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ا"! ا ي م"د د اذ :$ادإ

1

..................................................................... 3 ............................................................. 5 1.1 – ................................................................. 5 2.1 – C : .............................................. 5 3.1 – " # : ................................................................. 5 4.1 – "$# : ................................................................ 6 5.1 – %& ' : ............................................ 6 6.1 – () : ............................................................... 6 7.1 – *+ # : ................................................................. 7 8.1 – %& ,, : ............................................. 9 9.1 – "$# : .............................................................. 12

10.1 – "## : ........................................ 13 11.1 – " # : ............................................................. 15 12.1 – "$# : ............................................................ 17 13.1 – " # : ............................................................. 18 ............................................................20

1.2 – : ............................................................. 20 2.2 – ' : .................................................... 20 3.2 – % ' : ...................................................... 20 4.2 – - . : ..................................................... 21 5.2 – "/# ' : ........................................................... 22 6.2 – "$# : .............................................................. 22 7.2 – : .......................................................... 23

2

8.2 – "$# : .............................................................. 23 9.2 – : ............................................................. 25

10.2 – 0 & : .......................................................... 25 11.2 – "$# : ............................................................ 26 12.2 – ' : ..................................................... 27 13.2 – &% – 1 : .......................................... 28 14.2 – " # : ............................................................. 29 15.2 – : ............................................................ 29 ..........................................................38

1.3 – : ............................................................. 38 2.3 – : ............................................................. 38 3.3 – "$# : .............................................................. 38 4.3 – "$# : .............................................................. 39 5.3 – : ............................................................. 39 6.3 – "# : ............................................................. 39 7.3 – "$# : .............................................................. 40 8.3 – : ............................................................. 40 9.3 – ' , "' 23 : ........................................... 41

10.3 – "$# : ............................................................ 42 11.3 – "$# : ............................................................ 42 12.3 – ' 4 : ....................................................... 43 13.3 – ' ",, " : ............................................. 44 14.3 – ' " " :............................................. 45 15.3 – ' ),% : ( ................................... 48

3

16.3 – "$7 "8 : ..................................................... 49 17.3 – "$# : ............................................................ 49 18.3 – " # : ............................................................. 50 19.3 – "$7 "8 : ..................................................... 50 20.3 – ' 9 : ................................................. 52 21.3 – "$# : ............................................................ 52 22.3 – " # : ............................................................. 54 23.3 – " # : ............................................................. 54 24.3 – " " ' zw log= : ................................. 55 25.3 – : ............................................................ 57 26.3 – ".% "3 ' 3 : .................................. 58 27.3 – "$# : ............................................................ 59

1: 3 " ;: 1#< " => /. ;:" ?@ ; = # , @ .

%>& 1> A ;: " 1 % A # 3 :2 1 0+ =x =B 7,– " 4 " #% / 1% R. /. ): ' 1> " % 1@ A',: 1%<' D'E@ = "n '' ' n F7)? .( > ;> > + E

4

> > "#% A#% "+G " 3 H#8 ' ="' AI /. 8 ;: &' E .

' #8 # 1J =/ K' " I "4 L' K/+ + A 1% 1@.

% ; M7 ; =A 1 G) )>" A' 1 = ;:.(

5

1.1 – :

NG 8 1 4 " ),( ba. ) 1@ # ),(),( abba ≠ 1% ba =( "# 1 ' 8.

G# >, > >' ?>E ) , ,z w > %# 1>#@ @ = ),( baz =. G# G' " 4 % "C. 1@ @:

2( , ):( , )z a b a b= = ∈C R

2.1 – C :

#)+ ( )⋅( % 1 1: 1% :1 1 1( , )z x y= 2 2 2( , )z x y= 1O :

1 2 1 2 1 2

1 2 1 2 1 2 1 2 2 1

( ; )

( ; )

z z x x y y

z z x x y y x y x y

+ = + +

⋅ = − +

" 4 1 NG P#4 = Q' 8 /# %.

:1 1' 1 (1,3)z = 2 (4,0)z = #% /' /:

1 2 1 2(5,3) ; (4,12)z z z z+ = ⋅ = #G : > > "'#' 1 1E# >>')0,0(

)0,1( = ; 1O ( , , )+ ⋅C %& F7.

: "37 1:)≤ (R " ; R > 8# . = . P>#@ @ "' % 1%21 zz ≤ ' '1z 2z 1 #.

3.1 – ! :

6

# ;: #'# :xoy #G > >')0,1(1 = ;ox => >' ; =G ># >>'

)1,0(=i ;oy >#O =>) ' ; = > 1>% P# =F' % " :

1)0,1()1,0()1,0(2 −=−=⋅=i 1@ @:

2 1i = −

4.1 – ! :

@ @ 1( , )z x y= 1Oz %&' % :z x iy= + "#$ : ' %# 1@ 1% ="' :

( , ) ( ,0) (0, ) (1,0) (0,1)

1

z x y x y x y

x y i

x iy

= = + = +

= ⋅ + ⋅

= +

: (2,5) 2 5z i= = +

5.1 – % &'( % :

1% : ),( yxz = = ; 'z x iy= + ' %&'z .+# # x > SG'z >' P G#Rex z=.

# y > >) SG>'z #G P >'Imy z=. ; ( , )z x iy x y= − = − 0 z.

6.1 – )* :

7

% : 11 1 1( , )z x y= 2 2 2( , )z x y= 1O : 1 (1 2 1 2 1 2y y x x z z= = ⇔ =.

2 (

/' ;: " SG4 @.

/' ;: ") SG4 . 3(

0' .# 8.

0' .# 8. 4 (

7.1 – ,-#! :

1 (1% : iyxz += 1O iyxz −=. 1O :

1 2 1 1 2 2 1 2 1 2

1 2 1 2

( ) ( )

( ; )

z z x iy x iy x x i y y

x x y y

+ = + + + = + + +

= + +

1 2 1 1 2 2

21 2 1 2 2 1 1 2

1 2 1 2 1 2 2 1

1 2 1 2 1 2 2 1

( ) ( )

( ) ( )

( ; )

z z x iy x iy

x x ix y ix y i y y

x x y y i x y x y

x x y y x y x y

⋅ = + ⋅ +

= + + +

= − + +

= − +

1 1 1 1 1 2 2

2 2 2 2 2 2 2

1 2 1 2 2 1 1 2

2 22 2

1 2 1 2 2 1 1 2

2 2 2 22 2 2 2

( )( )

( )( )

( ) ( )

z x iy x iy x iy

z x iy x iy x iy

x x y y i x y x y

x y

x x y y x y x yi

x y x y

+ + −= =

+ + −

+ + −=

+

+ −= +

+ +

8

2 2 2 2 2

2 2 Re

2 2 Im

( )( )

z z x z

z z iy i z

z z x iy x iy x ixy ixy i y x y

+ = =

− = =

⋅ = + − = − + − = +

2( 1% : P#@ * # # " E'w i= α + β. 1O :

;z w z w z w z w⋅ = ⋅ ± = ±

( 0)z z

ww w

= ≠

1

y

(1, 2)z =

x

2

2B (1, 2)z = −

9

8.1 – &'( % :

' # ? ? = & % # % 0 : + T, #oxz :

cos cos

sin sin

yy r

r

xx r

r

= θ ⇒ = θ

= θ ⇒ = θ

>' #G Tr M& P ')0 ( P /#z= >' #Gθ

M& 8 /#E ox= Q )" "% %.(

;Rr > " " @ 'z >>' / G = :r z= . >% ; θ @ "G'z >' / G :arg zθ =.

1@ $7#tgy

xθ =

22 2 2

2 2

r z x y

r z x y

= = +

= = +

1O ': ( , ) cos sin (cos sin ) iz x y x iy r ir r i r e θ= = + = θ + θ = θ + θ =

@: cos sini

e iθ = θ + θ

) 1@ $> :2 2

cosx

x yθ =

+

2 2sin

y

x yθ =

+ 1@)0(arg

x

y y

x o

r

θ

),( yxz =

10

)(arg ∞ # 91.( "' % 1% 0' 'z &' %:

arctg2 2

yi

i i xz r e z e x y eθ θ= = = +

: ' ; :iz z e

θ= ,, %&'z.

: 1% :z x iy= + 1O 2 2z x y= +.

P# : 2 2 2

z x y z z= + = ⋅ @:

2z z z= ⋅

7 Q8? "E "3.

&# : "' %1 3

2

iz

+= ,, %& ; ' %& ;.

1@ $7# :1 3

2 2z i= + > ' %& 8z T1

2x = 3

2y =

T12

3

2

122

=

+

=z tg 3

y

xθ = =.

1@ '02

1>=x 0

2

3>=y 1O z 1 4 ' .

1:: 3

πθ =.

' ,, %& 1O > z 83i

iz z e e

πθ= =.

!: " 4 1 ",, N # 1%.

. : 1% :1

1 1i

z r eθ= 2

2 2i

z r eθ= 1O :

11

1 2 1 2( )1 2 1 2 1 2

i i ire r e z z r r e

θ θ θ +θ⋅ = ⋅ = P#:

[ ]1 1 1 2 2 2

1 2 1 2 1 2

(cos sin ) (cos sin )

cos ( ) sin ( )

r i r i

r r i

θ + θ ⋅ θ + θ =

= θ + θ + θ + θ

34 " G#O' ">' > >4 > 1 SG> ) SG = SG SG = =>) 1@ #:

1 2 1 2 1 2

1 2 1 2 2 1

cos ( ) cos cos sin sin

sin ( ) sin cos sin cos

θ + θ = θ θ − θ θ

θ + θ = θ θ + θ θ

:

1@ $7#: 1

1 2

2

1 1 1

2 22

ii i

i

z re re

z rr e

θθ − θ

θ= =

# H%:

cos sin cos2

cos sin sin2

ix ixix

ix ixix

e ee x i x x

e ee x i x x

i

−

−−

+= + =

−= − =

': cos( ) cos( )

cos cos2

cos( ) cos( )sin sin

2

sin( ) sin( )sin cos

2

x y x yx y

x y x yx y

x y x yx y

+ + −⋅ =

− − +⋅ =

+ + −⋅ =

⇒

12

N # 1% %8S&# @.

9.1 – ! :

1% : ( , )z x y= ( , )w = α β "> () 1O =1 1 ":

(1 0z ≥ (5 wzwz +≤±

(2 zz = (6 )0( ≠= ww

z

w

z

(3 (7 wzwz −≥±

(4 wzw.z ⋅= (8 zyxz 2≤+≤

"#$:

* #1 ( 2 ( 1 z.

@3 ( 1% 1 * # : 2 2 2

2 2 2

x x x y z

y y x y z

= ≤ + =

= ≤ + =

@ 4( 1% 1 * # zzz ⋅=2 14 :

2 2 2. ( )( ) ( )( )z w zw zw zw z w z w= = =

P#: zw z w=

# H%: 2 2 2

2 2

( )( )z w z w z w z zw zw w

z zw zw w

± = ± ± = ± ± +

= ± ± +

Re

Im

z z z

z z z

− ≤ ≤

− ≤ ≤

13

14 :zw zw=. T2 Rezw zw zw+ = 1O :

2 22 Rez w z zw w± = ± +

3 ( 1% :2 22z zw w≤ + +

3 4 ( 1%: 2 22z z w w= + +

P#( )22

z w z w± ≤ + ' # : z w z w± ≤ +

"377 ( 1 * # 5.( A',<8 ( #:

( )22 2 2 2 22 2 2z x y x y x y x y x y= + = + ≤ + + = +

P#: yxz +≤

# 2)@ "/ 1: ( )

( ) ( )22222

222

22

20

yxyxyxyx

yxyxyx

+=++≥+⇒

≥+⇒≥−

P#: zyx 2≤+

10.1 – !! / :

' # 0 %& : 1% #z "# A O 1

rz = arg z = θ. ' #3 O z )0 ( 1 F37#A

;: ,A 1O ? rz = A#% %= x

y

y

x o θ

A

z

),( yxz =

14

#'arg 2z = θ + π : = R% #1O "#, rz = >#' D'E arg 4z = θ + πU . . .%8 #R% :k 1O> rz = >#'

arg 2z k= θ + π Tk ∈Z )Z "E 4 ".( ) ' : "' "G z " Q % ( %>& 1% +#

,,z 8: 2 ;i k i

z z e kθ+ π= ∈

"3k A z )0.( : "## z >' ; :

( )21

; ,

ki

n n nnz z z e n k

θ+ π

= = ∈ ∈

)@ .% 0,1, , 1k n∈ −… 14 > 2)4k *+> # >.# ; >;4.

1@ DEzz =0. #$ : z ,, %& ; P ' % 1 ' .

15

11.1 – ! :

% @ 1z ∈C 1O :

nn zz =

"#$ : ,, %&> z 8: ( 2 )i k

z z eθ+ π=

> "## 1O 'z 8:

2 2( ) ( )

k ki i

n n nn n nz z e z e z

θ+ π θ+ π

= = =

141ie

β = % @ 1β∈.

&# : @ 11z = − 1O 1− 8 1 1 (0)i− = − + 1 1− = arg( 1)− = π

,, %& '( 1)− 821 i k ie

π+ π− = ': 2

( )3 31 ;

ki

e k

π+ π

− = ∈ > " 3 T7, )@ .%k 4 = > 2)4 "' 1k .# %

*+ # . )V# H 0,1,2k ∈ @ 1, 2,3k ∈ @ 2, 1,0k ∈ − −= ...W. > #k H 7) % ="' ?3 ?.

3 1:3 1− 8:

3i

e

π

@ 10k = ie

π @ 11k = 5

3i

e

π

@ 12k =.

3 1:3 1− 85

3 31 ; ;i i

e e

π π

−.

8 ' %&' :1 3 1 31 ; ;

2 2 2 2i i− + −.

"'% 1:( 1)− ". ) 3 T7, 8.

-1 x

y

16

") ( 1)− ". ) 3 ) 8. "## ( 1)− 8n ". ) "3.

*0 &# : "'% 1z =. #1 1= arg1 0=

,, %& '1 8: 0 21 i k i

e+ π=

1O ':

2

3 31 ; 0,1,2

ki

e k

π

= ∈

1O ' 33 1 "3 =k 8 : 2 4

3 31 ; ;i i

e e

π π

@ :1 3 1 31 ; ;

2 2 2 2i i− + − −.

% 1@ Tk ∈ %&' P ' % 1% :( ,0) (0)x x x i= = + = 8 % 1O . @⊂ .

@ ; %. # . # P#4– 7,– 2

2

2 2(1) 2

2 2( 1) 2

k i

i k i

e

e

π

π + π

= =

− = − =

1% : 2

2( )

2 2

2 2

ki

n n n

ki

n n n

e

e

π

π+ π

=

− =

3 7, 1% 2− 8:

2

( )22 2 ; 0,1

ki

e k

π+ π

− = ∈

@3

2 22 ; 2i i

e e

π π

.

1 x

y

17

82 ; 2i i−.

# 12 : – 7,B 5

2

3

2

1i+.

# " Q8 iz2

3

2

1+= %#z ; ,, %&:

221 3

12 2

z

= + =

arg3

zπ

= F' #@ %.

P#: 2

31 3

2 2

i k i

i e

π+ π

+ =

':

2

15 551 3

; 0,1,2,3, 42 2

k ii

i e k

π+

+ = ∈

π

3 1% 51 3

2 2i+ 8:

7 13 19 25

15 15 15 15 15; ; ; ;i i i i i

e e e e e

π π π π π

12.1 – ! :

1% nnn iyxz += iyxz +=.

/-! : " 1% nnz > # "' z :)⇔ ( 1> A>#%

nnx

nny ' # 1x y ;.

1@ @zzyy

xx

nnn

n

nn =⇔

=

=

∞+∞+

∞+ limlim

lim .

18

@0

00

nn

nn

nn

x x

z zy y

→+∞

→+∞

→+∞

− →

⇔ − →− →

"#$ : "$# 1.1.9 SG)8 ( #: zzyyxxyyixxzz nnnnnn −≤−+−≤−+−=− 2)()(

P# ; E# .

13.1 – ! :

" 1@ '"+G A " ;: X 7 / 1O ="$# "' " 7 " ∑

≥0n

nz " ;:1 1>

∑≥0n

nx ∑≥0n

ny SG4 " SGI".

" 14 " "+G ∑=

=n

k

kn zS0

Y %– 1.1.13 –

">) SG4 " SG4 0

n

n k

k

S x=

′ =∑ 0

n

n k

k

S y=

′′ =∑

;.

&# : " "1 log

n

n

i

n≥∑ " A.

T "37' 1 1 "# 1% P#@)≤ (" .

P@G ;: " E ;: V# ##O

% ) : #1i = arg2

iπ

=.

,, %& 'i 8 :

Z∈=+

keiiki

;2

2π

π

':

y i

x

19

22 cos( 2 ) sin( 2 )

2 2

ni nk i

n n ni e nk i nk

π+ π π π

= = + π + + π

':

2 2 2

cos( 2 ) sin( 2 )2 2

n

n n

n nnk nk

ii x iy

n n n

π π+ π + π

= + = +

T1

n

n

x≥∑

1

n

n

y≥∑ ' 1O 1

21

n

n

i

n≥∑ "' .

20

' "

1.2 – :

' Z ' ' % 8 f ; 3 )VP ) . 1 SG ; @ 1 SG P3 )V(

:

( )

f

z w f z

→

→ =

C C

2.2 – 3 # :

1 ( ' 1 #)z(fw = P#@ : "3)⇔ ( > "3 % A'3z = > "3w.

2 ( #1 ' )(zfw = P#@ ,% @ :)⇔ ( "3 % A'3 >z > 3 =w.

&#: • 2( )w f z z= = # P#4 "3 ' 8:

2 2 2 2( ) 2w z x iy x y xyi= = + = − + > "3 % z > "3 /' w. @ 1 7,1 2z i= + # :3 2w i= +.

3 "' @ 1 %8z. • n zzfw == )( ' 83 > "3 % 14z /' n >> ">3w ) $#10.1.1.(

3.2 – ' :

1%f g "3 1 1' .

21

>' G#fD fR E f ;. >' G#gD gR E g ;.

% #g >'f >' P G#gof ' P#V'( ))())(( zfgzfgo = > = P. " 1% gofD 8:

gfgof DzfDzD ∈∈= )(: 1 T ? 1% P#@ $ gof 1% [%f gR D∩ ≠ φ.

1% gofD 1 SG 8fD ? .D'E gof ;> F fD A#% : /%'gf DR ⊆.

' #$# :f 1> Q>#3 # 1@ 1% =? ? P .E' %# = )(zf ,, %&' ' %&':

( ) ( , ) ( , ) iw f z u x y iv x y w e

ϕ= = + = ( )iz x iy z e

θ= + =

TRR →2:v,u 1 1' 1 1Z '. ;),( yxu ' SG'f >' P G)(Re),( zfyxu =.

; %),( yxv > ) SG'f >' P G)(Im),( zfyxv =.

4.2 – 4 :

1%0z ∈C 0>δ K..

z Dgof

Df

f(z)

Dg

Rf

gof

Rg

Rgof g(f(z)) f g

22

> ? . ? #0z 3 E#'δ >' P G#)( 0zNδ >8 1> SG 1 ' : 0 0 0( ) : ( , )N z z z z D zδ = ∈ − <δ = δC . >8

QG% - . (30z Q3 E#δ. SG 1 #G ⊆C - . P#@ :)⇔ ( 1% :

GzNzNGz ⊂∃∈∀ )(:)(; δδ 7>) >&# . ? = "%' E 1\ 1 ##

H.

5.2 – # #! :

1%f ?' RF > ; 0z S#, '0z ,%4 ;. 1@ #=

→)(lim

0

zfzz

:)⇔ ( 0 :

εδδε <−⇒<−>∃>∀ )(:0,0 0 zfzz @0)( →− zf 1% %00 →− zz Tz > 0z.

A3 : P#@ # 8z 0z 1O )(zf > > . 3 1@ Tz 10z

( )0zz → P # 9 ' 3 8 14 =' 1

. "3 Z 1@ P#O ' Z ' P% z 1 8

0z. ) 3 1% x 10x ( )0xx → ">/# "> > > 1' '

".(

6.2 – ! :

ε

f(z)

δ

z 0z

23

1%),(),()( yxivyxuzf += > ; "3 '

0z S#, '0z ,%4 ;. iyxz += 000 iyxz += βα i+=.

# +#:

=⇔

=

=

→

→

→)(lim

),(lim

),(lim

0

00

00

),(),(

),(),(zf

yxv

yxu

zz

yxyx

yxyx

β

α

"#$: "$# 1.1.8 SG)8( 1O :

−≤−+−≤− )(2),(),()( zfyxvyxuzf βα ; E# P#.

7.2 – 56 :

1%f ?' F "3 > ;0z. 1@ #f # 0z :)⇔ ( A#%)()(lim 0

0

zfzfzz

=→

.

Y % 8: εδδε <−⇒<−>∃>∀ )()(:0,0 00 zfzfzz

@ 0)()( 0 →− zfzf #00 →− zz % @ 1z 0z. 1@ #f " ; ? ( )G G⊆ C :)⇔ ( 1%f >

1 "# % #G. 1O "/# " P#@ $f ># 1% 1@ ' 0z =

#' 1% 1@ "f # F 0z .

8.2 – ! :

24

1%),(),()( yxivyxuzf += >3 > "> >> ; ),( 00000 yxiyxz =+= . ]+#: f # ? 0),( zyxu ⇔ ),( yxv # 1 ),( 00 yx.

T:

),(),()( 00000 yxivyxuzf +=

1@ # 8 1O :

)()(lim),(),(lim

),(),(lim

000

),(),(

00),(),(

0

00

00 zfzfyxvyxv

yxuyxu

zz

yxyx

yxyx=⇔

=

=

→

→

→

"#$: "$# 1.1.8 SG)8( 1O :

0 0 0 0 0

0

( ) ( ) ( , ) ( , ) ( , ) ( , )

2 ( ) ( )

f z f z u x y u x y v x y v x y

f z f z

− ≤ − + −

≤ −

* # P#.

&# : ' iyxzf += 2)( ; 142),( xyxu = yyxv =),( "# % # 1 ),( 000 yxz =.

#'11

)(−

++

=x

yi

y

xzg ; \ (1, 1)−C.

#$ :

;>' ="> ' A/#' " A$# % ;' ' "'#' "E"/'& "' 18' =".

25

9.2 – :

1%f ?' ' "3 0z S#, '0z ,%4 ;. 1@ #0)(lim

0

=→

zfzz

:)⇔ ( A#%0)(lim0

=→

zfzz

.

1@ #∞=→

)(lim0

zfzz

:)⇔ ( A#%∞=→

)(lim0

zfzz

.

10.2 – 7#(6 :

1%f > ; "3 ' 0z. 1@ >#f >># 0> & >'0z :)⇔ ( ">>/# A>>#%

h

zfhzf

h

)()(lim 00

0

−+

→ @

h

zfhzf

h

)()(lim 00

0

−+

→ .

G' 8 "/# Q/ G#)(' 0zf 0 & /#f >#

0z T =0z z h− = ∈C ?. 1@ @:

0 00

0

( ) ( )lim ( )h

f z h f zf ' z

h→

+ −=

P#@ # 8:

0 00 0

( ) ( )0, 0: ( )

f z h f zz z h f ' z

h

+ −∀ε > ∃ > − = < ⇒ − <δ δ ε

'3 :f & "# % #z UGz ∈ 1@ # f ; 0 & ' "G.

' ) ' 0 & 1@ > ' 0 & 11. 1' ?'% ?8 7 ) H#8 1@ : =8$ %& T 1.

P 1% = ' 0 & 14x 1 0x 1'> >' :0xx →> 0xx →<.

26

>> 3 P> > => ' 0 & #'z 1>>0z )( 0zz → ' 1 P # 9 ' . : ">/# 1@ A> 1

P% '' 0 z 1 80z . 1@ > ">/# "3 1@ @ 3 ' Z ' Z .

&#: ' 1)( 2 += zzf ; 0> & >' >% . P>>#4z∀ ∈C

1O :

0

0

20

2

0

0 2lim)()(

lim00

zzz

zz

zz

zfzf

zzzz=

−

−=

−

−

→→

&#: ' @4)( += zzg "# @ # 0 & ' 0z 1 . 14:

0 0

0 0

( ) ( )lim limh h

g z h g z h

h h→ →

+ −=

1% O 0→h 'ox 1O xh = ' :

1limlim00

==→→ x

x

h

h

xh

1% :0→h ) 'oy 1O iyh = ':

1limlim00

−==→→ iy

iy

h

h

yh

"/# 1:h

h

h 0lim

→ > ' = 3 ' Z Z /#4 Ag

0 & ' .

11.2 – ! :

> 1%),(),()( yxivyxuzf += >>' > ">3 > >'

000 iyxz += # +# =: f # 0 & '0z⇔ 1 1& 0 :

27

1( #% 1@v u '3 # . 1),( 00 yx. 2( 1 0 1@:

),(),( 0000 yxy

vyx

x

u

∂

∂=

∂

∂

0 0 0 0( , ) ( , )u v

x y x yy x

∂ ∂= −

∂ ∂

) >' #@ G#0 0 0 0( , ) ( , )x

ux y u x y

x

∂=

∂(.

: 1% : P#@ $7#f "># ># 0 & '0 0 0( , )z x y= T> =

f u iv= + # P#O : 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

( ) ( , ) ( , )

( , ) ( , )

( , ) ( , )

( , ) ( , )

x x

y x

x y

y y

f z u x y iv x y

v x y iv x y

u x y iu x y

v x y iu x y

′ = +

= +

= −

= −

12.2 – # :

1%ivuf += 1' 1% :v u "#, "' 1 "+G A & . # +#:

yx

v

x

u

y

v

x

u

∂∂

∂=

∂

∂⇒

∂

∂=

∂

∂ 2

2

2

xy

v

y

u

x

v

y

u

∂∂

∂−=

∂

∂⇒

∂

∂−=

∂

∂ 2

2

2

Txy

v

yx

v

∂∂

∂=

∂∂

∂ 22

1O :

02

2

2

2

=∂

∂+

∂

∂=

y

u

x

uu∆

28

1@ # " .#' :0=v∆ '0=f∆ . "> ;> :

02

2

2

2

=∂

∂+

∂

∂=

y

u

x

uu∆ >?' =>/ ' % ; =". (7' "

? .

. : 22),( yxyxu −= 14 ' :

022 =−=+= yyxx uuu∆ #'22),( yxyxu += 14 ? :

2

24 0 ( )xx yy xx

uu u u u

x= + = ≠ =

∂∆

∂

13.2 – # ('– "# :

1 ; :

∂

∂−=

∂

∂∂

∂=

∂

∂

x

v

y

u

y

v

x

u

&% 1 +G 1 . 1 '– 1. 1> % . "'3 & ;: " <' =1 1 8 "8@ 1%

u v % (# "$# 1.2.14U '> 0 & "' G A.E /#%fivuf )( +=.

1' 4 1% P#O 8 ;u v ?SG> >? ?SG #% 1@ ' ?) f "$# & 0 &7 7'31.2.14 1>' 1: /

&% ' – 1 .

29

14.2 – ! :

1% :ivuf += 1%u v ># 1> &% 1 "#0 0( , )x y 1O =f "# Q8 # 0 & ' .

: ' U" ' " 0 &' " A$# % ;' ">+3 ;

"/'& "' 18' =" ' " .

. :

1% :f g 1O 0 & 17'gf ± fg f

g )( ) 0g z ≠ >#

8# ' #g 0 & ( 0 & ' #:

)()()()( z'gz'fz'gf ±=± )().()()()().( z'gzfzgz'fz'gf +=

)(

)()()()()(

2 zg

z'gzfzgz'fz

'

g

f −=

15.2 – :

1@ #)(zf # 0> & >'∞=z :)⇔ ( 1%1( )fz

'

# 0 &0=z.

30

"#

8 3#$ : P S3 '3 1 ' = 1@ V#.

1 – ' %& ; " 4 1 % % :

( )510

31;2

1;

21

21i

i

i

i+

+

+

−

&:

8 ' %& :iyxz +=.

T22yxz += θ=zarg U

x

ytg =θ.

1%31 iz += 1O 2=z 3

argπ

θ == z.

> ,, %& 1:z 8:

32

πθ

ii

eezz == P#:

( )5 5

5 31 3

1 3 2 32( )2 2

i

i e i+ = = −π

1:: ( ) ii 3161631

5−=+

.#' ":

' '22

1

2

1 iiz +=

+= 1O 1=z

4arg

πθ == z.

1:: i

ez 4

π

= P#:

31

ieei ii

===

+2

5

410

10

2

1ππ

1::

ii

+=

+0

2

110

%# =)4 "'#' @:

ii

ii

i

i

i

5

4

5

3

5

43

)21()21(

)21(

21

21 2

−−=−−

=−+

−=

+

−

2 – @ % 0 " ": 76 2;)5(;232 iii +−−

&: % @ 1 #z ∈C 1O θi

ezz = . 1O 'θinnn

ezz = P#: / 1

n nn in inz z e z e= = =θ θ

θinnn

ezz−=

1:: ( ) nnin

zezz == − θ 1@ @n

nz z=. 1: :

iii 232232;4412232 +−=−−=+=−−

( )7 7 7 7 72 2 2 ; 2 2 ( ) 2( )i i i i i= = = = −

( ) ( ) ( ) ( )6 66 666(5 ) 5 26 ; 5 5 5i i i i i+ = + = + = + = −

3 – " I ") :

32

( ) iii 2222;1;31;11

++ 1 "' :i267;1 +−−.

&: Q / +# ,, %& ; "' % 1%@ %.

@ 1i # :1=i 2

argπ

=i.

,, %& 1:i 8: i

kiiki

eiei 5

2

1052

2

πππ

π++

=⇒= TZ∈k . )@ .% 4,3,2,1,0∈k % 2)4 "' 14 .

3 1:5 i 8: iiiii

eeeeeπππ

ππ

10

17

10

13

10

9

210 ;;;; "'#' @1 1O ik

eπ21 = P#:

Z∈= kei

k

;1 5

2

5

π

)@ .% 4,3,2,1,0∈k 3 1:5 1 8:

2 4 6 8

5 5 5 51 ; ; ; ;i i i i

e e e e

π π π π

@ 1i2222 + #:

1,42222 ===+x

ytgi θ

022 >=x 022 >=y 1:4

πθ =.

P#: iki

eiπ

π2

442222+

=+⇐

4,3,2,1,0;42222 5

2

2055 ∈=++

kei

iki

ππ

SO'k ) # P3"' ".

33

?)@( )1131 i+ :

( ) 111111

23131 =+=+ ii

T )(arg)arg( znzn = 1O :

( ) ( )3

11

3.1131arg1131arg

11 ππ==+=+ ii

P#:

( ) iki

eiπ

π2

3

11

1111231

+=+

( ) 5

2

15

11

5

11

5 11231

iki

ei

ππ+

=+⇒ SO'k 4,3,2,1,0 #. #' "')1(− #:

Z∈=−⇒=−+

+ keeiki

iki ;11 22π

πππ

P#:

8*9i267 +− :

7

26;11267

−==+− θtgi

"G 1@ $7#θ /& G 1 A . "> Q>8 V# H ;:i267 +− ' " ":

1@ K.#iyxi +=+−⇔ 267

@ 1k G @ 1k

−=−

:

:1

i

i

34

xyiyxiyxi 2)(267 222 +−=+=+−

T1126726722

222 =

+−=+−=+=+ iiiyxyx

" A " ; E# P#:

=+

=

−=−

11

262

7

22

22

yx

xy

yx

1@ # " Q8 ':

2±=x 3±=y 1O P#:

( )ii 32267 +±=+− 4 – " :0)8()5(2 =−+−− iziz.

& :iii 68)8(4)5( 2 −−=−−−=∆ # Hiyx +=⇔ ∆

xyiyxiyxi 2)(68 222 +−=+==−− ∆ # 1 "'':

62

822

−=

−=−

xy

yx

T22 210 x y= = +∆.

" ; E# A:

=+

−=

−=−

10

62

8

22

22

yx

xy

yx

P#1x = ± 3y = ∓. '1 3i= ±∆ ∓ 8 1 P#:

35

1 2z i= +

2 3 2z i= −

5 – 1@ 18'z

z

z 0lim

→ 9.

& : "/# 1: ; '' 0 8 z # Q')0.( 1% :0→z 'ox 1O R∈= xz P#:

1limlim00

==→→ x

x

z

z

xz

A#% :0z → 'oy 1O iyz = P#:

1limlim00

−=−

=→→ iy

iy

z

z

yz

Q' ; ' Z ' AZ "/# Pz #)0 .( ">/# ' A.

6 – 1

1lim

6

2

+

+

→ z

z

iz.

& : 1 8 " 1:0

0

1

1lim

6

2

=+

+

→ z

z

iz.

" '016 =+z 1@ #:

3

1

)1)(1(

1lim

1

1lim

242

2

6

2

=+−+

+=

+

+

→→ zzz

z

z

z

iziz

7 – 9 ( 1@ 18' :)()log()( 2222 yxiyxzf −++= ; *

C.

( ' g >' :

36

2 4 : 22( )

3 4 : 2

z z iz ig z

i z i

+ ≠ −= + =

&:

@ (f ;>> ? >>*C 14)log(),( 22 yxyxu += 22),( yxyxv −=

; 1 *C.

( #g "# #iz 2= :

)2(4)2(

)2)(2(lim

2

4lim)(lim

2

2

22igi

iz

iziz

iz

zzg

iziziz≠=

−

+−=

−

+=

→→→

P#g # 9iz 2=.

8 – – 1 '– ' 0 & "'3:

1

1)(

+

−=

z

zzf

1R'1% 1: f ' # 0 &∞^

& : ;@ " :f >% # 0 &7 F7'3 / =0 &7 1'3 1' "3 8

"#)0,1(−=z.

"#, ":

2222

22

)1(

12

)1(

1

1

1)(),(),(

yx

yi

yx

yx

iyx

iyxzfyxivyxu

++

−+

++

−+=

++

−+==+

2# 1@ 22

22

)1(

1),(

yx

yxyxu

++

−+=

22)1(

12),(

yx

yyxv

++

−= 1> >&

$# "1.2.14 "# % #)0,1(−=z. 1O 'f ; 0 & ' \ 1G = −C.

37

"# "'#')∞ ( #:

1@ '1 1( )

1

zf

z z

−=

+ # 0 & ')0 ( 1O

1

1)(

+

−=

z

zzf # 0 & '

)∞.(

38

' "

1.3 – :

' % ;f " . " ; 0 & '( )G G⊆ ; = ; ? ?' G ) ; 8 @G.(

; " ' " G#G G'( )H G. 1@ @ :( )f f H G⇔ ∈ - . ; 0 & 'G. :( ) 2f z z= + ; C.

2( )

1

zg z

z

+=

− ; \ 1

2.3 – :

1@ #f "# # 0z :)⇔ ( 1>%f >> ; >? >0z.

3.3 – ! :

1% :)(; GHgf ∈. 1O )(GHgf ∈± ( )

fH G

g∈ )( ) 0g z ≠ 1% # #g ?

8#( )(GHgf ∈⋅. 1% :)(GHf ∈ )( 1GHg ∈ 1)( GGf ⊆.

1O )(GHfgo ∈.

39

4.3 – ! :

@

!: 1% 1@ R #f u iv= + 1> >% R1@ ;: RX ? u v . 1% : 'u @v ? R1O f ? .

5.3 – :

1@ #)(zf ' ∞=z :)⇔ ( 1%1( )fz

'0=z.

6.3 – :! :

SG 1 #( )G G⊆ C : ' P#@)⇔ ( 1 # % 4 1%

21 , zz 1G E ) 1z >'2z ) P%' G. 1@ @G 1>) >9 1+G ;: 8& 1% 1G 2G T>'

φ=∩ 21 GG GGG =∪ 21. ; G : "#)⇔ ("' " . A#%.

1 (u,v ; . 17'G. 2 ( &% – ; 1 1G. 3 (G 1 - . SG.

)(GHivuf ∈+=⇔

yxyx vvuu ,,, .

xyyx vuvu −== ; - . ;G )(GHivuf ∈+=⇔

40

7.3 – ! :

1% G "# . 1% : +#:

"#$:

ivuf += 1: ),(),()(0 yxivyxuz'f xx +==.

P#0),( =yxux 0),( =yxvx . ')(),( yhyxu = Th ' ' 0 & . 1 &% – 1 # )(),(),(0 y'hyxuyxv yx ==−=

# P#cyh =)( A', . 1:cyxu =),( .

1@ # " .#'1),( cyxv = A', 1O 'f A',.

8.3 – :

" #),( 0 rzD QG% . E30z Q>3 >E# r )0≥r( :

; 0 0( , ) :D z r z z z r= ∈ − ≤C FZ ?E3. ; 0 0 0( , ) \ : 0D z r z z z z r= ∈ < − <C > > . E3 G%0z.

# @z & 0 :rzz =− 0 ( ; / ),( 0 rzD 8G% + ; @0z 83 E#r.

)(GHf ∈ 1O )⇐ (f ; A',G.

Gzz'f ∈∀= 0)(

0z r

z 0 0( , ) :D z r z z z r= ∈ − <C

41

1 ' % 1' 0 :

rzz =− 0 iterzzt +=≤≤ 0;20 π

1@ # /# % z 8G% + ; 0z 83 E#r.

"' % @rzz >− 0 # 1@z

N) ' + N) (0( , )D z r. 0z z r− < 1@ # z ( ).

1% :00 =z 1r = ; )1,0(D ; (31=z @itezt =≤≤ ;20 π +.

9.3 – ;< 55 & # :

A#% :∑≥

−0

0 )(n

nn zza >/' >3 >E# ="' 23 "

RR )0( +∞<≤ "> 1> ">) "# % # =" Q/ 1% +# = /' ),( 0 RzD >' P G# # =)(zf 0>' ; %8 E# f

>' :

0

0

0

: ( , )

( ) ( )nn

n

f D z R

z f z a z z≥

→

→ = −∑

C

# ' 1@ +# f 23 "' ,∑≥

−0

0 )(n

nn zza 1@ ># % =

' ' ", =Q8 2 "f.

0z r

z

42

10.3 – ! :

1% :f 23 "' F7, ' ∑≥

−0

0 )(n

nn zza >3 E# "'

/' R . # +#:

1 = ( )),( 0 RzDHf ∈ f 1> P >& ;> ="' @ 1 0 & ' "' k >':

( )0( ) ( 1)( 2) ( 1) ( )k n k

n

n k

f z n n n n k a z z−

≥

= − − − + −∑

>' ,4 ; :( )

0( )

!

n

n

f za

n=.

2 – , f ? 1% 23 "'. 1% : @f >' 7, :∑

≥

−0

0 )(n

nn zza >' :∑

≥

−0

0 )(n

nn zzb 1O nn ba =.

11.3 – ! :

1%:f G →C :g →Ω C 1 . ; 1 1' G Ω T' =Ω⊂)(Gf :

( ) zzfgGz =∈∀ )(: '3 : +#g 1% 0 &0)( ≠z'g 1O f 0 & ' # ?@:

( ))(

1)(

zf'gz'f =

"#$:

1%Gz ∈0 Ghz ∈+0 T'0≠h. 1@ '( )0 0( )g f z z= ( ) hzhzfg +=+ 00 )( 1O :

)()( 00 hzfzf +≠

43

1% P#40 0( ) ( )f z f z h= + 1%( ) ( ))()( 00 zfghzfg =+ ># >8

00 zhz =+ @0=h K. K3# .

1:0)()( 00 ≠−+ zfhzf Tf 1O :

0

)()( 00

→

→+

h

zfhzf

':

( ) ( ) ( ) ( )

( ) ( )

−+

−+=

−+⇒

−+⋅

−+

−+=

−+=

)()(

)()(

1)()(

)()(

)()(

)()()()(1

00

00

00

00

00

0000

zfhzf

zfghzfgh

zfhzfh

zfhzf

zfhzf

zfghzfg

h

zfghzfg

=K. 0→h 1@ #:

( )0

0

1( )

( )f ' z

g' f z=

12.3 – 5 # :

G' 4 ' G#ze % @ 1 P # =z ∈C 2 "' =

:∑≥

=0 !n

nz

n

ze T10

!

)!1(lim

1

<=⋅+

+

+∞→ n

n

n z

n

n

z " 3 E# 1O

+∞=R "$# =2.2.3 , ' 1O ze >% > @ =

( )ze H∈ C. "' 1 0 & ;k >':

z

kn

knkz

ezz

zk

kz

k

k

kk

zn

knnne

=+++=

++

⋅+

++

++=

+−−=∑≥

−

2

2

)(

!2

1

!1

11

)!2(

1

!2

)!2(

)!1(

1

!1

)!1(

!

1!

!

1)1()1()(

44

@zkz ee =)()(.

PE) P'& ()' 4 ' R. %#– F7,– 1@ :2121 zzzz

eee+=

1 2

1 2z z

z z e e= ⇒ =

14ioiee

π2= 1%iπ20 ≠.

Z∈+=⇒= kikzzeezz ;221

21 π

13.3 – :

#R zsin zcos @ 1 z ∈C 2 ' U:

∑∑≥

+

≥ +−=−=

0

12

0

2

)!12()1(sin;

)!2()1(cos

n

nn

n

nn

n

zz

n

zz

/# % 3 E# 1 ' 1 +∞=R . ">$# >2.2.3 1O sin ( )z H∈ C cos ( )z H∈ C

2 1 2 1

1

1 0

(cos ) 2 ( 1) ( 1) sin(2 )! (2 1)!

n mn m

n m

z zz ' n z

n m

− ++

≥ ≥

= − = − =+

∑ ∑

!: # @ 1z ∈C :

∑ ∑

∑ ∑

∑

≥ ≥

+

≥ ≥

+

≥

+−+−=

++=

=

0 0

122

0 0

122

0

)!12()1(

)!2()1(

)!12(

)(

)!2(

)(

!

)(

n n

nn

nn

n n

nn

n

niz

n

zi

n

z

n

iz

n

iz

n

ize

1O "' 7 % 1@ T:

45

zizeiz sincos +=

1@ # " .#' :zize iz sincos −=− 1@ # ':

cos ; sin2 2

iz iz iz ize e e e

z zi

− −+ −= =(*)...

z

zzg

z

zztg

sin

coscot;

cos

sin==

# 1% P ' () 1 ",, ' () Nize @z

e. &' ' # ?@)(∗ #:

122121

212121

cossincossin)(sin

sinsincoscos)(cos

zzzzzz

zzzzzz

±=±

=± ∓

1)(∗ # H%:

1sincos 22 =+ zz

?)@:

14.3 – : :

+G @ #sh >+G @ > =ch 2 ':

∑∑≥

+

≥ +==

0

12

0

2

)!12(;

)!2( n

n

n

n

n

zshz

n

zchz

% 1 ' 1 ( )R = +∞ C $# ' "2.2.3 1O : ( ) ; ( )chz H shz H∈ ∈C C

1 2

1 2

1 2

2sin sin

(2 1)

z z kz z

z k z

= + π= ⇔

= + π − @

46

2 1

1

( )(2 1)!

n

n

zchz ' shz

n

−

≥

= =−

∑

'zcos zsin 1@ 2#:

chzn

z

n

iziz

n

nn

n

nn =−=−= ∑∑

≥≥ 0

22

0

2

)!2()1(

)!2(

)()1(cos

# " .#': zishiz =sin

1:izchz cos= izishz sin−= (>) * # # Pchz shz 1> ()zcos zsin () 1 'iz

e. 1@ 2#:

22cos

)()( zziziizieeee

izchz−− +

=+

==

22

sin)()( zziziizi

ee

i

eeiizishz

−− −=

−−=−=

@ 2

;2

zzzzee

shzee

chz−− −

=+

=

1@ * # # 1 )4 1 ' 1 8 1: z

eshzchz =+ 212121 )( zshzshzchzchzzch +=+

122121 )( zchzshzchzshzzsh +=+

&#: " :10cos =z %#:

yshxiychx

iyxiyxiyxz

sincos

sinsincoscos)cos(cos10

−=

−=+==

1 "'' P#:

47

=

=

)2(0sin

)1(10cos

yshx

ychx

" ')2 ( #0sin =x @0=ysh P#πkxk =∈ )( Z @0=y " K ')1 ( #:

@ 10=y 1O 10cos10cos =⇐= ychxx 1>% >9 8 141cos1 ≤≤− x 1%R∈x.

1% :x k= π 1O :10cos10)1( =⇐=− ychxychk . 1% O k @ 12 += mk 1O 1010 =−⇐−= ychych 8

14 1% 90>ych 1%R∈y.

1% :k G @ mk 2= 1O> 10102

=⇐=+ −

ychee

yy

P#

)9910log( ±=y 3 1% z 1% " 0 mz 8:

Z∈±+=

±+=+=

mim

ikiyxzm

;)9910log(2

)9910log(

π

π

P # 9 1@ $. 2)@ " H#8 :

2cos 10 10 20 1 02

iz iziz ze e

z e e−

−+= ⇔ = ⇔ − + =

P#: 9910 ±=iz

e Tikiziz

eeπ2+= 1O :

2 log(10 99)iz k i+ π = ± P#:

2 log(10 99) ;kz k' i k'= π − ± ∈Z

48

"' " 8 =P # 9 . 1% : P#4z 7 1O z 7.

15.3 – 3 1)3 1': (

•••• ? ? :#!:

1%)(zfw = 1%> => ' α ∈C =),( rDz α∈ T> =0>r # %&' Q ) 1% . A O z α Q ') >%

" ( ' )@ ="% f > ># "3z >.# ;>: "# +# # =P# @' 4 α ' M. "#f . )@ :f

> A ' 3 z α '> > > Q8 # Uf . > + M.' ;4 M ;f.

>E#' 3 .% P#O ="3 = ' # % 1>% Q> @ )V M. "# 1 @' +/# ) > />

1,>< 1> >4 ?G 8 E# )@ 1 x'ox @y'oy (# > Dz # 3 H' 1%# = P% S#,@ =?'@ P'

' M 1 ? . "3 ' Q' ' M. 8 " +# ##%.

&# : ' 1%f >' F :zzfw == )(. 1@ "' K.#rz = 0arg θ=z 0 %& %:

1% +#00 θθ iierezz ==

1%2

0

)(

θi

erzfw == > "% 'z 0=α

0θ z

x

y

o

49

1O> .# ;:rz = #' πθ 2arg 0 +=z 'iierz

πθ 20 += P#:

0

0 0

2( )

2

2 2

( )i

i ii

w f z r e

r e e r e

+

π

= =

= = −

θ π

θ θ

1:f .# # ;4 " .# )V z 1 '' 8 =z 0=α "% .

1:0=α ' M. "# 8zzfw == )( > P> ' 1

22

21

00

)(;)(

θθii

erzferzf −== . > E#' ' #3 O ox B 7,– M. "# 1 @' 0=α > D# =z +# ='@ P'

1O zarg 1@ /#% π2 "3 'f Z . D'E f ">3 ; [ [ \ 0,G = + ∞C.

16.3 – #$ :

=' Z "% ' # ; =/ 1 ' 8 1 '# 1@ =G Z #

,, %&'.

17.3 – ! :

1:0=α ' M. "# 8zzfw == )(.

"#$:

1@ 0' , #' 0=α 1@ A',< =M. "#0=α >8 2)@ "# @ 1@ A',: .% = M. "#β∈C 0≠β 1@ 1%

# 1% ' M. "f . 1 1@ @z β >.# ;>: "% ' "3 1 Z =4 f.

z

x

y

o 0θ

50

1: ' #0≠β )0 %&( 1% rz = 1arg θ=z.

# z β , "% : 1O =4 .# ;zarg

"/# (# , G , (# D'E 1arg θ=z 1 . 1@ @rz = 1arg θ=z 1 '3z β "3 ' =Q'f Z .

18.3 – ! :

: 1% )(zg > ,%z '> > M>. ># 1O> =n zgzf )()( = 0 # 80)( =zg.

:1)( 2 += zzf M. # Piz ±=. 5 )2)(1()( −+= zzzh # P M. 2=z 1−=z.

19.3 – #$ :

"# ; %α ∈C ' @ M. "# 8 8 =f G 1 R' ^α 1 S#,@ # z / . : P#4z 1 # α β 1_>'

Z = ' "3 AfM. "# 8 /# @ ; % # 7 =.

&# :

1%1)( 2 +== zzfw

2 2 2

2 1( )

2

1 0 1

; 0,1

i k i

ki

k

z z e

z e k

+

+

+ = ⇒ = − =

⇒ = ∈

π π

π

P#:

β α

z

−==

+

:

:2

2

i

iez

iki

k

ππ

K G K

o x

y

z β

θ1

51

i z

x 1θ

y

2θ

i−

8 M. # 1:iz ±= ' =)()(12 izizz +−=+ ">' % 1iz + iz − ;: N # ,, %&'iz + iz − E# 1@ =

z 1 # 1 %'i i− )%& %: (

9 ( "# G#i )(1 iN # =z "% i 1 i− ' = >>z 1O>> 4 >> >>.# ;>>: :1)arg( θ=+ iz >>#'

πθ 2)arg( 2 +=− iz 1O P# : 1 2 1 22

( ) ( )2 2

1 2 1 2( )i i

f z r r e r r e

+ + +

= = −θ θ π θ θ

3 "f ' AZ i=α ' M. "# 8f )%& $#@: (

( :z i−=β 1 "%i 1O .# ;: = :

2)arg( θ=− iz #'πθ 2)arg( 1 +=+ iz ' : 1 2 1 22

( ) ( )2 2

1 2 1 2( )i i

f z r r e r r e

+ + +

= = −θ θ π θ θ

1O 'i−=β > M. "#f )%&.(

@ ( 1 # 1 % G# :i± = #z = / "% z

i

i−

z

x

y

2r

1r

1θ

2θ

i z

x 1θ

y

2θ

i−

# 1% :2

2θi

eriz =− =1

1i

z i r e+ = θ P# :)(

212 211 θθ +=+ i

errz ' : 1 2( )

2 21 2( ) 1

i

f z z r r e

+

= + =θ θ

1 '3 8z .

A T7, @ 1%# +#:

o

o

o

52

1O 4 .# ;:: πθπθ 2)arg(;2)arg( 21 +=−+=+ iziz

1O ' :

1 2 1 24( ) ( )

2 21 2 1 2( )

i i

f z r r e r r e

+ + +

= =θ θ π θ θ

"3 1@ @f 1 '3 /# Z z. 1@ # 8f M. # P 1@ @ ="3 .

1 '3 "# G 1O 1:z > >/G 14 =/"Z /' "':.

20.3 – #A # :

' > Q# % =8$ %& T 1 9R = '>' ?>'% 7 ) H#8 =1 T 1 1% =4 ' % ' %

4 ' " P#4 =:

2: z z k iz e e + π∀ ∈ =C

: 1%,z w ∈C. #w ' 9 > z %#zw log= :)⇔ ( 1%w

ez =. .E ' 9 P#@ $7#)0 ( 14w

e=0 0 . % 1O '0≠z "' A 9 1 P # 9 P)3

k ( 14:

Z∈=⇔++= +kezzikizz

iki ;2loglog 2 πθπθ

' 8 9 ' P) 3 k.(

21.3 – ! :

z i

x 1θ

y

2θ

i− o

53

' zw log= M. "# P0=α.

54

"#$ : 1%1arg;1 θθ == zezzi 1 >'3z >0=α >+# =

1loglog θizzw +==. > "% 'z 0=α

D'E .# ;: arg 2z 1= +θ π P# :ii

ezzπθ 21+=

' :iizzw πθ 2loglog 1 ++==. ">3 Z $7#zw log= . 1:0=α >> M>>. "># >8

zw log= . # % 817.3.

22.3 – ! :

' M. # 1@ * # #)(log zhw = T)(zh > ,% 8 Z z# 8 = z 0 0)( =zh.

. :1log( )

1

zw

z

−=

+ 8 M. # P1± 14:

)1log()1log( +−−= zzw

23.3 – ! :

=>4 '> >% ' 8 9 ' 1% ; '4 ' () # P#O = : 4*

1z ∈C *2z ∈C.

1 2 1 2

11 2 1 1

2

log( ) log( ) log( )

log( ) log( ) log( ) ; log( ) logk

z z z z

zz z z k z

z

= +

= − =

1% : P#4:

222

111

log

log

2

1

zwez

zwez

w

w

=⇔=

=⇔=

z

x

y

1θ r

o

55

zw log=

v

u L

–π

π

o 0u

)log()log()log( 212121

212121

zzwwzz

eeezzwwww

+=+=⇒

=⋅= +

# P# :

1 11 2 2

2 2

log log( ) log( ) log( )z z

z z zz z

= ⋅ = +

': 1

1 2

2

log( ) log( ) log( )z

z zz

= −

H% #: 1 1 1 1log( ) log ( . )k

z z z z= …

1

111

log

logloglog

zk

zzz

=

+++=

24.3 – # 5zw log= :

' #logu iv w z+ = = 1 1%& ' #:

%5w %5z

k

y

x

o

wez =

56

#w & ;: /# % K +π2 A > ' H =

"Gou & ' # = :

+∞<<∞−

≤≤−

u

v ππ =>4 >& 8

" O' P# * # )J & % 14πk2. ' Tw

ez = Z P 3 1O "&4 "' .

# 7 5#5 :( 12wez = : " )V#L )%& $#@ ( /

=

≤≤−

0uu

v ππ 8 / E ':

0uw u iv ivz e e e e

+= = = ⋅ P#0u

ez = . E 'L 8G% + 8)o (83 E#

0ueR = . > " 1@ TL /

=

=

0uu

v π " E '

8: 00 uiu

eeez −=⋅= π

> ;. " L / :

=

−=

0uu

v π.

8 / E ': 00 uiu

eeez −=⋅= −π " E 1@ @ > ;. " L > ;> 1 1'# =

. ' " L P#O 4 & D H +∞→0u 1O>

−∞→−= 0uez.

−∞→0u 1O 00 →−= uez P#@ @L & D 4

w E 1O L ) 8+ ( D z 7% .

57

" " E 1% . ; 1 '# & ;P% .

> " E /#4 %&' 8 # /$ ' 1_' ;. " E.

1:zw log= @ => > ; ". 3 )V1 & 1 3 P /$ ... ' /$ ' 8 8zw log= > '> %

. # O ] ]0,∞− '> 1O zw log= >3 > D'>E " ; ] ] \ ,0G = −∞C.

"$# 11.3 P#O :

zee'e'zGz

zww

111

)(

1)(log:

log====∈∀

: + # )'] ]ππ ,− P> > 1@ # P G @ = ; '# 1%π.

+ # ) 8] ]π2,0 >8 P 1O =[ [∞+,0 P G = @ =2π= ...%8.

25.3 – :

% 4*z ∈C α ∈C # =:

zez

logαα =

7#$ 8 1 1@ :

αθαα

θθ

izz

izzezzi

+=⇒

+=⇒=

loglog

loglog

P#:

αθααθααα iizzezeez ===

+loglog

58

yz

1x

-2 -3

1r

2r

1θ

2θ

Q#3 81.1.12. : :ii)1( +.

#iki

eiπ

π2

421+

=+ P#: log 2 2 2

log(1 ) log 24 4(1 )i i k i k

i i i ii e e e e

−π+ + − + + = = = ⋅

ππ π

26.3 – 3< # < #5 ':

1 ' 1 ' ,, %&' " Q8 ' ) . ">8@ 2#>A7% 8.(

: 1%2)( 2 +−−== zzzfw T'( 3) 2f i− =. A# :z 1)3(− "#' Q ' Z ## "

2−=α "#' 1=β. "3 V )3(−f #z "# ;:)3(− >' > 1

"%.

& : ' M. #f 82−=α 1=β. 14)2)(1(22 +−=+−− zzzz E# z 1 #>'α β > %# ;>

)(zf " / ,, %&' . Tf 3 .

2

22)2( θierzz =+=−−

1

11 θierz =−

P#iierz

πθ +=− 1

11 ':

)2(21

2 21)2)(1(2 ππθθ kierrzzzz

+++=+−=+−− 1:)(zf 8 ,, %& ;:

59

1 2 2( )

21 2( ) ;

ki

f z r r e k

+ + +

= ∈θ θ π π

(*)... Z

+ ' & ' "37 Q8 1if 2)3( =− 1> >k ' # D'E =+ ' & 0 M. ' ; E

M. H. E #z ;:)3(− 1O :

1232

4311;,

2

121

=+−=+=

=+=−===

zr

zrπθπθ

# '3−=z #)(∗ : 3 2

( )22 ( 3) 4 2

ki

k ii f e i e

+

= − = = −π π

π 1% 1@ P#k )@ .% = 1=k. 8 + ' & 0 M. 1O ':

1 2 2( )

21 1 2( )

i

f z r r e

+ + +

=θ θ π π

O z "% )2(− )1( ;: )3(− 1O

π=− )1arg( z. #'π3)2arg( =+z 41 =r 12 =r M. " '1f >#

)3(− 8: 7

( )2

1( 3) 2 2i

f e i− = = −π

27.3 – ! :

" 1 " %n :

60

11 1 0( 0) 0n n

n n na a z a z a z a−

−≠ + + + + = ' n 7)? ( C.

61

"#

1 – ' @),(),()( yxivyxuzf +=.

1@ A : :2

),(z

yyxv = 0)2( =f.

&:

1:22

),(yx

yyxv

+= ; *

C.

1% ; f > 1% 1@ v u "+G A & ;*

C 1 &% ' ' 1@. 1# 1 &% 1 1:),( yxu %:

222 )(

2

yx

xyvu xy

+=−=

> "'#' 1 "%'y ' 'x 1@ # :

)(),(22

xhyx

xyxu +

+

−=

T A', )(xh > "'#'y 0 & ' Q )#.

1@ # "#, 1 &% " 1:

222

22

222

22

)()(

)( yx

yxx'h

yx

yxvu yx

+

−=+

+

−⇒=

P#0)( =x'h Th 1O cxh =)(. 1% %8:

62

czyx

yic

yx

xzf +

−=

+++

+

−=

1)(

2222

T0)2( =f 1O 2

1=c '

2

11)( +

−=

zzf.

1@ ;: E#*( )f H∈ C 1 ':

1@ 8 ;4f ; 1 1' "3 1 '*C / .

1 % 1@ 8 "#,u v G A & / ;> ;4 "' 1 "+*C 'u v 17..

1: u v "$# 14.3 '*( )f H∈ C.

2 – ' 1% 1: @23)( iyxzf += .

& : #xyxu =),( 23),( yyxv = 1% ; f 0 1@ u v "$#4.3 . P#:

yyxvyxv

yxuyxu

yx

yx

6),(;0),(

0),(;1),(

==

==

# % # 1 1 &% P#z %& 16

ixz += @

;:6

iG z z x

= ∈ = +

C.

1%G (3 % 14 " . A 1 "# QG%G 2 1%

G . 1O Pf ; 0 & 'G ; P#%G.

3 – 1>%

=

y

xFyxu ),( >F 1>> 0> & >'*

R∈x

*y R∈.

x

y

G 1

6y =

63

' @f ')y,x(u SG% P. & : 1% ; ),( yxu ?SG ? 1% 1@ ' u > = @

1@ :0=+= yyxx uuu∆.

2

2

2 4 3

1( ) ; ( )

1 2( ) ; ( ) ( )

x y

xx yy

x x xu F' u F'

y y yy

x x x x xu F" u F" F'

y y yy y y

−= =

= = +

' P#y

xt = 1@ #:

ktctFt

t

t'F

t"F+=⇒

+

−= arctg)(

1

2

)(

)(2

P#ky

xarctgcyxu +=),(.

1# 1@),( yxv 0 ),( yxu 1> &% 1) 1 @4.3.(

)()log(2

),( 22

22xgyx

cyxv

yx

ycuv xy ++=⇒

+==

A', T)(xg > "'#'y0 & ' Q )# = . H%:

)('2222

xgyx

xcvu

yx

xcxy +

+

−=−==

+

−

P#0)(' =xg Tg 1O 1)( cxg = A', . 1@ ># 'u v "$# 14.3 8 ' P#:

122 )log(

2)( ciyx

cik

y

xarctgczf ++++=

4 – 1 : 4ch; == ziez

3 1 ,z zsin ?.

64

&:

• " "'#'iez = 1O iki

zeie

ππ

22

+==.

8 P# :( 2 ) ;2

kz k i k= + ∈Zπ

π.

• " "'#'4ch =z 1 H#8:

;4 " : %#2

ch4zz

eez

−+==.

8 " P#( )log 4 15 2 ;kz k i k= ± − π ∈Z. "#, " : %#:

xyixy

iyxiyxiyxz

shsinchcos

shshchch)(chch4

+=

+=+==

P#:

=

=

0shsin

4chcos

xy

xy

" '0shsin =xy "#, " K # 1@ 8 ":

( ) Z∈+±= kikzk ;2154log π • 3 1 z sin z %# =?:

sin sin( ) sin cos sin cos

sin ch cos sh

z x iy x iy iy x

x y i x y

= + = +

= +

P#: cos 0

sin cos sh = 0sh 0

xx y

y

=∈ ⇔ ⇔

=Z R

1:2

x kπ

= + π @0y =.

3 1:z 8:

@

65

2z k iy

π= + π + @z x= Tk ∈Z

5 – 1@ 18': ∑

≥

+−=0

1)1()(n

nnzzf )1,0(D.

& : @ 1 "' 2 " 1@ '1<z 1O ( ))1,0(DHf ∈ >10.3.

6 – 1% G "#)(GHf ∈. 1% :z z G= ∈ ∈Ω C.

#:g →Ω C >' :( ) ( )g z f z=. 1@ 18')(ΩHg ∈.

& : 1% Gz ∈ Ghz ∈+ +#Ω∈z Ω∈+ hz P#:

0 0

0

( ) ( ) ( ) ( )lim lim

( ) ( )lim ( )

h h

h

g z h g z f z h f z

h h

f z h f zf ' z

h

→ →

→

+ − + −=

+ −= =

14f P# =0 & ' / g - . ; 0 & 'Ω > =1.3 1O )(ΩHg ∈.

7 – " 1@ 1R'wz tg= >' P G# zw arctg= 8:

1 1arctg log( )

2 1

izw z

i iz

+= =

−

&:

66

iz

iz

zi

zieeziezi

ee

eeiwz

iwiwiw

iwiw

iwiw

−

+=

+

−=⇔−=+

⇔+

−−==

−

−

−

1

1)()(

tg

2

P#: 1

2 log( )1

iziw

iz

+=

−

1 1arctg log( )

2 1

izw z

i iz

+= =

−

8 – @n

n

zn∑≥1

2 / @.

& :zzn

znn

n

n=

+ +

+∞→ 2

12)1(lim > ">' " '1<z . > @

)1,0(D. #∑

≥0n

nz "'1<z @)1,0(D.

∑≥

=− 01

1

n

nz

z "' @ 1 0 & ' 2 " , ' ) >

10.3.( 1@ # 1 0 &' 1::

∑∑≥

−

≥

− −=−

=− 2

2

31

1

2)1(

)1(

2;

)1(

1

n

n

n

nznn

zzn

z

P#:

∑∑≥

−

≥

−=− 2

1

2

2

3

2

)1( n

n

n

nznzzn

z

z

'2

23

1

2

)1(

13

−

−+−=∑

≥ z

zzzn

n

n.

67

9 – 1%)4)(4()( 2 +−= zzzf 4)0( =f. @)6(f 1@ A :z E 6 1 0# 1# ')0(

E 1 4 '6.

&B : ' f 0>> >> >8 M>. ># T7>, P> >3 0)4)(4(4 2 =+−⇐= zzz iz 2±= . ' "3 " f 1> R>'

G Z ' 3 Z "'3 ##% ; ,, %&' P ' %. ' # z "# E# z M. #')%& %.(

%# 1@ ##% +#: iki

erzπθ 00 2

04+=− P# :

ikiierz

ππθ 00 204

++=− iki

erizπθ 22 2

22 +=− iki

erizizπθ 11 2

1)2(2 +=−−=+ 1% :)2(

2102 210)4)(4(

ππθθθ kierrrzz

++++=+−. TZ∈++= 210 kkkk.

0 1 2( 2 )

20 1 2( )

ki

f z r r r e

+ + + +

=(*) ...θ θ θ π π

&4)0( =f P#@ #0=z "3 1O )(zf 84 . ># P>z

E)0( 1O )%& $#@: (

2;2202

2;2202

;4404

22

11

00

πθ

πθ

πθ

−==−=−=

==+=+=

==−=−=

iizr

iizr

zr

' Q8 K ')(∗ #:

0θ i2

i2−

y z

x 4 6

0r 1r

2r

1θ

2θ

68

)1(2

)222

(

416)0(4 +

++−+

=== ki

k

i

eef π

ππππ

π

1% 1@ P#1+k G ? . > "G % 1+k

"3 '@ )V# =9 0 01 =+k. "3 1O P#k 8 "'1−=k.

' M 1:)(zf & 0 4)0( =f > 0 M. 8

1−=k 1>% =1f 1O = :2

)(

2101

210

)(

πθθθ −++

=i

errrzf . a' #z 6 # T' =G 4 Z T P#O 4 '' ? >Ez

6 1O :

0 0

1 1

2 2

4 6 4 2 ; 0

2 6 2 40 ;

2 6 2 40 ;

r z

r z i i

r z i i

= − = − = =

= + = + = =

= − = − = = −

θ

θ α

θ α

"3 1O ')(zf #6 @)6(f "3 8)6(1f :

21(6) 2 40 80

i

f e i

−

= ⋅ = −π

α "G/

y

x

i2

i2−

4 6 o

α

- α