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