Cf z z zz i z i z != +- +_ ^ ]i h g Re z 0!_ i .
f z f z1 =d _n i.
f iz1 R!-c m .
Re Im Ref z f zi z1 2= +dd cc _nn mm i.
Rezz zw w2 1 2- = -] g -z i1 21 = + z i3 42 = +
z1 z2 z z1 22 ziz
2
1
zz
1200421002
zz z
15
23
12-
z i ii i i
2 3 12 5 3 1
1 4 2
6 8 3
= - ++ - -] ]] ] ]g gg g g
-
25
z i21 3 v
2 = - -c mz i3 5 3 53
8= + + -_ iz i
i1 21 2
v
v
4 2
2
= +-
-
+
]] ggz i
ii i ii i i
11
23 7 1 10 26 2 1 3
196 194
20 198
= ++
- + - -+ - - -^ ^ ^ ^^ ^ ^ ^h h h hh h h h
z 1=z i2 3 3- - =z i2 5+ - =z i1 3 7- + =z i2 1- =z 3 2+ =z i2 2+z i3 4 3H- +
z i1 3 4<
-
26
z z4 3>2 -z zi1 22 G-z i z i22 G- + -
z 1= z 1 1- = .
Rez w z w zw22 2 2+ = + + ] gRez w z w zw22 2 2- = + - ] g
zw z w z w1 1 12 2 2 2- - - = - -^ ^h hImz iz z z2 22 2 2+ = + ] g
z 3=w 2= z w 5+ = .
Re zw 6=] g .z w- .
z z4 2 1+ = +
,z z C1 2 ! z 02 ! zz 0>2
1 z z z z1 2 1 2+ = + .z C!
z i iz z2 2 1+- = + = .Re Imz z z2G+^ ^h h .
,z w C!z w z w z w22 2 2 2+ + - = +^ hz w z w1 12 2 2G- + +^ ^h h
, ,z z z C1 2 !z z z2 1 2=
z z z z z z z z2 212
22
2 12
22
21
22
2+ + + + - = +
,z w
zwz w z1 1 1+++ = = w 1=
,z wz w z w2 2 2+ = - w 0! Re w
z 0=b l .z wi z w2 2 2- = + w 0! Im w
z 0=b l ., ,z z w1 2 z z 11 2= =
u z zw z z w z z2
1 2
1 2 1 2= -+ - +] g z z1 2! .
-
27
,z z1 2 z z1 2= v N*!
w z zz zv v
v
1 2
1 2= +-_ i
.
,z z1 2 z z 11 2= = v N*!
w z zz z1 v v
v
1 2
1 2= -+_ i
, , , ,z z z z1 2 3 f o
z iz i
z iz i
z iz i 1+ - - .,M x y^ h
z x y i2 3 2= - + -^ h z z i1 1 2
-
29
z x yi= + ,x y R! y 0!
w zz12
= -r
,M x y^ h -z C! z z1 2 1- = + -
w i35
34=- + -
z it vyoa hna= +^ h ,0 2!a rh6 0>t -z z 2 2 02 2 2vyo i hn i vyo i- + - = -
2 2< ^ h .z z3 2 3- = + + .
z x yi= + y 0!z 11 =- z z 12 2= + z z23 =
,M x y^ hz.
z x yi= + y 0!w z
z22= +
+r ,M x y^ h -
z i z i4 3 4 3 6+ - - - - = .z i
iaa= -+ w i
i11aa= -
+R!a -
z i31a= + R!a
z i1 21
aa= + +
+R!a -
C!a b 1a b= = .1 0 1 0+a b ab a b ab+ - + = + + - = .a b^ h
1 0a b ab+ - + = .1 0!a b ab+ - + 1
1 1a b aba b ab+ - ++ + - = .
C
-
34
z z 15 9 = .z z=o r .z z=-o r .z z=-o .z iz=o r .
ab bc ca= =
a b c= = .
z z3 7 107 7+ =
z z 17 7= = .-
z 1! w zz
11= -+
z1 z2z z i4 41 2+ = + z z i2 5 51 2- = + -z1 z2 .
z i1 3 2G- - w i3 2G- - .
z w= .z w- .
z1 z2 z3 z z z 11 2 3= = =z z z 01 2 3+ + = .
z z z z z z1 2 2 3 1 3- = - = - .z z 41 2 2 G- Re z z 11 2 H-] g .
, ,z z z1 2 3
, ,z z z1 2 3
z zz zz z
z z z zz z z zz z z z
2 2 2 022
1 2
1 2
1 2
1 3 2 3
1 3 2 3
1 3 2 3
m
m
---
+ - - - - =- - = - -+ - = -
^^h
h R!m, ,z z z1 2 3
-
35
z iz 1 02 + - =z i3 =- .
z iz i1 11998 2 1998+ + = -^ hz 1 111998 1998+ =-^ h .
Rez z z z22 0 2 0 2a+ = +r^ hz0 0>a .
z x yi= +z z z z2 1 3+ + = +r r
,z z C1 2 !z z z z
z z4 3 5 9 16 01 2 1 2
12
22+= =
-+ = .
z z1 2! z i z iz1 21
21
3 2 1= + -` j, ,z z z1 2 3
z z z z 02 2- + = .,K 1 1-^ h
2t =w z
z1
2= - .,K 1 1^ h
2t =w z
1= r .
2 31a bc= = z i21 a b b a c= + + -^ ^h h
z i52 a= +z z 1 02 - + =
z z z 050 70 90+ + = .z z 1 040 40+ + =- .z z z1 1 1 031 30 29- + - + - =^ ^ ^h h h .
wz wz1 R!-- r
w ia b= + 12 2 !a b+ .
w z z2 2= - rw 1= .
w z z z2 22a= + -r -,1 0!a -^ h.
-
36
zi z2=r
z w zw2 1 27 7+ = + r^ ^h h -
z i z i z i z i41
41
4 410
- + + = - + -
w z z4= + r .
Re z z z z1 1- + = +r r^ ^h h6 @ ,z z1 2z z 2 21 2 G- .
M z^ h ,A 4 0^ h ,B 25 0` j
,A 1 1-^ h ,B 2 0^ hz 2
2> .
,z i8 0 2!+ 6 @ ,z 6 8 12!- 6 @.z0 Imz wz2 02 2t+ + =r^ h
0>t C!
z0 .N*!o z w 004 2 4 2+ =o o+ + .
z z 11 2= = I!Y z z wz z w1 2 1 2+ + =,z z1 2 .
z z z z z z1 2 2 3 3 1 l- = - = - = -
z z z z z z1 1 1
1 2 2 3 3 1- + - + - .
z iw z w- = + z w z w2 42 2 2+ = + .z wu2 = z 0!
w u z w u z w u2 2+ + + + - = + .
z z z2 4 2 01 2 f+ + + + =o o o o- -
w zz
22
I!= +- o` j
z z3=r .
z z2 5 02 2 2 2vyo i vyoi hn i vyo i- + + =^ ^h h ,2 2!i r r-` j.
-
37
z i1 2 12m m= - + -^ ^h h R!m .C z z z2 23= - +r Re z Z!^ h .1a b c= = = 0a b c+ + =
a b b c c a- = - = - .
z i6 8 5- - = .
z5 15G G .z .
z .z .
,z z1 2z z1 2- .C!a b c
02 2 2a b c a b c+ + = + + =
2a bc= .0ab bc ca+ + = .
2a b b c c a- = - -^ ^ ^h h h.
-w iz
z i2 1
2= +-
z w- .C1
i z1 2 2- - =^ h .C2
w iw i2 42 1- +
+ = .z w- .
ia b+
-
38
: x y2 1 0f - + = .Im Rez z2 1 2+ = -^ ^h h.
y x42 = .-
f x x 12= +^ h .z i2 2 4m m= - + -^ ^h h R!m .
z1 .z2 .
w z z5= +
z z2 2 02 2m m+ - + + =^ h .R!m
,z z1 2 z z 412 22+ =R!m ,z z1 2 .
,z wz w w z
1 1 2+ = + =r rz w= .z 1= .
z w w z1 1+ = +r r z w w z
1 1 2 02
+ - - - =r r` j
Re Rez z z4 2+ =b ]l g 1] g
Re z 0!] gw z z
4= + -w4 4G G- .
c z i3 4= + + .ii] g
, ,z z z1 2 3 1] gz z z z z z z z z21 2 2 3 1 3 1 2 3+ + = + +
z z6 4 9 02 2 2hn a hna vyo a- + + =^ ^h h ,0 2!a rb l
-
39
-
6ar= v N*! z N*v ! .
, ,z z z1 2 3 z z z 11 2 3+ + =z z z 11 2 3= = =
z z z1 1 1 11 2 3+ + = .
z z z z z z 11 2 1 3 2 3+ + = .z z z z z z 41 2 2 2 3 2 1 3 2+ + + + + = .,z w C*! ,z z1 2 z z w w z1 = +
z z w w z2 = -z z1 2zw R!r z z 01 2 = .
,z wz z1 2= .
z w 6- = z w 10+ =
z2 8G G .
wz
41 4G G .
z i w i6 18G G- + - .z w i8 2 12G G+ - .
z i3 1- = w 4 2- =z w2 8G G- .
, ,z z z1 2 3 z z z 01 2 3+ + =z z z 11 2 3= = = z C!z z z z z z z31 2 3 H- + - + - .
z 0!
z z z i z i1 1
-
w zz
59= -
- z 5!
z i2 1 2 1m m= + + -^ ^h h R!m .--
R!m .
z i10 = -
w w z122 0+ - =r .z i z2 4- = + .
z 6+ .
z i2 3- - .
z i z i z31
31
9102- + + = + .
z C! R!a b 0> >a b .z z2 2
a b a bab
+ - - =r -
z z7 48- =r
z ., ,z z z1 2 3 -
z z z z z z z z z12 22 32 1 2 2 3 1 3+ + = + + .Rezz z zi2 1 0+ - + =r ^ h
,z z1 2 z z 21 2 G- .,z z1 2 z z 21 2- =
z z 81 2 2+ = .z i 3+ =
w z i z i zi1 21
22= - + + - +
w 1= .
-
41
wz zi3 8- =r .
-
min z w 311- = max z w 5- = .
z i1 a b= + z zz
22
21
1= +-
R!a b !b z z R2 1 !- .z z 12 1- = .
z1 .z12 0>ab
z1 z i z i1 1 01 120 20+ + - + - =^ ^h h .,z z1 2 1>o z i21 = +o
z i1 22 = +o .w z
z2
1=
z ww
11= -
+
f c c w c w= + + -^ h C! .z 4= w z
z28= -
- -
w 2= .w w
w 41
2
= +
z ii
4 325 m
= ++^ h
R!m .
R!n
w z z R!n= + .
R!m z R!Y z z zm + =r .1m = .
min z i- .z R3 ! .
, ,z z z1 2 3z z z 01 2 3+ + = z z z 012 22 32+ + =
z z z z z z 01 2 2 3 3 1+ + = .z z z 01 2 3 ! .
-
42
z z z13 23 33= = .z z z1 2 3= = .
z z z z z z1 1 1
R1 2 31 2 3
!+ + + +^ ah k .z z z 0113 213 313+ + = .
, ,z z z1 2 3
z w i2 4 4+ = + .-
: y x 2f = +
: y x 1h = - w i- ., ,z z z1 2 3
Re Re Rez z z1 2 3= =^ ^ ^h h h z z z i1 1 1 61 2 3= = = -
z z z i1 1 1 61 2 3+ + =- .
, ,z z z1 2 3 .Im Im Imz z z2 31 2 3= =^ ^ ^h h h
, ,z z z1 2 3 .Imzz z2=r ^ h.M z^ h
z i 1- = .w z
1=
f z z i2= +^ h z C! .f z iz 2= +r^ h f z f z- = r^ ^h h.
f i z f i z1 2 3+ + + +=^ ^h h.f z f z 102 2+ =r^ ^h h z 1= .
f z f z= r^ ^h hz R! .
-
43
z f z i f i if i2 12+ - = - +^ ^ ^h h h.u f z= ^ h
z C! f z f z z z2 1 3 7 142+ - = - +^ ^h hz C! .
f z z z5 62= - +^ h z C! .Re f z^^ hh Im f z^^ hh.
f z z i5 9 4=- + +^ h .f z z5 4= -^ h -
zf 0>^ h .f z f z 0>r^ ^h h z C! .f z i z i3 4= + +^ ^h h z C!
Re Re Imf z z z3 4= -^^ ^ ^hh h hIm Re Imf z z z4 3 1= + +^^ ^ ^hh h h .
f z^ h -: y x 1f = +
, , ,z i f z f i^ ^h h -f z f i i z i3 4- = + -^ ^ ^ ^h h h h.
5CD AB=^ ^h h.-
z 2 1- = w i2 1- = .
-z w= .
z 3G w 3G .z w- .
z x yi 0!= + ,x y R! w z zz1= - r .
ia b+ R!a b .z 1= R! .
Re Imz w z22 2=^ ^h h.f z z z= +r^ h z C!
f z f z=r^ ^h h z C! .f z R!^ h z R! .
z 1G f z 2G^ h .f z z i2 3= -^ h .
-
44
z z9 1- = - .z .
w i w i5 5 1- + =r^ ^h h
z w1 9G G- .z i3 5 21 m n= - + -^ ^h h
z i2 1 12 n m= - + -^ ^h h A w1^ hw z z1 1 2= + : y x 21f = +
B w2^ h w z z2 1 2= - -: y x2f =
,z z1 2z z 11 2= = z z z z z z z z1 11 2 1 2 1 2 1 2+ + - = + - + .
z ii
22aa= +
+R!a .
-
,z z1 2z i
i2
2aa= +
+ 0a = 2a =-
,z z1 2 .z z1 2 2= -o o^ ^h h
o.,z z1 2 z z8 813 23= = .z z
11
1= z z
12
2= .
z z z z2 21
1 2 1 2+ = + .z z1 31 2G G+ .
A z1^ h z1,O 0 0^ h 2t = B z2^ h -z2 z iz z
42 1
1= -
z2 -
z i1 a b= + z zz
22
21
1= +-
R!a b !b z z R2 1 !- .z z 12 1- = .
-
45
z1 -
z12 0>ab -z1 z i z i1 1 01 20 1 20+ + - + - =^ ^h h .
, ,z z z1 2 3z z z2 2 2 21 2 3= = A z1^ h B z2^ h z3C^ h
,z z1 2 z3A 90c=t
ReRe
z zz z
21
2 3
1 2 =^ hh .,z z1 2
Rez z z z z z2 1 3 2 2 3 61 2 2 1 2 1 2G+ - + + + -^ ^h h6 @ .,z z1 2 .
w zz
zz
23
32
2
1
1
2= -
w 1= z w i4 2 1= + - .
-
,z z1 2z z i4 41 2+ = + z z i2 5 51 2- = +,z z1 2 .
z i1 3 2G- -w i3 2G- -
z w= .z w- .
z iz2 = r .
z z4 4 4 3 02 m m+ + - = R!m .R!m
-
46
m
z i1 21
0 =- +f z z
z i12= +
+^ h z 1!- .f z i3 2= -^ h .
Re f z 0=^^ hhf i z1 2 1+ + =^ h
z1z i1 2 1+ + = z2
z i3 2- - = -z z1 2- .
Re z1
41=` j
Im z 1=^ h Re z^ h.
z .
,z z1 2 Re Rez z1 1
41
1 2= =a ak k -
z z1 2- .C -
z i 1- =Im z 1H^ h .
-C w z i z i2
1 1= - - -` j'y y.
z iC! - -" , w z iiz 1= ++ .-
Im z 0
-
47
C2 -
M z C1 1!^ h 'M z C2 2!^ h -z z1 2- .
,z z C1 2 ! z z 02 a- + = R!a,z z R1 2 !Y .
41>a .
z 1=
z z z z 01 5 2 5- - - =^ ^h h .w z
i= r .w w1+ = z i 1- = .w i 2- = z 1 2+ = .
C1 C2
M1 M2 z1 z2C1 C2
z z1 2- .
z i z i1 2 2 1+ - = - +^ h .z i iz1 2 3+ - =^ h
-z
,z z1 2 -z z1 2- .
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48
Ekswfyllo_Mathimatica_G_Migadikoimigadikoi_new
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