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..................................................................... 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