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1
1i23
1+ 2 i1
3ii1
3( )( )
2i5i24i3 + 4 i4ii 218 +
1 ( )( ) ( )2i3131
2i32i32i3
2i31 =+
=+
133=
+ i231Re
132
2i31Im =
+ ,
( )2i3131
2i31 +=+ 13
13133
133
2i31 22 =
+
=+
k2i23
1argi23
1Arg +
+=
+
",2,1,0,232arctan =+= kk
2 ( )( )
( ) ( ) i,25
233i3
21i
i)(1i1i13i
iii
i13i
i1 =+=+
+=
,23
i13i
i1Re =
25
i13i
i1Im =
25i
23
i13i
i1 +=
234
25
23
i13i
i1 22 =
+
=
k2i1
i3i1arg
i1i3
i1Arg +
=
",,,=,+= 210235arctan kk .
3 ( )( ) ( )( )( )( )( )( )( )
42i7i26
2i2i2i5i24i3
2i5i24i3 =
+=+
13i27
226i7 ==
( )( )
27
2i5i24i3Re =
+
( )( ) 132i
5i24i3Im =
+
1
( )( ) l3i27
2i5i24i3 +=
+
( )( )2295
2i5i24i3 =+
( )( ) ( )( ) kk 2726arctan22
i2i52i43arg
i2i52i43Arg +=+
+=
+
( ) ",2,1,0,12726arctan =+= kk .
4 ( ) ( ) ( ) ( ) ii141iii4ii4ii 10410242218 +=+=+ 3i1i4i1 =+=
{ } { } 3i4iiIm1,i4iiRe 218218 =+=+ 3i1i4ii 218 +=
+ 10|i4ii| 218 =+
( ) ( ) ( ) 2k3i1arg2ki4iiargi4iiArg 218218 +=++=+ = .2,1,0,k2karctan3 "=+
2 ( ) i13i5
3yi1x +=+++ x, y
( ) ( )[ ]( )
( )( )3i53i53i53yi1x
3i53yi1x
+++=+
++ ( ) ( ) ( ) ( )[ ]
343y51x3i3y31x5 +++++=
[ ] ( ) i1185y3xi43y5x341 +=+++=
=+=+
34185334435
yxyx
=+=+
52533835
yxyx
11,1 == yx 3 i-i=i-1= i 4
21) | |
1 16) Re( ) ( ), Im( ) ( )2 2i
z zz
z z z z z
=
z= + =#
2
iz x y= + 5
z 2 | |z z= 2
2
2
z
iz x y= + 2 | |z z=2 2 22ix y xy x + = + y 0 2 2 2 2 ,x y x y xy = + = z
6 na 1|| z || azn +
|a||a||z|az nn +++ 1 na
ezargi=
( ) |a|ea|a|eea|z aann an +=+=+
=+| 11 argiargi
argi
||1 a+8 1i 2-1 31+ 3 i
4 ( )0isincos1 + 5i1
2i+ 6
( )( )3
2
isin3cos3isin5cos5
+
1 2i
e2isin
2cosi =+=
2 ieisincos1 =+=
3 3i
2e3isin
3cos2
23i
2123i1 =
+=
+=+
4 21 cos isin 2sin i2sin cos 2sin sin icos2 2 2 2 2 2 + = + = +
)(0,e2
2sin2
isin2
cos2
2sin 2i =
+=
5 ( )
===+ 2
1i2
12i1i12i21
i12i
=4isin
4cos2
= 4i
e2
6 ( )( ) ( ) ( )2
2 3i5 i3 i10 i9 i193
cos5 isin5e / e e /e e
cos3 isin3
+ = =
=
3
isin19cos19 += 9
1 1 11 1
,;
x x ay y b= + = +
2 1 11 1
cos sin ,sin cos .
x x yy x y
= = +
1 1iA a b= + 1 1 iz x y1= + iz x y= + 1 21z z A= + i1 1(cos i sin ) ez z z = + = 10-i
zezz Argi||= ( ) =||= 2Argi2iArgi zz |z|eeeziz
2
11 2 2 21 2 1 2 1 2| | | | 2(| | |z z z z z z+ + = + 2| ) 2 21 2 1 2 1 2 1 2 1 2 1 2
1 1 2 22 2
1 2
| | | | ( )( ) ( )(
2( )2(| | | | )
z z z z z z z z z z z z
z z z zz z
+ + = + + + = += +
)
12
1 ( )( )( )
P zR zQ z
= iX Y+ X Y x y
2 ( )R z 1 ( ) iR z X Y= 3 ia b+ 10 1 1 0
n nn na z a z a z a
+ + + + ="
ia b 1 ( ) ( ) ( ) Re( ( ) ( )) Im( ( ) ( ))( )
( ) ( , ) ( , )( ) ( )P z P z Q z P z Q z P z Q zR zQ z q x y q x yQ z Q z
= = = + ;
2 ( ) ( ) ( )( ) i i( ) ( )( )
P z P z P zR z XQ z Q zQ z
Y X Y= = = = + = ; 3
( ) 10 1 1n n n nP z a z a z a z a = + + + +" 4
( )zPzazazaa nn =++++= "2210 13 itez =1 nt
zz n
n cos21 =+ 2 ntz
z nn sini21 =
1 nteeeez
z nn sin21 intintintint =+=+=+
2 nteeeez
z nn sini21 intintintint ===
14
1 ( 5i3 ) 2 ( )6i1+ 3 6 1 4 ( ) 31i1 1 ( ) ( ) 6/5i56/i55 322
2i
232i3 ==
= ee
5 532 cos isin 16 3 16i6 6
= + =
2 ( ) ( )6 66 i /4 3 i/21 i1 i 2 2e 8e 8i2 2
+ = + = = =
3 ( ) ( )1 i 2 1 /6i+26 61 e e , 0,1,2,3,4,5kk k + = = = 6 1 6 ,
2i
23e /6i += ie /2i =
2i
23ei /65i +=
2i
23e /6i7 = i23i =/e
2i
23411i =/e
4 ( ) ( ) 0,1,2=,== 2212=13
+
31
/31
31
keek
2
4i
64i 22ii
3 ( )1/31 i
,127sini
127cos22
,12
sini12
cos22
612/7i6
62/i6
+=
=
e
e
+=4
5sini4
5cos22 64/5i6 e
15 (1 i) (1 i)n n+ = n
5
i / 4 i / 4 i / 4 i / 4( 2e ) ( 2e ) ,e en n n n = = sin 04n =
4 , 0, 1, 2,n k k= = " 161 083 =+z2 08''' =+ yy
1 ( ) ( )1 i 1 2338 2 kz e += = k=0,1,2
,3i1+ ,2 3i1 2 083 =+ i311 += 22 = i313 =
( )xCxCeeCy xx 3sin3cos 3221 ++= 17 z
1 1 1, , , , ,z z zz z z
o x
y
z
-z z
z
1z
1z
1z
18 1z 2z 321 ,, zzz
1 ( )2121 zzz +=
2 ( ) 21 1 zzz += 3 ( )3213
1 zzzz ++= 1,2,3i =+= ,kyxz kkk 1
2i
22121 yyxxz +++= z 1z 2z
6
2 ( ) ( )[ ]122122 i yyyxxxz += 1z 2z |z|z |z|z 12 1 =
3 ( ) ( 321321 3
i31 yyyxxxz +++++= ) z
321 zzz
19 1 2 3, ,z z z 0321 =++ zzz , 1321 === zzz z1z2z3 1=z
1321 === zzz 321 zzz
23 31 z z= = 3z ( )[ ] ( )[ ] 212322112121 zzzzzzzzzzzz +++=++=
21212 zzzz ++= 12121 =+ zzzz
)())(( 1221221121212
21 zzzzzzzzzzzzzz ++== ( ) 32 2121 =+= zzzz
321 = zz 33231 == zzzz 321 zzz 1=z
20z1z2z3
32
31
13
12
zzzz
zzzz
=
321312 zzzzzz ==
)arg()arg()arg()arg( 32311312 zzzzzzzz = 231312 zzzzzz =
( ) ( )( ) ( ) 12
32
3213
3112
13
12
zzzz
zzzzzzzz
zzzz
=+
+=
123312 zzzzzz = 321 zzz 321312 zzzzzz ==
7
21 z 1 | 5 2|z = 6 1|i2| +z 3Re( 2) 1z + = 4 ( ) 3iRe =z 5 |i||i| =+ zz 6 4|1||3| =+++ zz 7 Im( ) 2z 8 1
23
zz
90 arg z < < 10 ( )4
iarg =z 1 0 5z = 6a
2 i20 =z 1b 3Re( 2) 1 3z x+ = = z 3x = c 4 xyyxz i)i(ii +== .33)iRe( == yz z y=3
d 5 2 2i i i i i)( i) ( i)( i)z z z z z z z z+ = + = + = +
2 2i i 1 i i 1 i i 0 2Re(i ) 0 2 0z z z z z z z z z y + + = + + = = = 0.y = ze
6 2222 14)2(122)14(3413 +=+=+=+=+++ zxzxzzzz
134
)20412322
22 =++=++ yxyx z -3,0-1,0
2 3f 7 g 2y 8 +
933)2)(2()3)(3(23123 222 zzzzzzzzz
zz
2 52 2 4 5 . z 2
z z z z z x + + 25=x
25=x
h 9i 10 i 0,1 >+= xxy j
8
x
y
-2
(b)
Oi
x5
(a)
yO
-3
(d)
y
3i
x O
(c)
x
y
z
-i
i
y
x
i3
-2
O
y
5/2 x x
y=x+1 i
y
O
( e)
(f) (g)
(h) (i)
2i
(j)
22
1 20Im >z 41 >z 3 Re0 z
y
O x
2 41 >z
x
y
5 O 1
16)1( 22 =+ yz3 0 1Re
2 3 5 131 >+
O x
D
8/15 -17/15
y
11
1517=z
158
8 | 2 | | 2 |z z + + 6
x
y
o 3
5
2 2
19 5x y+ =
9 2 242 | 2 | 1 4 1, 0z z x y x + > > <
D
1/2
y
2 24415
x y 1 = 10 (2 2i) ( i) 4zz z z +
y
o 2-1
12 15x
x
9 23z
2 2( 2) 1x y + + =
Czaza =+ aC CByAx =+
)(i2
1Im),(21Re zzzyzzzx ==+==
CzBAzBA =+ )i(21)i(
21
)i(21 BA a = + (
21a = )iBA Czaza =+
24 0, (zz z z c )c + + + =
2 2( ) 2( ) 0z a z a R zz az az aa R+ + = + + + = c aa R=
25t 1 2 tbtaz sinicos +=tz i)1( += 3
ttz i+= 4 22 ittz +=
5 67 z e a b= = +
1
t
ch i shz a t b t= + i it tz ae be= + , ( i )
2 2
2 2
chch i sh 1
shx a t x yz a t b ty b t a b== + = = 5
2 2
2 2 1,( ) ( )x y
a b a b+ =+ 6
2 arctan2 2a y
b xx y e+ = 726
zw 1= z w
( )ivuwiyxz +=+= , 1 2622 =+ yx xy = 3 ( ) 11 22 =+ yx 1x = 4
222211
yxyi
yxx
iyxzw ++=+== 2222 , yx
yvyx
xu +=+=
1 )(2 2
2 2 1 1x yu vx
++ = = =
2
w2 2 22 2 4x yy ++( ) v
yxyx ++ 2222yy
yxxu ===+= 22
x =
w
3 1 21 ,u =
1 y+ 21yvy
= 2 2 211u v uy+ = =+ +
2 2
21 12 2
u v + =
4
w
( )21211 22
2222 =+=+=+ yxxxyxyx
21=u
27
w
v 3zw =
1 i1 =z i12 +=z i33 +=z
2
w
3arg0
w
i3/3i1 == ew
2i22
1323
i2
1222 42 += +==
ew
i82 2i3
3 ==
ew
2 = zf 0>
00 | |z z < < | ( ) | 1f z A 00 | |z z < < | ( ) | | ( ) | | ( ) | | | 1 | |f z f z A A f z A A A= + + +
31 1( ) , ( 0)2i
z zf z zz z
= 0z ( )f z
2x 21 2( )2i
z z xyf zz z y
= = + 32 )arg(arg
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