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Author: Араманович И.Г. ВолковыскийЛ.И. Лунц.Г.Л.
Tags: физика решение задач задачи по физике издательство физматлит теория физики
ISBN: 5-9221-0264-8
Year: 2006
Text
Сборник задач по теории функций комплексного переменного.
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СБОРНИК ЗАДАЧ ПО ТЕОРИИ ФУНКЦИЙ
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jak as j2 .
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0 ( , ( (& "
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12
x 2.
exp = z = x cos + sin
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sin = e 2ie
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z
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5 sin sin 2 + + 1n1 sin .
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1 cos + cos + + + cos +
2 sin + sin + + + sin + .
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1 sin + cos2 = 1 2 sin = cos 2
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, 5 1 - #2 %:
1 sin 2 cos 3 tg 4 sh 5 ch 6 th .
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1 cos 2 + 2 sin 2 3 tg 2
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% 5 , ' :
1 % (
2 ( .
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1 tg = 1 2 th = 1.
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= 0 1 2 ,
ln = ln +
6 ln ( ( Ln .
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1 Ln 4, Ln 1, ln 1 2 Ln , ln
3 Ln 1p i 4 Ln 2 3i, Ln 2 + 3i.
2
1.72.
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Ln z = Ln z .
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z = 2. 0( , ( f z )% + ( z , ) & ) ( Im f z ! ) 1 , !:
1 f z = 2 Ln z 2 f z = Ln 1z
3 f z = Ln z Ln z + 1 4 f z = Ln z + Ln z + 1.
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p
p
1 1 2 & 2 2 2 & 3 2i & 4 1i & 5 ii &
6 1p2i 1+i & 7 3 4i1+i & 8 3 + 4i1+i .
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= m=n $ z " :
p
z m=n = n zm
. # 1.6.
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, " " w = Arccos z 3" " " " z = cos w. 4 0 5/ Arcsin z , Arctg z ,
Arcctg z 0 5/ Arch z , Arsh z , Arth z ,
Arcth z .
1.77. 6 # $ " " ! "
# :
p
1 Arccos z = i Lnz + z 2 1&
p
2 Arcsin z = i Ln iz + z 2 1&
&
3 Arctg z = 2i Ln ii + zz = 21i Ln 11 + iz
iz
p
4 Arcctg z = 2i Ln zz + ii & 5 Arch z = Ln z + z 2 1&
p
6 Arsh z = Ln z + z 2 + 1& 7 Arth z = 12 Ln 11 + zz &
8 Arcth z = 21 Ln zz + 11 .
1.78. 6 # , 0 # Arccos z # Arcsin z , 3 # ! " =2. 6 # 0 " Arctg z Arcctg z .
. 7 " " Arcsin z + Arccos z = =2 Arctg z +
+ Arcctg z = =2 "0 , # ! " $!
# .
1 1 ez = exp fz Ln eg = exp fz 1 + 2ik g. ,
, k = 0 . . -" ez = exp z .
3.
15
# , " # Arccos z " 5
p
Arccos z = i Ln z + z 2 1
p
0 z 2 1 - 0 # .
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Arctg z ! " ?
2 6 z 5/ Arsh z # ?
1.81. ! " # $ 5/ !:
1 Arcsin 21 & 2 Arccos 12 & 3 Arccos 2& 4 Arcsin i&
5 Arctg 1 + 2i& 6 Arch 2i& 7 Arth 1 i.
1.82. ! " $ " !:
1 sin z + cos z = 2& 2 sin z cos z = 3&
3 sin z cos z = i& 4 ch z sh z = 1&
5 sh z ch z = 2i& 6 2ch z + sh z = i.
1.83. ! " $ " !:
1 cos z = ch z & 2 sin z = i sh z & 3 cos z = i sh 2z .
1.79.
x 3.
1
X
6 # , cn
n=1
.
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1
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cn
c2n . 6 # ,
1
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n=1
n=1
0, jcn j2 $ .
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Re cn
j arg cn j 6 2 ,
1
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n=1
7 cn "! ", 0 ,
n=1
$
# ", $ " # " , . 6 # , ! .
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1.86.
n
X
k=m
ak bk =
nX
1
k=m
Sk bk bk+1 Sm1 bm + Sn bn
0 1 6 m 6 n, Sk = a1 + a2 + ::: + ak k 1, S0 = 0.
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1
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an bn, bn 0,
1
n
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n=1
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n=1
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n
3 pn ,
n
X
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1
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cn
n!1
n=1
, q 1, , q 1.
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1
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n=1
Sn =
k=1
1 + 21 + 31 + 41 + ::: 1
+ 2 + 3 + 4 + ::: 0 1
cn . ,
c
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n!1 cn
1
lim cn+1 1.
n!1 c
n
$
cn , nlim
n ccnn+1 1 1
!1
/ .n=1
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n
a n a 1, .
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3.
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n=1
n
n
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in
1.95. cn =
. 1.96. cn = inn . 1.97. cn = e .
2in
ein . 1.99. c = ein' . 1.100. c = ein .
1.98. cn =
n
n
n
n
n2
1 i=n
1.101. cn = e .
n
+ 1::: + n 1 + 1::: + n 1 1.102. cn =
n! + 1::: + n 1
$ , Re + 0.
cos in
n sin in .
1.103. cn = n . 1.104. cn =
2
3n
1.105. 5$ . :
1 z = 1 + 1 n n +n 1 n = 1 2 ::: )
2 z = m1 + ni m, n | 6 )
3 z = mp + i nq m, n, p, q | 6 )
4 jz j 1.
1.106. , $ $ fzn g . % .
1.107. % . :
1 fzn = xn + iyng 7
$ $ fxn g fyng)
2
% nlim
!1 zn 6= 0,
, % nlim
!1 jzn j 6= 0 % arg zn nlim
!1 arg zn . 8 nlim
!1 zn 6 , ., , , arg zn 6 .
! fzng 7 fjzn jg?
1.108. 5 . $ 1.107 :
z n = ex cos y + i sin y )
1
+
1 nlim
!1
n
h
i
p
n z 1 = ln r + i' + 2 ik k = 0 1 2 ::: .
2 nlim
n
!1
2 ..
.
. I.
18
x 4.
_
1.109 1.115 " # , #.
1.109. z = 1 it, 0 6 t 6 2. 1.110. z = t + it2 ,
1 t 1.
1.111. z = t2 + it4 , 1 t 1.
1.112. z = a)cos t + i sin t*,
6 t 6 32 , a 0.
2
z = t + ti , 1 t 0.
p
1.114. 1* z = t + i 1 t2 , 1 6 t 6 1+
p
2* z = t + i 1 t2 , 1 6 t 0 )"#
1.113.
#*.
1.115. 1* z = a)t + i
ie
1 t 1,
a 0+
2* z = ia + at ibe , 0 6 t 6 2, a 0, b 0.
it
,
it
*,
w = z 2 " #:
1* " x = C , y = C , x = y, jz j = R, arg z =
1.116. -# ". #
# , " 0# +
2* " ) z -* u = C , v = C )w =
= u + iv*.
1.117. -# ". # w = 1=z :
1* " x = C , y = C , jz j = R, arg z = , jz 1j = 1+
2* " u = C , v = C .
1.118. -# ". w = z + 1 w = z 1 "
z
z
. jz j = R.
1
1.119. -# " # w = z +
z - z
" # )u = C , v = C * w.
1.120. " # . jz j = 1 ".
w = z=)1 z *2 ?
1.121. -# ". # w = ez :
1* " x = C , y = C , x = y+
2* " = )0 6 1*.
4.
19
1.122. ' x = C , y = C
z () * + :
1 w = z 2 + z $ 2 w = cth z $ 3 w = ez2 .
1.123. ) () * +
w = ez + z
- x = C y = C , ( 0 6 y 6 ?
1.124. . z - jwj = R,
arg w = -: 1 w = e1=z $ 2 w = ez2 ?
'
1.125. / + f z , 0
z0 , z0 , 0 ) 0 fzn g, -0( z0 , nlim
!1 f zn =
= f z0 $ 1 * + , 0
) " 0 ( " 0, jz z0j
0 , jf z f z0 j ". 1 - 0 ., , 21, . I, . 3.64 .
Re z
z Re z 2 z Re z
1.126. / +
,
, z 2 , z 0 0 z 6= 0.
z
z
5 - 0 0 z = 0 ,
1 ?
1.127. 6 0 * + :
1 1= 1 z $ 2 1= 1 + z 2 $
0 jz j 1 ? 6 0 ?
1.128. 1 , * + e1=jzj
jz j 6 R z = 0.
2 6 0 1 * + e1=z2 ?
3 6 0 * + e1=z2
0 jz j 6 R, j arg z j 6 =6?
1.129. / + w = e1=z 0 )0 ,
z = 0.
, :
1
0 jz j 6 1, j arg z j 6 =2 1 * + ,
$
2 1
* + , $
3 0 jz j 6 1, j arg z j 6 6 =2 * +
.
1.130. / + f z
jz j 1.
, 0 )
jz j = 1 ) j j
2
j j
j j
20
. I.
zn ! , jzn j 1 nlim
!1 f zn .
, fzn g , ! !
" ! #, " "" ! jz j 6 1.
x 5.
$ % & '( "
1.131. ) * %& '( " * z n, ez , cos z , Ln z ,
z n 0 = nz n1 ez 0 = ez cos z 0 = sin z Ln z 0 = 1z :
1.132. ,* a, b, c, # f z
*:
1 f z = x + ay + ibx + cy/
2 f z = cos xch y + a sh y + i sin xch y + b sh y.
1.133. ,* , #
f z = jx2 y 2 j + 2ijxy j
*.
1.134. f z = u + iv = ei | . ,
* u, v, , *,
f z .
1.135. ) z = rei' f z = ur ' + iv r '. 2 %& '( " # #.
1.136. , f z = u + iv |
s n | , " s
n "* ! & * ,
@u
@v
@u
= @n
= @v
@s
@n
@s
@=@s @=@n | * # * # " # 3" 3.
1.137. ,
f z = z4
! ".
1.138. , w = z Re z " z = 0/ * w0 0.
p
1.139. , f z = jxy j z = 0
3 %& '( " , .
5.
21
1.140.
3 :
w = f z z
h
i
w
lim
Re
z
z !0
ux vy
3
" */
h
i
w
Im
2 lim
,
3
z
z !0
uy vx , " uy = vx /
3 , u v ", 3 ! , # . 1
2, !! , , " f z .
1.141. 7 w = f z z 3 " * " :
w
1 u, v "/ 2 lim
.
z
z !0
, f z , f z " z .
1.142. 7 w = f z z 3 " * " :
1 u, v "/ 2 lim
arg wz .
z !0
, f z " z .
1.143. ) w = f z = u + iv ux y v x y " z . , " " # # * &
8w=8z 8z ! 0 ,
.
1
7"
%&
9 &
z + z z z
w = wz = f x y = f
2 2i = 'z z4
" z z4 " " ,
" " "
@
1 @ i @ @ = 1 @ + i @ :
=
@z
2 @x @y @ z 2 @x @y
: *&"
1.144.
@w
@z
= wz
@w
@ z
= wz
3 & :
1 dw = wz dz + wzdz4/ 2 wz = 21 ;ux + vy + iuy + vx </
3 wz = 21 ;ux vy + iuy + vx <.
. .
. I.
22
1.145. ,
= 0.
1.146. ,
= 0 !
@2u
" @z @ z = 0.
1.147. ,
= #, #z = z , #z = z $% , '" !
"(( ) *.
1.148. , " ( ) $ *, % +
$ *,
= w 2 wz jw j2 + jw j2 wzjw j2 #
z
z
z
z
1.149. , % % $ *
@ u v
= j z j 2 j z j2
w=z =
@ x y
wz
u
dw
dw
w
w
w
w
z w
w z
dz
j
dw
j
dw:
w z
J
w
w
:
1.150. " - :
1*
dw
dz
=
wz
dw
dz
2* max
+
wz e
2i
, 0"
1*
d
d
2*
0
max d
d
= arg 1
dz
dw
dz
= j z j + j z j1 3* min
w
w
1.151. ,
0
= arg ,
dz
= jj z j j z jj.
w
0
Jw=z
w
= arg
dw
= u cos + u sin 2 + v cos + v sin 2 =
x
y
x
y
= , min
p
d0
d
= 1p , 0"
4 (
p
=
max
min
dw
dz
dw
dz
, :
dz 2 1
1
dw Jz=w
= jjjwwz jj + jjwwz jjj .
z
z
)
5 ) $ *, %"- + %
" " 0 " " -
@2u @2u
@x
+ @y2 = 0
2
u x y
u
$ * $
u x y
.
0 ( )
*,
@u
@v
@u
@v
= @y
= @x
@x
@y
v x y
.
5.
1.152. " - "! :
X
23
n
1* + % ) 0 ' ( )+
ci ui $x y*
i=1
( ) 0 .
2* 6 0 0 + ( ) u$x y* " 0
% x = 2 + 2 y = 2 + 2 % ( ) % " 0 +.
3* 6 0 0 + ( ) u$x y* " 0
% x = '$ *, y = $ *, 0" ' | !
0 ( ), % ( ) % " 0 +.
$8 ", , " "" - !" .*
4* : u$x y* v$x y* | ! 0 ( )
u v
% @@ x
+ %
y
. ;0" %
( ) x$u v* y$u v* ! % "
0 ! .
1.153. 1* , 0 " + % G ( ) u$x y* + ! ' 0 ' ( )+, -' " 0 " 0 0
v$x y* =
Zxy
x0 y0
@u
dx + @u
dy + C:
@y
@x
2* , % G 0 0 <0 <1 , <2 , :::, <n $!"+
' ! !" 1 . 1*, ( ) v$x y*
! 0 + %- ( " + % " "
n
Zxy
@u
v$x y* =
@u
dx
+
dy
+
mk k + C:
@y
@x
X
k=1
x0 y0
= 0 % , !- % G, mk | )
Z
@u
dx + @u
dy
k =
@y
@x
0"
k
k | , !"+ ' " !
% " 0 ) $<k * $ k
*.
" ( ) v$x y* %'" " , % k % .
. I.
24
. 0 , .
! " # , , # # . # 1.152, 2,
. ' # , "
n
k = 0.
u x y
U
X
u x y
k=1
1.154. # *, , # !+ *:
1 =
= + ! * .
2 # * !
! .
1.155. 1 / 2 , | ?
2 | . 2 3 ?
1.156. / j j, arg , ln j j , | ?
@2u @2u
1.157. # 4 5 = 2 + 2 @x
@y
6 4 5 = 0, #+
.
1.158. 7* = 1, 2, 3, 4 *
n , * n = n + n . :
n
+ n n .
# # 1.153, # 3 1.159<1.163 , * *
#*3 3.
1.159. = 2 2 + , 0 6 j j 1.
x
1.160. = 2 2 , 0 j j 6 1.
x +y
1
1.161. = ln 2 + 2 :
2
, # : = 0,
1 6 0.
* * : 0 j j 1.
1.162. = 1 fln 2 + 2 ln > 12 + 2 ?g:
2
* * = 0 = 1.
* # : = 0,
0 6 6 1.
* : = 0, 1 6
1.
f z
u
iv
u
u
u
f
f u
f z
f z
f z
f z
u
r '
u
r
n
p
x y
q
x y
p
u x y
z
x
y
x
u x y
u x y
p
iq
q
z
z
x
y
y
x
u x y
x
y
x
z
y
z
z
y
x
y
x
5.
1.163.
u x y
= 12
X
n
k
k=1
ln > 2 + 2 ?:
x
xk
y
* *
zi
25
6= .
z1
yk
, 2, ,
z
=
::: zn
zk
xk
+
iyk
,
zj
. . #
, !+ * .
1.164. @ + = + ,
: 2 2
1 = xx2 + yy2 2
v
x2 y2 x2 y2
u ey=x
f z
u
2
= ln
1.1651.168
!
!
+
+
y
.
x2 + y 2
y2 + 5x + y
1.166. u = ex x cos y y
1.167. v = 3 + x2 y 2
sin
3
"#! $$!
1.165. u = x2
y
1.168. v = ln x2 + y 2 + x
iv
=
?
f z = u + iv
.
y + 2 sin x sh y + x3
2x2 + y2
u
xy2 + y.
3
.
y
2 .
1.1691.176 "'(#, *" + , -$!
!,! "
!
' !
!!(!, "
-
*"!" ( .
1.169. u = 'x.
1.170. u = 'ax + by a b | "#' .
1.171. u = ' y . 1.172. u = 'xy . 1.173. u = 'x2 + y 2 .
x
2
2
p
1.174. u = ' x + y . 1.175. u = 'x + x2 + y 2 .
x
1.176. u = 'x2 + y .
1.1771.180
!
# *"!"
f z !
!$ $! + -, $ .
2
2
2
x
1.177. = x + y e . 1.178. = er cos 2' .
1.179. = xy . 1.180. = ' + r sin '.
1.181. 1! #: !3' $"! 'x y = C , ,
-
" 4 ' --'"!
-!"( !!-! , -$!
!
7
!,
'
|
- - $ ( (, 3'! $"!$
! , !3! $!
'=grad '2
!3' !!5 6
"! !# ! !
. 8- " -#! !"#,
( ( $ "
u = f 9'x y:.
'.
!-
! !$ ( , -$!-
26
. I.
1.182 1.186 ,
! " # $# # , " "# ,
" , !" .
1.182. x = C . 1.183. y = C . 1.184. y = Cx.
1.185. x2 + y 2 = C . 1.186. x2 + y 2 = Cx.
(" " "# !"
$
1.187. ) * +# $" w = z 2
w = z 3 . , ! $ -#. $ #, # ! z0 , / #* # -k. :
1. z0 = 11 2. z0 = 1=41 3. z0 = 1 + i1 4. z0 =
3 + 4i.
1.188. 4# $ *"#, # #! #,
* ## :
1. w = z 2 1 2. w = z 2 + 2z 1 3. w = 1=z 1
4. w = ez 1 5. w = ln -z 1.?
1.189. ) G *# $" f -z . " " G . 7 " #
# $ S G L !,
*# # ! l, $ * # G.
1.190. , L $, $" ez *# y = x, 0 6 x 6 2.
1.191. , $ , $" ez *# $#"! 1 6 x 6 2, 0 6 y 6 4.
1.192. , D, # ez * $#"! 1 6 x 6 2, 0 6 y 6 8. $ D
$" ", $ $ + 1.189, :# , $" / " $ .
0
0
( ; < < II
,
!
x 1. "#$%#&$ '(#)*""
=
2.1. , , * ! + " 0, 1, i $ " !
+ " 0, 2, 1 + i.
2.2. , $ $ * 1 + 2i, $ # i i.
2.3. ># $ $ * z0 - ., ! $ ! #
/ #* # k. ? / $ # " w z0 = -z z0 .:
1. w = 2z + 1 3i1 2. w = iz + 41 3. w = z + 1 2i1
4. w w1 = a-z z1 . -a 6= 0.1 5. w = az + b -a 6= 0..
2.4. , " ! ! $ #, $ # !:
1. $$ #1
2. $$ * $$ 1
3. $$ $ $$ 1
4. $ $$ #.
? , # $ $## " $
$ ! .
2.5. , " ! ! $ #, $ # !:
1. $ 0 x 1 #1
2. $ 2 y 1 #1
3. $, ! $#"" y = x y = x 1, #.
# , $ "! $ / * # ! ! " /
$# * .
2.6. , w-z ., * $, "* " $#"", $ 0 u 1
$ " :
.II.
28
1 x = a, x = a + h wa = 0
2 x = a, x = a + h w a + h2 = 12 + i, Im w a + h2 + i 1
3 y = kx, y = kx + b w0 = 0
4 y = kx + b1 , y = kx + b2 wb1 = 0.
2.7. , jzj
1 jw w0 j R , ! ! " #"#""
$ $ %&! $' ( )$ " $' , % # ( "' $#"& # .
+ - !
2.8. +, w = 1=z %! #$ ) :
1 #'#" # x2 + y2 = ax
2 #'#" # x2 + y2 = by
3 ( ( &!) ( ,'!) y = x + b
4 ( ( ,'!) y = kx
5 ( ( ,'!), ( )$, ) % %$ z0 6= 0
6 ( ! y = x2 .
1
2.9. 0!,#&, " , w =
+ h ( "$:
z z0
1 ( ,' & # x = C , y = C
2 (, # jz z0 j = R, arg z z0 = .
z z1
2.10. + , w =
.
z z2
1 +%&, ( %' #'#" jwj = 0 1 ,",#, #'#" # # 1(,. +, $ $# ( #"#" #
" z -(##.
2 ( %! arg w = .
3 2# & # " z -(##, #"#" (,
# " w-(##.
4 #& z -(##, #"#" ( jwj
1, Im w 0.
0 %$ ) 2.1132.15 "!,#&, " ( %#, %!
# ( %$!) ) ,).
z i
2.11. 4"$ x 0, y 0 w =
.
z+i
jzj 1, Im z 0 w = 22z+ izi .
5 0 ' 4 w = z z 1 .
2.12. 2
2.13.
1.
2.14. 2# 0 x 1: 1 w = z
2.15. 4& 1 jzj 2
z
1
w = z z 1.
29
2 w = zz
1
.
2
2.16. 6 %& " & (# 0 Rew 1:
1 ((##& Re z 0 # "!!' ' z d2 6 d2
2 $" &, % ! ' $ #,'
d
d
d
d
z 21 = 21 z 22 = 22 d1 d2
3 "7#& " z + d21 = d21 , z d22 = d22 , !
wd2 = 0.
2.17. $ -! , ( "$, 1, i,
1 + i #"#" " : 1 0, 2i, 1 i 2 i, 1, 1.
2.18. $ -! , ( "$, 1,
1, i #"#" " :
1 i, 1, 1 + i 2 1, i, 1 3 0, 1, 1.
2.19. $ -! ( #$ ' #",':
1 1 i ($" !, 0 ( )$ " 1
2 21 2 ($" !, 54 + 34 i ( )$ " 1
3 i ,",#, $" ($" , 1 ( )$ " 1.
2.20. $ - , ( "$, 1,
0, 1 #"#" " 1, i, 1, "!,#&, " ( 8'
( )$ " ),, ((##&.
2.21. "$ $ - ( %",, ( "$, :
1 " ) ((##& #,
2 " ) ((##& ((##&
3 " ) ((##& ( " ((##&.
2.22. " ) ((## #, ( % ' ":
1 w0 = 1, w1 = 2, w2 = 1 2 w0 = 1, wi = 2i.
2 ' . 6 " ) ((## #,
( $ ' " #'. %$ 2.34.
2.23. wz , jzj R ( " ((##& Re w 0 , wR = 0, wR = 1,
w0 = 1. 4" ( 8' % " ) ( ?
.II.
30
1 , 2
,
, ,
2
1
2 =
P
P
K
O
R
O
OP
OP
R :
2.24. , 2 +
: 1" = 1# 2"
= 3.
2.25. & :
1" = 12 # 2"
1 = 1# 3" = 2#
4"
0 = 0 ) 0 = 0 + 0 "#
p 2 1 ) 1"# 6" ,-& 2 2 = 1#
5"
0 =
0
0
7" , - , , / 1 , 2 , 3
) i = 0".
2.26. , 1 2
& - , :
1" 1 , - 1 , 2 , , #
MP1
2" MP
= const ). .
2
1- 1 2 ".
2.27. 2
= i zz ) = + ,
0" &
-- ,:
1" arg ) " = ) "# 2" ) "#
3" , -- - 3 & , .
2.28. 4& -- Im
0
,
1 , & :
1" ) " = 0, arg ) " = 2 # 2" )2 " = 0, arg )2 " = 0#
3" ) + " = 0, arg ) + " = ) 0".
2.29. 4& -- Im
0 ,
, & -/ , , -
0
3 & - .
2.30. 4& ,
2 -- Re
0 ,
&
)0" = 1, arg )0" = 2.
2.31. 4& ,
4
2 --
,
& , -/ 4, 2 |
.
i
jzj
jzj
jz ij
jz
jz z j
jz z j
jz j
j
z
jz j
y
x
iy
jz j
x
y
z
z
z
z
6
P
P
K
K
P
P
K
M
w
K
e
w x
x
a
w
0
P
P
ib
b
K
z
jwj
w i
w a
w
0
i
bi
w
0
a
bi
w
i
b
w
0
i
z
w j R
jzj
w
jw
i
w
0
w
=
jz
ij
v u
i
1.
31
2.32. & & - 6 ) ", & ,
1 - -- Re
0 , &
) 1 " = 0, ) 2 " = , , 1 , 2 |
= 1 , arg 1 arg 2 .
7 ,
1,
- -- Re
0.
8
. 9- & 6 & - ,
-& -
i'
arg eei' zz1 .
2
2.33. 0 , 6 = zz zz1 & )Im 2 = 0".
2
2.34. 6 , & --
& , ) " = , arg ) " = )Im 0, Im 0".
8
. 4& - & 3 - -- , .
2.35. 4& --
, &
) " = arg ) " = 2 )Im 0".
2.36. 6 = i z a )
1", & 1 a
z
, &:
1" arg ) i' " = ) "# 2" )0" ) "#
3" , , , - 3 &
, #
4" max dw
min dw
6 1.
dz
dz
2.37. 4& ,
1 ,
1 , & :
1" 21 = 0, arg 12 = 0# 2" 12 = 0, arg 2i = 2 #
w z
jzj
w z
w z
w
1
jzj
z
z
z
z
jzj
w
w
R
w
z
w a
w a
a
b
w
0
a
w
a
w
w e
0
6
a
b
a
e
jaj
'
w
0
w
0
a
jzj
jzj
w
w
jwj
0
w
w
0
3" )0" = 0, arg )0" = 2 # 4" ) " = , arg ) " = .
2.38. 4& ,
,
1
2 , &
) " = , arg ) " = )
,
".
1
2
2.39. 4& ,
1
,
1 1 , &
)0" = 1 2 )1" = 0.
2.40. 4& ,
2 1 ,
2 2 , &
)2" = arg )2" = 0.
2.41. & & - 6 ) ", & ,
& - :
1" ) " = 0 )
"# 2" ) " = )
,
"#
3" ) " = .
w
w
0
w a
a
jzj R
w a
b
w
0
a
jaj R
=
a
jwj R
jw
j
w
jz
w
0
jbj R
jzj
w
w
i
w
j
jw
ij
0
w z
jzj R
w a
w R
jaj R
R
w a
b
jaj R
jbj R
.II.
32
2.42. jz j 1 , z1 , z2 a 0 a 1 a.
! . " # 2.41, 2 &# 1.10.
2.43. jz j 1 , y = 0, 0 6 x 6 a a 1 , ## . (
.
2.44. ) , & # .
2.45. * & , z0 6= 0
+ , . ) , -# & & & ,
, & #, +.# z0.
2.46. / & , # , + , 0 6
6 ' 6 =2 & & 0. :
1 0 6 6 2 2 0 6 6 3 3 2 6 6 76 .
x 2.
3 4# +
)- & z0
. / #&
4#
1
= z 1 z + h
w z0
0
z0 6= 1,
w = z + h
z0 = 1.
)- # # &# # z1 z2 4# #
w z1
= k zz zz1
w z2
2
z1 6= 1, z2 6= 1, w z1 = kz z1 , z2 = 1 # # &# #
, k 0, , k = ei 6= 0,
, k = aei , # a 6= 1 6= 0 a |
, a 0 .
2.
2.47. ) 0. &:
33
+b
1 . - w = az
#&
cz + d
z +
= 1
w = z + ,
2 + | , - #, j + j 2, #,
j + j 2, #, j + j = 2
3 Im + 6= 0, # .
2.48. ) , # & , + -+ +
,.
2.49. ( &, # w 1 z = z 1 z + n + # .
0
0
2.50. ( .
jz j R , R & .
2.51. ) 0. :
1 0 & , +. & ,
+ # , # + +
2 0 & , &#, +.#
& , + & , 0. 0 #
& #. 7 1 .
! . / # , & 0 1.
2.52. ) , - # :
1 0 & , &#, +.#
& , + # +#
+
2 &, 0. & , +
&, 0. 0 & 0. 0 = arg k .
2.53. 1 ) , # # +0 2 #. 2.51 - #. 2.52 .
2 ) , # # . &+ & , 6= = arg k .
) , = , &, +. & , + # ##
+.
2.54. ) , # # w =
= aei z 4# r = Aeln a=' A 0 +
# .
3 ..
.
.II.
34
w = ei 1z aza a =
= jajei , jaj 1,
,
,
, . # , $
$ a
$ . %
.
2.55. ,
%
&
$ $
2.56. # $
, z = hi h 0 $ ( . % )
*0 a+ a 0
& ) a=h h=a.
2.57. ,
,
( | x0 | $ . %
) 0 6 ' 6
6 .
.
)= x0 ?
2.58. # $ 2.57 & :
1 ) = 11 + xx0 + O 3
0
2
=
" ctg
x0 .
=1
"2
2
2
ctg 2
+ O "3
2
",
p o
"
=
2.59. , z0 =
= r0 ei'0 ! " # . $ z1 = ei'1 z2 = ei'2
, ! % & z0 '0 '1 '2
'0 + .
' , z0 & ,
& (,
& '1
' '2
"" )
6
6 6
h
=
"
ctg
'1
2
'0
ctg
'2
2
'0
i
+
"=1
r0 .
*
+
"2
2
h
ctg
2 '1
2
'0
+! " "
ctg
2 '2
2
'0
i
3
+ O "
&!
jj
2.60. ,& (, z 1
" " , # (#
# " ! " # .
- . . ( /% .
2.
&
35
(
r1 jzj r2 ( R1 jwj R2 , $
R2 =R1 = r2 =r1 . : < & ( $ : w = az
w = a=z . 8 < $
$ ., , *3, . II, x 3+.
2.61. 1 8 ( 2 jzj 5 ( 4 jwj 10
, w5 = 4.
2 8 ( 1 jz 2ij 2 ( 2 jw 3 + 2ij 4
, w0 = 1 2i.
? < .
. , (
<
, & (
(
9
9
| .
2.63. : Re z 0 jz hj R
h R ( jwj 1 ,
9 jwj = 1. % .
@
. : ( < jz hj = R5
, < <
< , , ( (. , ( (
,
0 1.
2.64. : Re z 0 jz hj 1,
h 1, ( 1 jwj 2. % h.
2.65. ; ( (,
jz
3j = 9, jz 8j = 16, ( jwj 1. % .
2.66. < ,
< jz
z1j = r1 , jz z2 j = r2 jz2 z1 j r1 + r2
jz2 z1 j jr2 r1 j,
( ( ( . % .
@
. % , $ $ , $ 0, <
1.
:
. % , 9 , $ 2.63 2.66,
.
2.67. : 9
,
$ ,
$ :
1 jz ij = 2, jz + ij = 55 2 jz 3ij = 1, jz 4j = 2.
3
.II.
36
-
T z = T2T1 z T1 T2 T = T2 T1
, , , T2 T1 6= T1 T2. " G T ,
# # # T # T 1. ,
T n T n T , . $ G T1 , T2 , :::, Tn #
# # , %
& G. '( , ( # ) ( z # *& G, & * G.
G * , ( , % * , ( ( #& ( , %
( # .
2.67. * Ti | :
Ti z = ac zz ++ db -i = acii dbii 6= 0 i = 1 2 ::::
1 * & :
1 T = T1 T2 | # - =
= -1 -2 2
2
3 , . .
T3 T2 T1 = T3 T2 T12
3 Ti # Ti 1, . .
Ti Ti 1 = Ti 1Ti = J
J z z | 2
4 , , ##
#.
2.68. 1 *, (
T1 = z T2 = 1z T3 = 1 z T4 = 1 1 z T5 = z z 1 T6 = z z 1
& .
2.69. 1 *, ( # , &( & ( ,
, 3 ( & . 6 # ( % * ( ( ?
i
i
i
i
2.
37
2.70. 1 1 *, ( #
w=
az + b
= cz + d , a, b, c d | 3 * ( ad bc = 1,
% .
2 1 *, ( a, b, c d ( * 3# # #
( # . . ( # m + ni, m n | 3 * ( , & # & ad bc = 1, # . 1 !.
2.71. 9 ) # * , #
#:
1 T z = e2i=n z n | * ( 2
2 T1 z = e2i=n z , T2 z = 1z 2 3 T z = z + !2
4 T1 z = z + !, T2 z = z 2
5 T1z = z + !1 , T2 z = z + !2 Im !!2 6= 0
1
2
6 T1 z = z + !1 , T2 z = z + !2 , T3z = z 2
7 T1 z = z + !, T2 z = iz 2
8 T1 z = z + !, T2 z = e2i=3 z 2
9 T1 z = z + !, T2 z = e2i=6 z .
2.72. 9 , &
)( 3 & ):
1 * # 2
2 # , * * 2
3 # , * ## 2
4 # , )( 3 3 * ( a.
? . $ z1 , z2 | # * ( ), z1 z@2 = 1 #. ( 1.49.
2.73. 1 1 *, ( , & & ) ( )( # 3# a b , A#
w b
i z a
1 + bw = e 1 + az :
2 1 *, ( ))3 ds = 1 +dzz 2 * % )( &
%# dz . . % %# ).
j
j
j j
.II.
38
jz j 1 ! , # |
, z1 z2 | %z1 ,z2 & = 12 ln %, , z2 , z1 &
| ( ", z1
z2 , % : , z1 , z2, &,
%, , z2 , z1 & | + . -
, %., , .2,
. II, x 41&.
2.74. 2 , %z1 z2 & 0, z1 6= z2 %z z & = 0.
2.75. 2 , %z1 z3 & 6 %z1 z2 & + %z2 z3 &,
, z3
( ", z1 z2.
2.76. 2 , z1 z2 % | &, %z1 z2&
%. .
&.
dz
2.77. 2 , 55 ds =
jz j 1 1 jz j2
j
j j
.
,
z 1
dz
. ! " #
,
$
$
j
a
2.78.
j
,
-
b a 1, b 1.
j
z 1
j j
j j
" :
1 $ & ", $
$ 0 (
2 & ", $
$ 1 2 (
3 & "
$
$ , & "
& "(
4 ,
$ & ".
z
2.79.
z
z
1 , , $
& "
'1 , '2 , '3
'1 + '2 + '3 :
2 , , $
& "
&. "
1, 2,
& "
.
' ' '3 .
$
/
"
3.
x 3.
39
8 w- w = '.l%z &1, '%z & | , l | -
% z = l. %w&1&.
9 , . . , . : , 5.
5 5
%., , .1, . IX,
. 5 . X, . 71 .3, . II, x 1 x 31&.
D1 C % , &, 5 w = f1 %z & 5 ! D1 ,
C C . = 5 f2 %z &, , C , , f1 %z & C 1 &,
D2 ,
D1 C D2 ,
D1 C .
>
f 1 %z &
D1
w = f1 %z & = f2 %z & C
f 2 %z &
D2
5 D1 + C + D2
D1 + C + D2 2 &.
D D | C C ,
D . :
5 w = f %z & D, D
C C , 5
D D .
+ ! 5 , 5, 1 C C |
, ! " -
" #,
f2 z = f1 %z .
2 ' " "
(, " D1 D2 ,
D1 D2 ) #.
40
.II.
.
2.80. w = z 2 ! :
1 $ % x2 y2 = a2
! &'
2 ) % y2 = 2px, p 0 . . , $
! &.
. , , $ %! % $ , . 2.110, 2.111, 2.136/2.140, 2.175.
2.81. , %! % ,
&:
1 & r = a cos ' a 0 &
% = 12 1 + cos '
2 & p & % = cos 2'
3 $ jz j 1 & % = A1 + cos , A 0,
, % w0 = A=8, w 0 0.
2.82. 5 &, $ jz j 1
w = Rz + mz 2 , R 0, 0 6 m 6 1=2. 5 % z - .
2.83. 5 &, $ jz j 1, Re z 0,
w = z + z 2.
2.84. 1 5 &, $ jz j 1
n
w = R z + zn , R 0, n | , n 1.
2 5 &, ) & $ $ jz j 1 w = R z + nz1n , R 0, n |
, n 1.
. , !,
)%!
1
w=R z+ z
7 $ , . 2.106 &) .
2.85. 1 8% &, ! m w = Rz +
+ mz n, $ n | & , $ jz j 1 &, &.
2 8% & % ) $
jz j 1 w = R z + zmn
$
$ w = R z1 + mz n .
0
3.
, $%!
41
2.86. 1 , & $ 0 arg z 0 6 2 !
&.
2 , & $ 4 arg z 2 ! &
, % w1 i = 2, wi = 1, w0 = 0.
2.87. 5
wz , $ jz j 1,
Im z 0, ! & !:
1 w1 = 0, w0 = 1, w1 = 1'
2 w1 = 1, w0 = 1'
3 w 2i = i, arg w 2i = 2 .
0
2.88. 5
wz , $ jz j 1,
Im z 0, $ jwj 1 !:
1 w1 = 1, w0 = i' 2 w 2i = 0, arg w 2i = 2 .
2.89. 5 wz , & jz j 1, Im z
0, ! &.
2.90. , & !
&:
1 jz j R, 0 arg z 0 6 2'
2 & jz j R, 0 arg z 0 6 2.
2.91. , & !
& $% $ & :
1 jz j 1, jz ij 1' 2 jz j 1, jz ij 1'
3 jz j 1, jz ij 1' 4 jz j 1, jz ij 1'
p p
5 jz j 2, jz 2j 2.
2.92. , & !
& ) & $ !$ $.
8 ! 2.93/2.105 & % !
&.
2.93. & :1 1;.
2.94. & :i i;.
2.95. & :z1 z2 ;.
2.96. & 1 R;, :R 1 R 0.
2.97. & , %!
i & y = x.
0
.II.
42
2.98.
,
1 1 ih, 0 h 1.
2.99. Im z 0 0 ih , h 0.
2.100. Im z 0 ih 1 "h 0#.
2.101. Im z 0
jzj = 1 z = 1 z = e , 0 .
2.102. % 0 arg z , 0 2,
jz j = 1 z = 1 z = ei , 0 .
2.103. ' (
0 i " ( ) "#.
% . + , - ,
.
2.104. 1# . jz j 1 0 1 /
2# ( 1 1#.
2.105. 0 - i jz j 1 w- 1 41 , w"0# = 0, w "0# 0.
i
0
12 3
jz j = R, arg z =
1
1
42 3 w = 2 z + z .
2.107. 0 - , , 42 3 - :
1# jz j R 1/ 2# - jz j R 1/
3# jz j 1/ 4# - jz j 1/
5# Im z 0/ 6# Im z 0/
7# jz j 1, Im z 0/ 8# jz j 1, Im z 0/
9# - jz j 1, Im z 0/
10# - 1 jz j R, Im z 0/
11# - R jz j 1, Im z 0/
12# - R1 jz j R, Im z 0, Re z 0/
13# 2 arg z 2 + 0 2 .
2.108. 0 - 42:
2
1# w = 21 z 1z / 2# w = 12 z + az "a 0#/
2.106. 0 -
3# w = 21 z + cz , c = jcjei "0 6 #.
2
3.
43
42 3 , - :
1# ( c c "c 0# (
, w"1# = 1, arg w "1# = /
2
2
2# ( > xa2 + yb2 = 1 ( , -, w"1# = 1, arg w "1# = 0.
2.110. ?- ,, 2
2
> xa2 + yb2 6 1, y 0, .
2.111. ?- - , 42
2
2
2
, > xa2 + yb2 = 1, a2 x+ k2 + b2 y+ k2 = 1 "a b#, 2 2 2
" . . 37# - .
2.112. 0 - , 42 3 - jz j 1 a 1 "1 a 1#. @ a 0 a 0.
' 2.113A2.117 - , -
.
2.113. . jz j 1 1=2 1 .
2.114. . jz j 1
1 0 a 1
"0 a 1#.
2.115. ' (
a 1 1 1#, a 1.
2.116. ' jz j 1
0 i "0 1#.
2.117. ' jz j 1
i i "0 1#.
2.118. ?- jz j 1 ,, "1
h#ei ei , w.
2.119. . jz j 1 a 1 , 0 a 1, - jwj 1 , -, w"0# = 0, w "0# 0. 0 w "0#
, . a
?
% . C - - - , jwj 1 ( .
2.120. . jz j 1
a 1 , 1 b "0
a 1, 0 b 1# - jwj 1 , -, w"0# = 0,
w "0# 0. ? w "0# , , .
2.121. 42 3
w 1
z 1 2
=
w+1
z+1
2.109.
0
0
0
0
0
0
44
.II.
:
1
C , z = 1 1, ,
! " #
2
C , z = 1 1,
, ! " .
2.122. 1 &
' z - ,
' 2.121, ! ( ) w z
w 1
z 1 2
=
0 2 w 0 z 1:
w+1
z+1
0
2 + , C ?
2.123. . "
jz j 1
1 = 0 j j ,
w- arg w
w+1
' w 1 = 1, arg w0 1 = .
/ 2.12412.127 , ' ' ' ( ) .
2.124. + jz j 1# w = 2 z .
z +1
1 .
2.125. 3 jz j 1, Im z 0# w = 2
z +1
1
1
2.126. 4 0 arg z # w = z n + n .
n
2
z
z
2.127. 6 arg z , jz j 1# w =
n
n
1 + zn 2=n w z 0
z 0.
4 . 3 !! ( )!
p
w = F ff 7' z 8g ' t = tn f t = 1 +t t2 F t = n t:
3 )
2.128. 1 3 " 2.127 ) z
, w =
1 + zn 2=n .
2 & ( )!, !! "
" 9 '":
jwj 6 1 arg w = 2k=n k = 0 1 2 ::: n 1:
3.
2.129. .
45
" :
1 ! 71 18 7i i8 " #
2 ! 1, 18, 71, +1,
i1 i8 7i +i1.
2.130. 1 3 ( ) 2.126, jz j 1, 0 arg z =n n | ) , ' jz j 6 , arg z = 0 jz j 6 , arg z = =n 0 1
" ! '.
2 . " 1 6 jz j 6 , arg z = 2k=n k =
= 0 1 2 ::: n 1 " .
2.131. . !! "
"
, 7a, b8
7ci, ci8
a 0, b 0, c 0, a2 + b2 + c2 6=
6= 0.
4 . & ( )!, !! !! 70, ci8 !! , )
, "
!
.
2.132. 3
7a, +1 a 0 7ci, ci8 c 0
!! .
4 . 6 . 2.131.
2.133. . "
)
.
4 . > '
2.129, 1.
2.134. 3
7i, 08 1 . 2 !! .
4 . > '
2.131.
2.135. . !! 71 b8 b 1 )
ei , z = 1 . 3.
.II.
46
2.136.
: i bi , bi i , 1 a , a 1
a 1, b 1 .
" . # $% & '
2.131.
2.137 .
* +", ' - . 4.
2.138 .
+ cosx2 y2 = 1:
sin
2.139.
2
2
x
y
+ cos2 sin2 = 1:
2.140. , 2
2
' % + xa2 yb2 = 1.
2
-
2
0 - + '
%
1 2.14122.142 % ' %, %3
+ % + % 3 .
2.141. 4- , + ' 3
5 $ w = z 2:
1 $ r1 jz j r2 , 0 arg z + 0 6 8
2 jz 2 1j a 0 a 1 .
2.142. 4- , +
' 3
1
1
5 $ & w = 2 z + z :
1 jz j R R 1 . " . 9 ' jz j 1 $ 1 jz j R .
2.107 8
2 zz + 11
R
0
R
1
.
3
x 2 . VIII.
. -
4.
47
1 2.14322.145 + + 5 $-. r
p
2.143. 1 w = z 1 8 2 w = z 2 1.
z+1
w = z z + 1 8 2 w =
p
w = 3 z 2 1.
2.144. 1
2.145.
p
x 4.
2
r
z2 + 1 .
z
+ $ + 5 $
2.146. 1+% , % ' w = ez :
1 % % x = C , y = C 8
2 %+ y = kx + b8
3 y 0 6 6 2 8
4 ' %+ y = x, y = x + 28
5 x 0, 0 y 6 28
6 x 0, 0 y 6 28
7 % x , y 6 2 .
2.147. =
n - '
w = 1 + nz ? = +- -
n ! 1?
2.148. 1+% , % ' w =
= ln z :
1 % % jz j = R, arg z = #8
2 5 r = Aek' A 0 8
3 0 arg z 6 28
4 jz j 1, 0 arg z 6 28
5 $ r1 jz j r2 r1 r2 .
z = x + iy a a 0 + % - % % 5 $
w = + i = ln aa + zz :
2.149. 1 ? , 5 $% w ' z - 1, a a, 1 w: 6 6 ,
%% =
= pi, ' = . 5 .
.II.
48
2
= ch a+shcos
a sin
= ch + cos
p 2 + 2 = = r ch cos
ch + cos
3 ,
= 0,
! :
x
x
y
y
r
a
:
6 6
2
# cth 0 2 + 2 = sha
# = 0, 6 6 # . 6.
a
x
a
y
0
4 ,
= 0 ( , )!* )
,
2
2
+ # + ctg 0 2 = sina
a
x
y
a
0
* )
0
0
0. +
= 0 0
, , -. ( , .* ! = 0
= 0 # 0 0, !.
( ( # . 7.
5 1 2 # . 6 # . 7.
3 . 3 ! !
- 2 !
.
2.150. 42! , .!
=
= cos :
1 ! ( ! = , = 6
2 0
,
06
3 0
2,
0,
4 2
2,
06
5 0
6
6 ! ( 0
,
# 0.
2.151. 42! , .!
=
= arcsin :
1 )!! 6
a
a
b
l
z
w
z
x
x
C
x =
=
y
C
y
y
x =
y
x
x
h y h
h
w
z
4.
49
2
#1 1-, ,1 16
3 1- 6
4
0
#1 1-.
2.152. 42! , .!
=
= ch :
1 ! ( ! = , = 6 2 0
6
3
0, 0
.
2.153. 42! , .!
=
= Arsh :
1 1 6
1
1 6 16
2 2 .
2.154. 42! , .!
=
= tg :
1 ! ( ! = , = 6
2 0
,
06 3 0
6
4 0
46 5 4
4.
2.155. 42! , .!
=
= cth :
1 0
6 2 0
,
0.
x
w
z
x
x
C
y
C
y
y
w
z
y
y
w
z
x
x
C
y
C
y
x
x =
=
x =
w
z
y
y
x
7 !,
!* ! !
4 ) 2.15682.163 2 )..
.
2.156. 3 , ( . ! 2
= , = + .
2.157. 3
1, 0
.
2.158. 9 ( . , ( . ! j j = 2,
j 1j = 1.
2.159. 7 , ( . ! j j = 2, j 3j = 1
# 2 2 ( .
2.160. 7 , .
j 1j 1 j + 1j 1 Im 0
# )!! 2 2 ( .
2.161. 7 , . . : 2
2
= 4# + 1 2 = 8# + 2
. < . 2.80, 2.
2.162. 1 : = . # , .* . , ( . . j j = 1
! Im = 1 # Im 1
y
x
x
y
x
h
y h
z
z
z
z
y
x
z
y
z
z
x
:
w z
z
4 ..
.
z
.II.
50
:
1 w 1 w 3i = 0, arg w 3i = =3
2 jwj 1 w 3i = 12+ i , arg w 3i = 2
3 w 3i = 1 + i,
arg w 3i = .
! "
2.163. # $ % :
1 0 x 1 % % & ' x = 1=2, h 6 y 1
2 0 x 1 % % & ' x = 1=2, h1 6 y 1
x = 1=2, 1 y 6 h2 h2 h1 .
( % . )' $ % 0 x 1=2 . # $ +,- ."- $&,
" , $ + %& $ % % .
/ %&' 2.16422.174 $ % % $ .
2.164. ! 0 x 1
% % & % 0 6 x 6 h, y =
= 0 h 1 .
2.165. ! 0 x 1
% % & % 0 6 x 6 h1 ,
y = 0 1 h2 6 x 6 1, y = 0 h1 + h2 1 .
2.166. ! 0 x , y 0
% % & % x =
= =2, 0 6 y 6 h.
2.167. ! 0 x , y 0
% % & ' x =
= =2, h 6 y 1 h 0 .
2.168. ! 0 x , y 0
% % & %
x = =2, 0 6 y 6 h1 & ' x = =2, h2 6 y 1 h2 h1 .
2.169. #$ ,
' + - jz 1j = 1, jz + 1j =
= 1, % % ' 2 6 x 1, y = 0.
2.170. #$ ,
' + - jz 1j = 1, jz 2j =
= 2, % % & % y = 0, 2 6 x 6 a a 4 .
2.171. #$ ,
' + - jz 1j = 1, jz 2j =
= 2, % % & % y = 0, 2 6 x 6 a y = 0, b 6 x 6 4
a b .
2.172. #$ ,
' + jz
1j = 1, % % & % y = 0, 2 6 x 6 a & '
y = 0, b 6 x 1 a b .
2.173. #$ ,
' + - jz 1j = 1, jz + 1j =
= 1, % % % x = 0, 6 y 6 0, 0 .
2.174. #$ jz 1j 1, jz + 1j 1, Im z 0 -- % % % x = 0,
0 6 y 6 h.
j
j
0
0
0
4.
51
2.175. # $ % $ y 2 = 42 x + 2
&' .
( % . ! % % $ ,p $% $ , ." z
, % % - "
.
2.176 . # $ %
% % % 0 6 y 6 a, x = =2 + k k = 0 1 2 :::
. 8 .
2.177. ! % % a 6 x 6 a, y =
= =2 + k k = 0 1 2 ::: $ % % %
% & 5k b k + b6 k = 0 1 2 ::: 0 b
=2 .
( % . ! & % % , & % $ % 7 - $ $ % % - " .
2.178. # $ %
% % ' 1 =26,
5=2 +1 % a 6 y 6 a, x = =2 + k k = 0 1 2 :::
7 &' . 9 .
( % . 9"-, &,- 7 %&' 2.176, $ +
%& $ i %h % '
1 arcsin11=ch a arcsin11=ch a +1 :
4
.II.
52
2.179.
1
p
, +1 2 6
q
=
6 6 , = 2 + = 0 1
. 10.
2.180 .
0 6
1, = 2
= 0 1 2 .
!"
# $ % &
' ( 2.181*2.184 % & (& # $ & . . 49.
2.181. . ! , $ % & /
0 1
= z:
1 & # 0
, 0
3
2 0
,
03 3 0
.
2.182. . ! , $ % & /
0 1
= cos :
1 2
23 2 0
2 .
2.183. . ! , / 0 1
= tg
% & 0
2 .
2.184.
, 0 1 & = 1=z % -.
6
2
p q
:::
6 2
=
a
y
a
x
=
k
y
k
k
x
k=
:::
w
e
x a
x a
w
y b
y
x a
z
=
x =
x
w
x
w
x 5.
e
z
z
,
= | 0 1 &, & ( ,
0 = 0.
' ( 2.185*2.193 1 $! (
# 1 ( , 0 1 & = ( 3 2 | $! ( # 1 ( , $! 0 1 ! = % & (
$ , 3 | $! ( #
1 ( , % # 0 1 ! = ( , ( = 0. 7
w
f z
f
r
w
f z
r
w
f z
r
w
w
f z
5. ,
53
$ & ( ! ,
% ( ( ! & ! $ , 1 % / . 7 ( , 2 6 3 6 1 .
2.185. 8& 0 1
= 1 z z ! 1 , 2 , 3
$ # j j 1 , j j 2 , j j 3 .
2.186. . ! 1 (& %(! ( / 0 1 !:
1 = + 2 3 2 = + 2 | ( ! 3
3 = 1 2 .
2.187. 8 , %
=
z f z =f z
% j j = $ % & 0 ! = 1 + Re zf
.
z
2.188. 8 , & 0 1 & % % j j =h $
#(
#( , #(
i
h
i
zf z
@
+ + arg 1 + Re f z
0
@' 2
r
r
w
z
r
z
r
r
r
z
r
r
r
r
w
z
z
w
z=
w
z
az
a
z
w
f z
00
z
r
k
0
j
0
j
f z
z
r
'
f
00
0
z
0
(& = .
2.189. 8 , # j j
% & !
0 1 ! 0 = 0 , (
i
h
@
zf z
= 0, #( #( , #( @' arg = Re f z
0 (&
i'
= .
2.190. 8 :
1 0 1 & = 0 = 0 % # j j 1 Zz
, ( = 0, 0 1 & 1 = f tt
0
( % 9 % # $ 3
2
= % # j j 1 $ ,
1 = " % 9# # , (
= 0.
2.191. . ! 2 (& %(! ( / 0 1 !:
1 = + 2 3 2 = + 2 | ( ! 3
3 = 1 2 .
2.192. . ! 1
= z 1.
2 (& 0 1
2.193. . ! 3 (& %(! ( / 0 1 !:
1 = + 2 3 2 = + 2 | ( ! 3
3 = 1 2 .
: . "
( 2.193, 3 ( ( @
( @' + arg 0.
'
i' 4
re
z
z
f z
r
f
0
w
f z
'
z
re
w
f z
f
z
w
w
w
zf
0
f z
w
z
z
w
r
w
z
w
z=
z
w
z
az
a
z
r
r
w
e
r
w
z
w
z=
z
w
z
az
a
z
'
4
.,
, 4, . XIII, x 2.
f
0
z
dt
III
, , ,
. . " #$
" % " .
x
1. !
3.1.z & "z "" " % :
Z1
1$ dz = z1
z0
z0 *
Z1
2$ z dz =
z0
1
2
z12
z02 $.
3.2. , % C | " ,
%
Z :
Z
Z S . . %
1$ x dz = iS * 2$ y dz = S * 3$ z dz = 2iS .
C
C
C
Z
Z
3.3. % I1 = x dz, I2 = y dz " #":
1$ - z = 2 + i*
2$ jz j = 1, 0 6 arg z 6 |
z = 1$*
3$ jz aj = R.
Z
3.4. % jzj dz " #":
1$ - z = 2 i*
2$ jz j = 1, 0 6 arg z 6 |
z = 1$*
3$ jz j = 1, =2 6 arg z 6 =2 |
z = i$*
4$ jz j = R.
Z
3.5. % jzj z dz, C | " ,
C
# jz j = 1 1 6
6 x 6 1, y = 0.
Z
3.6. % zz dz, C | 3 %3,
C
. 11.
1.
55
3.7.
an dz n | + :
1
jz
aj = R, 0 6 arg z a 6
|
z = a + R
2
jz aj = R
Z
z
3
+
a
,
!
.
( * 3.8/3.11 ! !
(
() 0 +
! ! ( ( !
) !. 1 ( , ) )
! ! ,
) ( ( )
0 +
, (
, 2 ) ) .
3.8.
Z
pdzz
jz j = 1, y 0,
jz j = 1, y 0,
jz j =p1, y 6 0,
jz j = 1, 1 = 1
jz j = 1, p
1
2
3
4
5
3.9.
Z
C
p
p1 = 1
p1 = 1
:
1 = 1
1 = i.
Ln z dz , :
C | ! Ln 1 = 0
C | ! Ln i = i=2
C | jz j = R
Ln R = ln R
C | jz j = R Z Ln R = ln R + 2i.
3.10.
z n Ln z dz , n | + :
jzj=1
1 Ln 1 = 0
2 Ln 1 = i.
Z
3.11.
z dz , | (
j
z
j
=1
1 =
Z 1.
3.12. 4( ,
az dz = 0 , ,
( ! 0 +
az . jzj=1
3.13. 4! * 0 6 2 :
1
2
3
4
1
I1
Z
=
e1=z dz
(!
2
-
Ip
Z
=
e1=z dz p
z = ei ?
p
|
. III.
56
3.14. ,
1
2
a 6= R,
jR22Rjaj2 j :
j j
Z
3.15.
dz j
ajjz + aj
j
z
jzj=R
j
:
f z ,
Z2
lim f rei' d' = 2f 0
r!0
0
f z Z z = a,
f z dz
lim
= 2if a:
r!0
3.16.
jzaj=r
z
a
.
1 " f z x x0 , 0 6 y 6 h xlim
f x + iy = A, % y & %
Z!1
y, xlim
!1 f z dz = iAh, ' x | % &%
x
0 6 y 6 h, ) ' & % *.
2 " f z 0 jz aj 6 r0 , 0 6 arg z a 6
6 0 6 2
lim +z af z , = A
z!a
Z
lim
f z dz = iA
r !0
r
' r | * & ' jz aj r,
) ' & & .
3 " f z ) jz j R0 , 0 6 arg z 6 0 6
6 2 zlim
zf z = A,
!1
Z
lim f z dz = iA
R!1
R
' .R | ' jz j = R, % ) , ) ' & & .
3.17.
& .
1 " f z ) jz j R0 , Im z a a | / %
0% ) f z ! 0 z ! 1,
)' '
m
Z
imz
lim e f z dz = 0
R!1
R
2.
57
' .R | ' jz j = R, & &% ) .
1 . 2 3 & '
jz j = R, Im z 0 & sin 2 =
0 6 6 =2, 3 '&, & % a 0, | &, % * & jaj
R ! 1.
2 " f z Re z
| / %
0% f z ! 0
z ! 1, )' Z 3 '
t
zt
lim e f z dz = 0
R!1
R
' .R | ' jz j = R, Re z . " f z
Re z 6 , ,
t , .R | ' jz j = R, Re z 6 .
2 & . ) * & , & , +3, ' . V, x 2, . 73,.
x 2.
3.18. 2,
,
Zz
1
1
* d
= ln r + i' + 2ik
' k | 3
, , '
)* z = rei' .
3.19. 2,
* i,
Z1
d
= 4 + k
1
+
2
0
' k | 3
.
3.20. 2,
C | % % & %
, * % a, n | 3
,
0
n
= 1
6
Z
n = 1 a C
z an dz = 2i
0
n = 1 a C:
C
3.21. 5 ' & 67 & & :
f z &% ) G, ' % & & & & & C , G,
1
,
,
. "#$
x 4 . IV,
%
.
. III.
58
Z
C
f z dz
= 0.
2 .
. C ,
C : = z 0 1 z 2 C | "
# $ #% # ! 1 ., # , &1, . V, #. 8( &3,
. I, x 4, #. 12(.
3.22.
,- ..
1 / f z # 0 6 y 6 h, x!1
lim f x + iy = 0
1
Z
1
f x dx
- ,
1
Z
1
f x + ih dx
.
- , . 1$.
2 / f z 0 6 arg z 6 0 6 2,
Z1
Z
lim
zf z = 0 f x dx - , f z dz
z!1
0
z = rei , 0 6 r 1, . - ,
. 1$.
. 2#
3.16.
3.23.
,
1
Z
cos 2bx dx =
2
e x
p
2
0
2
e b .
. 4 5 6, f z = ez2 # 6 # jxj 6 R, 0 6 y 6 b # 8 Z1
p
2
et dt = :
1
3.24.
1
Z
1
Z
p
cos x2 dx = sin x2 dx = p
2 2
-
0
0
.
. 4 5 6, f z = eiz2 # 6 0 6 jz j 6 R, 0 6 arg z 6 =4 # 3.17, 1 #. z 2 = .
3.25.
,
1sin x
Z
0
x
dx
=
2
.
. 4
5 6, f z = eiz =z # 6
1 r 6 jz j 6 R, 0 6 arg z 6 #
3.16 3.17.
3.26.
, # 0 s 1 # :
2
,
,
, .
1
1
Z
3.
xs 1 cos x dx
= :s cos
s
;
2
2
1
Z
xs 1 sin x dx
59
= :s sin
s
2
.
. 4 5 6, f z = z s1 eiz # 6
1 r 6 jz j 6 R, =2 6 arg z 6 0; #
3.16, 2 3.17, 1 # :-5 6:
Z1
:t = xt1 ex dx:
0
0
0
x 3.
2, % # 5 C # $ $
# $ .
Z
dz
3.27. 2
, :
z2 + 9
C
1 3i . C , 3i | ;
2 3i . C , 3i | ;
3 3i . C .
Z
dz
3.28. 2 .
z z 2
C
1
# % #.% C . 8# , C #% 0, 1 1.
3.29.
< % $ . # Z
! dzz , !n z = z z1 z z2 :::z zn zi 6= zj C
n
C #% zi ?
Z
z dz
3.30. 2
, a 1.
z4 1
jzaj=a
3.31. 2
1
2 i
Z
C
ez dz
,
z + a2
2
C
-
. 1 jz j 6 a.
Z
zez dz
3.32. 2 1
, a .
2 i z a3
C
C .
. 2# 5 # % <".
Z
ez dz
3.33. 2 1
, :
2 i z 1 z 3
1 0 . ,
C
1 |
C;
. III.
60
2 1
, 0 | C
3 0 1
C .
3.34. f z , !" #"!" " C , $ % " $ . &#',
(" ! Ln z
Z
1
f z Ln z dz = f z0 f 0
2 i
0
C
$ z0 | ' .
* # . + ' ".
Z
1
3.35. ,! '
z 2 Ln zz + 11 dz , Ln a = ln a
2 i
C
$ a 0, C :
1 ' jz j = 2
2 ' jz 1j = 1 '
z = 1 + i.
3.36. / " 0
, 1 f z ,
Z, . &#'
f z dz
2 ", !
jaj R jbj R
z az b
jzj=R
# $ R ! 1.
3.37. 3 ' f z #" ,
" C z1 z2 ::: zn | # ! # '! C !n z = z z1z z2 :::z zn . 3#',
Z
P z = 21 i !f !n !z n z d
C
n
' " n 1- , $(% f z 4 z1
z2 ::: zn " P z #!
.
3.38. &#' $(%( " -
.
3 ' C | #"! , (% ( ' D. f z 5 D
lim f z = A. 6 $
z!1
f z + A z $
Z
f
1
5 D
d
=
2 i z
A
z $
D:
C
7 C 4$ '" '
D.
* # . / " ' A = 0.
4.
61
3.39. - ( ) f z C 0
1* .
+ ,
" . D,
Z
1
f
z 2 D
d = f z0=z
z 2= D:
2 i z 2
C
x 4.
7 2
1 2 3.40:3.51
1
1 2 .
1
X zn
X nn
Xn n
:
3.42.
n
z
:
3.43.
z:
n!
2n
n=1
n=0
n=1
n=1
1 n!
1
1
1 n
X
X
X
X
n
n
!
n
n
!
3.44.
z : 3.45. z : 3.46. 2 z : 3.47. z 2 :
nn
n=1
n=0
n=0
1
1
1 n=0
X
X
X
3.48.
<3 + 1n =n z n : 3.49. cos in z n: 3.50.
n + an z n :
3.40.
:
n
n=0
3.51. 1 +
X zn
3.41.
n=0
1 + 1::: + n
X
n=0
1 + 1::: + n
n! + 1::: + n 1
n=1
1
X
3.52. > 2
n=0
1 n
.
z
cn z n R 0 R 1.
7 2 :
1
1
1
1
X
X
X
X
1 nk cn z n$ 2 2n 1cnz n $ 3 cnn! z n$ 4 nn cn z n$
1
X
n=0
5
n=0
1
X
n=0
c z $ 6
k n
n
n=0
n=0
1 + z0 cn z .
n
n=1
n
3.53. > 2
1
X
n=0
an z
1
X
n
n=0
bn z n -
r1 r2 . @ " 2 2 :
1
1
1 an
X
X
X
n
n
1
an bn z $ 2 an bnz $ 3 b z n?
n=0
n=0
n=0
n
3.54. - jz j 1 01 :
1
1 zn
1 z2n+1
1
n
X
X
X
X
n
n+1 z
1
n=1
nz $ 2
n=1
n
$ 3
n=0
2n + 1
$ 4
n=1
1
n
.
. III.
62
3.553.61
.
3.55.
3.58.
3.60.
X1
X1=
X1
z
n:
n 0
1
n
n
n=1
1
ln
n=2
X1
3.56.
z
n
z
n
n=1
n:
n
zn
:
X1 z
n
X1 z
n=1
pn
3.59.
3n 1
X1 z
3.57.
:
n=1
3.61.
n=1
p
n
n2
:
| ".
n
!
n
:
2
#$
% & #$ ,
lim
r !1
X1
n=0
cn r
=
n
X1
n=0
0
cn
X1
n=0
r
1"
n
c
,
:
X
, , $ & #$ ,
1
, . .
*
n ,
1
n=0
n
* rlim
!1 n .
3.62.
X
c
c r
n=0
3.63. & & #$ + 3.54,
-* :
1
cos n'
1"
= ln 2 sin '2 0 j j 6 "0
n
X
1
n=1
2"
3"
4"
5"
6"
X
1
n=1
X
1
n=0
X
1
n=0
X
1
n=1
X
'
n' = '
n
2
sin
0
'
2 "0
n + 1' 1
'
n + 1 = 2 ln ctg 2
0 j j
n + 1' =
n+1
4
"0
cos 2
'
2
sin 2
2
0
'
1"n+1 cosnn' = ln 2 sin '2
1"n+1 sinnn' = '2
"0
'
'
n=1
".
"0
5.
3.64 . 1 ,
X1
n=1
1
pn
63
n
zn
.
. z = 1, , ! " # $ . z = 1 z 6=p 1,
# 1.90, ' an = (1) nz n ,
bn = 1=n.
3.65. * ,
# ' fang 1
an z n 1, ! 'n=0
jz j = 1, , ' , z = 1.
. + *
( . 1.88).
j j
X
X1
* , cn z n = Rei
n=0
' , #
# G, ' $ # , ' $ #
' - ' jz j = R,
$ , ' $ # ' jz j = R, .
- . . ' $ # /#
# 0 ( ., , 12, . III, x 74).
3.66.
x 5.
5 ' /# 6 #
X
+ 1 3.6793.81 / ' #
cn z n # .
n=0
3.67. chz: 3.68. shz: 3.69. sin2 z: 3.70. ch2 z:
p
p
3.71. (a + z ) (a = e ln a ): 3.72. z + i
i=
3.73.
1
az + b
1+z
(b 6= 0):
3.74.
z
:
z 4z + 13
2
3.75.
z : 3.77. Arctg z (Arctg 0 = 0):
3.78. Arsh z (Arsh 0 = 0): 3.79. ln (z 2 3z + 2):
3.76. ln
1
1+i
p
z
:
2
2
z + 1
2
:
. III.
64
3.80.
Zz
0
ez2 dz:
3.81.
Zz
0
sin z dz:
z
3.82 3.87 !z 1# $ .
2
3.82. z : 3.83. 2 z
:
3.84. z 2 :
z+2
z
2z + 5
z + 1
p
p
p
1 + i 3 : 3.86. ln z: 3.87. sin !2z z 2#:
3.85. 3 z 3 1 =
2
3.88 3.92 $
z sinzz p $.
3.88. e
: 3.89. cos z !pcos z z= 1 z = 0#:
3.90. !1 + z #z = ez ln 1+z : 3.91. ee : 3.92. ez ln !1 + z #:
3.93. 1# ) z * ln !1 + ez # !$ + , #.
- . . $ $ $ .
2# /, , 1
* z , 2 z=2.
- . ln !1 + ez # ln !1 +
+ e z # = z:
3.94 3.98, $
, z .
3.94. 4ln!1 z #52 : 3.95. 4Ln !1 z #52 !Ln 1 = 2i#:
3.96. !Arctg z #2 !Arctg 0 = 0#:
3.97. Arctg z ln !1 + z 2 # !Arctg 0 = 0#: 3.98. ez=1 z :
3.99. /, 1= cos z
1
X
cos1 z = !1#n 2En2n! z 2n, E2n ! 6$# n=0
* +
E0 = 1 E0 + 22n E2 + ::: + 22nn E2n = 0:
/, z=!ez 1#
1
X
z ez z 1 = Bnn! z n , Bn !
n=0
# * +
n+1
n+1
n+1
B 0 = 1
B
B
0 +
1 + ::: +
0
1
n Bn = 0:
3.100.
5.
65
3.101. /, 7 ,
B1 , *.
- . ez z 1 ez z 1 = z .
3.102. ) z * z ctg z $
8 .
- . *1 6$ z ctg z = iz + 2iz=!e2iz 1#.
3.103. ) z $
:
1# ln sinz z : 2# tg z : 3# ln cos z : 4# sinz z .
3.104. /, , cn
1
1 =X
n
1 z z2 n=0 cn z
* +* cn = cn1 + cn2 !n 2#. ;$ cn
.
. . < cn * .
1
A + Bz + Cz2 = X
n
3.105.
+ z + z2 + z3 n=0 cn z ! 6= 0# $
c0 , c1 , c2 , + cn , cn1 , cn2 ,
cn3 !n 3#.
. 1 8
= 8 jtj R1
X
F !t z # = fn!z #tn
n=0
F !t z # ffn!z #g. < $
$ ffn !z #g , $ 1$ .
3.106. . 7 'n !z # *
1
tz
X
t eet 11 = 'nn!z tn :
n=0
/ *1 $ :
1# 'n !z + 1# 'n !z # = nz n1:
2# m | ,
'n+1 m = 1 + 2n + 3n + ::: + !m 1#n :
n+1
5 ..
.
66
3 'n z =
3.100.
X
n 1
k=0
. III.
n B z n
k k
k
Bk |
,
. -
p1 21tz + t Pn z :
1
=
Pn z tn :
p 1
1 2tz + t
n=0
:
1 n + 1Pn+1 z 2n + 1zPn z + nPn1 z = 0$
2 Pn z = Pn0 +1 z 2zPn0 z + Pn0 1 z $
3 2n + 1Pn z = Pn0 +1 z Pn0 1 z .
' . (( ) () t z .
3.108. * + ( ,(( - , , / Z1 s 1,
n d
Pn s = 21i
3.107.
X
2
2
C
p
1
2s + 2
+ C | R 1 / = 0.
4 t
3.109. , / (
4 4tz + t
1 23
Tn z = 2n1 cos n arccos z :
* + 3 ( ,(( - , , / 4Tn+1 z 4zTnz + Tn1z = 0 n 2.
3.110. * 3 5 61 23 Hn z )
1
e2tzt2 = Hn z tn :
2
2
1
X
n=0
n!
) :
1 Hn+1 z 2zHnz + 2nHn1z = 0 n 1$
2 Hn0 z = 2nHn1z n 1$
3 Hn00 z 2zHn0 z + 2nHn2 z = 0 n 0$
n z
4 Hn z = 1n ez2 d dze n .
3.111. * 3 1 23 6+
n n z
Ln z = ez d zdzne :
5.
67
8" !. - . ' fLn z g
!'. ' . - , 4 .!.
Ln1 z Ln z
Ln+1 z :
/ . 9 : 3.10773.111 4 (' 4 4 " 4:
. < : $"( : : " :, .! : 3 '(. ' (
4: " - , ., , =3, . VII, x 2>
:
? @., A
' 3 &. B 4 " - . | B.:
A : , 1951. | 1. 1, . II, VII.
@( -- '4: "
9 : 3.11273.114 " ( 4: -- '4:
", .! w0 = 0, w0 0 = 1.
3.112. w00 z 2 w = 3z 2 z 4 .
3.113. 1 z 2 w00 2zw0 + nn + 1w = 0.
3.114. 1 z 2 w00 4zw0 2w = 0.
1
X
3.115. @ $ '
cn z n - . cos m arcsin z
=0
n
arcsin 0 = 0,
-- ' , ( " 0 - .
3.116. & -- '
z 1 z ddzw2 + =c a + b + 1z > dw
abw = 0
dz
2
4 .
8" z = 0 ( wz , .! . w0 = 1, ,
c .
'
.
3.117. & ', 3! (
c
-
w = C1 F a b c z + C2 z 1cF a + 1 c b + 1 c 2 c z
F a b c z | - , 4!"
" .
3.118 . & ',
'
5
-
c .
,
dF a b c z ab
=
F a + 1 b + 1 c + 1 z :
dz
c
. III.
68
x 6.
3.119. , z0
k f z , z0 f z = z z0 k ' z ,
' z z0 ' z0 6= 0.
3.120. z = 0 :
1 z 2 ez2 1 $ 2 6 sin z 3 + z 3 z 6 6 $ 3 esin z etg z .
3.121. ) z0 k f z
l ' z . * z0
+, :
1 f z ' z $ 2 f z + ' z $ 3 f z =' z ?
. 3.122/3.136
0 .
z2 + 9
2
3.122. z + 9: 3.123. 4 : 3.124. z sin z:
z
z2 2 sin z :
z
2
3
3.125. 1 e z 4 : 3.126. 1 cos z: 3.127.
z7
1
ctg
z
3
tg
z
3.128.
: 3.129. e : 3.130. sin z:
z
3
sin
z
3.131.
: 3.132. sin z 3 : 3.133. cos3 z:
z
p
p
3.134. cos z 3 : 3.135. z 2 3 : 3.136. 1 2 2 cos z 2 :
)
3.137. 34 5, A-
, 4
, + ?
3.138. 6, , z = 0
+, z = 1=n n = 1 2 ::: :
1 0 1 0 1 0 1 ::: 0 1 :::$ 2 0 21 0 14 0 16 ::: 0 21k :::$
3 21 12 14 14 16 16 ::: 21k 21k :::$ 4 12 32 34 54 65 67 ::: n +n 1 :::?
3.139. 6, + , z = 0
+,
n |
1
1 :
1
1
1
1 f
= f = $ 2 f
=f = 1?
n
n
n2
n
n
n3
3.140. 8 sin 1 5 +
1 z
, ,+ z = 1, 9 . 9 ?
6.
69
23
) # #
3.141 . . " f z = ux y + iv x y z0 =
= x0 + iy0 f z0 = c0 . 5 ) ,
f z = 2u z +2 z0 z 2iz0 c0 :
3.142. 5 ) , "( 3 )
z+z z z
f z = 2iv
2 2i + c0 :
2 ) ( 3.14363.145 # # f z =
= ux y + ivx y .
3.143. ux y = x2 y 2 + 2.
3.144. ux y = ex x cos y y sin y 2 y 2 .
x +y
3.144. v x y = x + y 3. 3.145. ux y = cos x sh y sh x sin y .
1
3.147. :
0
0
8 9
) f z ; jz j R
1
X
f z = cn z n :
n=0
1
Z2
X
1 5 ) , 21 jf rei' j2 d' = jcn j2 r2n r R.
n=0
0
2 5 ) , max jf z j = M r, / 3 cn
jzj=r
"#
jcn j 6 Mrnr r R:
3 5 ) , ( " 3 ) 8 9 " , . . jck j = M r=rk , " f z f z =
= ck z k .
< ) . 2 ) " # ) . 1
1
X
jcn j2 r2n 6 =M r>2 :
n=0
3.148. % # 8 9 ) :
" f z ; ,
" .
: . 5 ; ) 3 ?"
) ) 3.36.
. III.
70
1
X
=
3.149. , z = 0 -
f
z
n=0
cn z n
,
M
c
j 0j
+
c
j 0j
, | !
, " #$!% & , M = M =
= jmax
f z .
zj=
( . ( #, f z # , f z c0 j jc0 j, jf z c0 j, # "##
# # ).
j
j
j
3.150. *
,
f z
' z =
=
X1
X1
n=0
cn z n
cn z n
n!
" jz j 6 r. -
&
# # "
$ " # #$ j' z j M ejzj=r , j'k z j M
ejzj=r
rk
M | " .
n=0
+ $ "% $&
X1
# jz j 6 1
.1 G "% S . , S = njcn j2 .
n=1
( . /" #$ "% S # " $& &.
, . 0 " # , $ G , " ! # jz j 6 1 # #!% f z .
3.152. , # # " $ % f z # $ jz j 1 "
. % # $ " rlim
!1 Sr = S , Sr | "% 3.151. , f z =
X
jz j 6 r 1,
n=0
X1 j
n=1
cn z n
n cn j2
cxo # S .
X1
2
, rlim
!1 Sr = 1, n=1 njcn j & .
, . 1 ., " , 24, . XIII, x 14.
3.153. 1 , 3.151, ,
f 0 0 = 1 f z #
6.
71
jz j 6 1 ! G, "% G
"%
.
2 , # & f z , $& # jz j 6
6 R ## !%& #!
Z2
0
jf Rei' j2 d' = M ,
"% . 8 . "% , f 0 = 0.
, "
9 & 3.154:3.157 # "# " " .
3.154. , f z , $,
# G % # , jf z j
# G.
3.155. 1 , # , " # f z . . , # # &
& jf z j = const % # #
f z , "
. f z 6 C .
2 , P z | " n, # jP z j = C " ,
n # $& " .
3.156. : f z #
jz j 1, f 0 = 0 jf z j 6 1, # # jf z j 6 jz j.
, & $ # # jf z j = jz j, f z = ei z | # .
( . = ! f z =z "
" " .
3.157. , # " $ % # f 0 =
= 0 # f = 0 jj 1 , " jz j 6 1 " # #
# # jf z j 6 1z z
.
( . = ! 1z z
f z .
IV
.
.
x
1.
4.1 4.18 !, !## $ . %#!! #! ! %!! #, $ !!
!! !#.
4.1.
1
2
1
z
!## !
z=0
z = 1.
4.2. z ak (a 6= 0, k | ! #* !## ! z = 0 z = 1.
4.3. z1 1 z !## ! z = 0, z = 1, z = 1.
4.4. z a1 z b (0 jaj jbj* !## ! z = 0,
z = a, z = 1 ! jaj jz j jbj.
4.5. zz 22zz ++51 z = 2 1 z 2.
1
4.6.
z + 1 z = i z = 1.
p
4.7. z az b jbj jaj z = 1 2
2
2
j j
2
.
r
3
z
4.8. f z =
Im
f
0
1 jz j 2.
z 1z 2
2
4.9. z 2 e1=z z = 0
z = 1.
4.10. e1=1 z z = 1
z = 1.
2
z
4z
4.11. cos
z 22 z = 2.
z = 1.
4.12. z 2 sin 1
z 1
4.13. ez+1=z 0 jz j 1.
1.
73
4.14. sin z sin 1z # 0 jz j 1.
4.15. sin 1 z z !## ! z = 1 z = 1 ( %#!!
#! ,# !- %! - ! *.
4.16. ctg z !## z = 0 ! jz j 2.
4.17. ln zz ab !## z = 1.
4.18. z 1 2 ln zz + ii !## z = 1 !
1 jz j 2.
4.19. - #, %# -! !!
!## $ :
1* cos z1 , z = 0/ 2* cos 1z , z = 1/ 3* sec z 1 1 , z = 1/
4* ctg z , z = 1/ 5* th z1 , z = 0/ 6*
z2
, =
sin1=z
9* ln z 1 1 ,
z 0/
z = 1/ 8* ln z , z = 0/
z = 1/ 11* z (= e ln z *, z = 0.
z
,
sin z 3
10* ln zz + 1i ,
7*
z = 1/
4.20. - #, ! -! ,-! -! ! , %#5! !! ( ##, 6!$* !## $ :
r
p
p
z
1* z, z = 0/ 2* z (z 1*, z = 1/ 3* z 1z
, z = 1/
2
p
p
4* 3 (z 1*(z 2*(z 3*, z = 1/ 5* 4 z (z 1*2 , z = 1/
p p
p p
p p
6* 1 + z , z = 1/ 7* 1 + z, z = 0/ 8* z + z 2 1, z = 1/
r q
p
9* z + z 1, z = 1/ 10* 1 + z +z 1 , z = 1/
z
11* Ln 8(z 1*(z 2*9, z = 1/ 12* Ln z
,
z z
p
2
3
z = 1/
13* Arcsin z , z = 0/ 14* Arctg (1 + z *, z = 0/ 15* Arsh (i + z *, z = 0/
p
p
p
16* =2 Arcsin z, z = 1/ 17* =4 Arcsin z, z = 2=2.
1
X
4.21. : f (z * =
cn z n , !# ! r 6
n=1
6 jz j 6 R, # ! ; ! # D.
<, %5 S ;$ #
1
X
S=
njcn j2 (R2n r2n *:
n=1
74
. IV.
.
2 , S
, f z r jz j R !
.
# . $. 3.151 3.152.
1
jj
4.22. () f z z 1
c
c
!* + f z = z + 1 + 22 + :::
z
z
,
1
n cn 2 6 1
n=1
. * . *.
# . 0 , Sr , * 1 z = r 1, f z = u + iv
Xj
0 6 Sr =
x 2.
jj
Z
jzj=r
u dv =
j
Z2
f
0
+f 1
2 2i
@f
@'
@f
@'
d':
0 4.2344.58 * . )*, .
)* 1 .
1 . 4.24. z4 . 4.25. z5 . 4.26.
1 .
4.23.
z z3
1 + z4
1 z2
z z 2 + 42
ez
z2 + 1
1
1.
4.27.
.
4.28.
. 4.29. zez . 4.30. z
1 + z2z
ez
e
1
z
e
1 ez . 4.33.
1
4.31.
. 4.32.
.
4.34. th z .
z 1 ez
2 + ez
z 3 2 cos z
2
4.35. e 1=z . 4.36. ze1=z . 4.37. ez= 1 z . 4.38. ez 1=z .
e1= z
1
cos z
4.39. z
.
4.40.
.
4.41.
.
4.42. tg z .
1
e
4.43. tg
4.47.
4.50.
2
z
1
4.44.
.
1
sin z
z2
4.53. sin
sin a
z
+
4.56. etg 1=z .
1
.
z2
.
z2
z
4.45. ctg z
1
cos z + cos a
2
1
z
z
1
z
sin 1=z
.
.
4.46. ctg z
4.49. sin
.
4.51. ctg 1 .
.
1
4.54. e z cos .
4.57. sin
z2
.
4.48.
.
7
42 cos 1=
1
1
z
sin z
ctg z
1
1
z
z
.
.
4.52. ctg 1
4.55. ectg 1=z .
1
4.58. sin
2
1
z
cos 1=z
z
.
.
-
2.
75
0 4.5944.68 1* .
* * * ) . , * *
* .
z
1
4.59.
1 + pz 3 , z = 4. 4.60. pz + p3 z , z = 1.
2z + 3
1
4.61.
1 + z 2pz , z = 1. 4.62. cos 1 + pz , z = 1.
1
2 + pz sin 2 pz , z = 4.
1p , z = 1 + 1 2 , k = 1 2 ::: z = 1.
4.64. ctg
k
1+ z
1
, k = 1 2 :::
, z = 21 + k
4.65.
q
2 1
1
+
k
sin 1 +
2
z = 1.
4.66. sin
q1
, z = 1.
1+
2
1
1 , z = 1.
4.67. 1
, z=1
2
Ln
sin 2
sin Ln4
4.68. sin ctg Ln z , z = 1.
4i
4.69. 8 Pn z Qm z | . n-*
m-* *. : )*:
P z
1 Pn z + Qm z
2 n
3 Pn z Qm z .
Qm z
4.63.
z
z
z
z
z
i
z
i
4.70.
*:
1 z0 .
n ) f z ,
1 f z z0
1
X
cn z z0n cn 6= 0 cn+1 = cn+2 = ::: = 0
n=1
2 z0 .
n ) f z ,
* !* f z = ' z = z z0 n , ) ' z ' z0 6= 0.
f z =
4.71. 8 . )*, *
:
1
2 z = 0 * 1 c2 =z 2 *
3 . . zk = ! k , ! = e2i=n k = 0 1 2 :::
::: n 1.
76
. IV.
.
4.72.
, :
1 2 n
3 " # z = 0 " # &' 1=z 2
4 n # z = 0 m #
5 n " .
4.73. + f ,z | & #
, G ". , . /&, #
f
z
f z A ," # &
f ,z A 0 0 . .
f ,z . A-#. 1
. ". 0. #.
4.74. 2 # z = z0 , # z0 = 1 F ,z = f 3',z 4, ',z 1 #
# , # 0 = ',z0
f , " :
1 # 2 n
3 #?
4.75. 6# z0 , # z0 = 1 & #
f ,z , ' " ' , 0 & " '
, 0 & . 2 .
f ,z # z0 , # z0 , f ,z
' #& , # z0
f ,z :
1 n 2 #?
4.76. ' : # z0
f ,z , , 0 0 # A , # A = 1 ,
# fzn g, . # z0 , # nlim
f ,zn = A. /&, #
!1
7." & #, 2 . ,8 "
# # 0.
4.77. 0:
1
1 y!1
lim ctg 2 z 2 y!1
lim sin1 z 3 x!1
lim ch1 z 4 xlim
.
=0 sin 1=z
0
0
y!0
He # 1. 7."?
2
,
.
-
3.
77
' :
# #
# " & # & # , & # , 0 ', ",
&0 nu apo . 9 0
, &' # #0. & # , # 1 . . ,., , 32,
". VIII, x 84 .
4.78. + +
:
1
z
1
=z
2
1 e 2 e 3 cos z 4 tg z 5 tg z .
# 0 & # ' & 1.
&, # 1 & # , #, . . # ' & . 0
, # #,
# & # .
x 3.
; & #. 4.79>4.99 0#0 & 0.
. & 0. 0. # # # , 0. # .
1
z2 .
4.79. 3 5 . 4.80. 2
z z
z + 12
2n
z
4.81.
,n | # . 4.82. z 1 1 z2 .
1 + z n
2
z
4.83. z 2 + z 1 . 4.84. sin 2z 3 . 4.85. 2 e2
z z 1
z + 1
z z + 9 . 4.86. tg z .
2
ctg 3 z .
z . 14.88. ctg z . 4.89.
1
1
3
z+1=z .
4.90. 1 cos
4.92. sin z sin .
z 2 2 z cos z2 2 . 4.91. e
z
z
+ 4z
1
1
z
4.93. sin
z + 1 . 4.94. cos z + 3 . 4.95. z 1 ehz ,h p6= 0 .
1
4.96. z n sin 1 ,n | # . 4.97.
. 4.98. sin pz z .
z
sin 1=z
tg z
4.99. n ,n | # .
z
; & #. 4.100>4.107 0#0 ' & & # 0. . "& # 0.
&p 0. #.
z # z = 1.
4.100.
1z
4.101. p 1
# z = 1.
2z+1
4.87.
1
sin
78
. IV.
za
p
4.102.
pz
1
.
z a = ea Ln z
z
z = 1.
a z b z = 1.
z = 1
4.104. 1 Ln
z
z z = 1.
2 ez Ln
z
1
4.105. 1 Ln z sin
z = 1
z1
1
2 Ln z cos
z = 1.
z1
4.103.
4.106.
4.107.
1
z
z
Arctg z
z
z n Ln zz n |
6
z=0
&
4.109.
6
! "# $
f
z = c0 + cz1
:::
+
'zfz
(% res )
*z =a ,
+%
:
1 % - &
f z &
2 -
4.110.
z = 1.
=
,
= 0,
= 0.
4.108.
k
h f 0 z i
(% res
f z
(% res )
A
%
z=a
,
:
' z
z=0
-
z=0
#
2
*z = .
ff z g
1
%
a,
c1 + ::: + ck .
za
z ak
a | # n "# f z
a | - n "# f z .
f 0 z
4.111. (% res ' z
,
' z f z z =a
a :
1 a | #
n "# f z
2 a | - n "# f z .
4.112. (% res ff )' z *gz=a , "# ' z
a '0 a 6= 0, f & -
= ' a & A.
4.113. .# ' z &
a -
& A, f & $ -
% - B . (% res ff )' z *gz =a .
4.114. .# f z , &-/ # l #! jz
aj = R % , !
+# # # # && . 0# z = 6= a
1
2
h
i
4.
79
X
k
f z
(%
cn
k % :
z
n
n=1
| , &&
-&
res )
z =
f z *z=
l .
,
x 4.
%
( & &
1
2 1 4.11554.124 , , $1 & #1
# 1 ! & .
4.115.
4.116.
4.117.
Z
CZ
dz
,
z4 + 1
C|
z dz
#!
x2 + y2 = 2x.
,
z 1z 22
CZ
C|
#!
z 3z5 1 ,
C
C|
#!
dz
jz j
jz j
2 =
1.
2
= 2.
6 . 2
&, #&& 1 $1
- $ # #- # -.
Z
jj
z 3 dz
2z4 + 1 , C | #! z = 1.
C
Z
ez
4.119.
dz , C | #! z = 1.
2
z z 2 9
C Z
1
1
4.120.
2i C sin z dz , C | #! z = r.
1 Z sin2 1 dz , C | #! z = r.
4.121.
2i C
z
Z
1
n 2=z
4.122.
2i C z e dz , n | , C |
= r.
4.118.
jj
jj
jj
jzj
4.123.
4.124.
4.125.
Z
jzjZ=3
jzj=5
1+
z + z 2 e1=z + e1=z1 + e1=z2 dz .
z dz
sin z1 cos z .
2
& #%
#!
1 Z f z dz ,
2i C zgz
#, -/% $
G,
C
| %
!/#- -
80
. IV.
.
= 0.
,
,
!
"
1 2
n
! ! .
4.126. #
= 0 + 1 + + n n.
% ,
Z
1
n1
2
= 0 &n 2n
2 i
z
f z
g z
G
g z
C
G
a a ::: a
f z
jzj=R
z
jf z j
a
a z
dz
a a
:::
R
a z
:
' ! " 4.127,4.130
. . Z
1
4.127.
p 2 dz , .! | 02 i
jzj
|
4.129.
y
1
2 i
Z
C
z
p1 = 1 ,
dz
4
p
+ 1 z
2
+1
x
y
e
a
C
" = , 0 Z 1.
dz
2 . 3 21 i az sin
, .!
z
x
4.130.
5
Z
C
!
! 1.
.! . . 13.
ez dz
,
cos z
C
4! .
, f x
c b
Zb
f x dx
a
lim
!0
"
" c"
6
Z
f x
a
. 12,
1
2 i
.!
= , "! .
dz
z = z ln a , .!
0,
| "!
az sin z
2
a
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z +z+1
r
4.128.
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6 Z 1.
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c
a
#
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c+"
x
f x dx
:
7 ! . . .
5
, . f x
.
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1
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N
f x
.
dx
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81
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4.131.
4.132.
4.133.
4.134.
4.135.
4.136.
4.137.
4.138.
Z2
0
d'
a + cos '
Z2
a 1. . 34 ei' = z .
d'
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0
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0
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1
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0
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0
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a+b
a
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:
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2
2
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1 a b1
. z + z z dz ,
C
e C | , 4 $! . 14,
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8 4 $ + 4 7! $
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& ! 9p91 p = sin p .
p + q
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82
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.
4.140 4.146 .
4.140.
4.142.
1
Z
1
Z1
0
4.143.
4.145.
1
Z
1
Z1
x dx
.
x2 + 4x + 132
dx
x2 + 1n
dx
0
0
x2 dx
x a2 2
2 +
a 0 .
n | ! .
dx
x2 + a2 x2 + b2
1 + xn
1
Z
4.141.
a 0, b 0 . 4.144.
1x
Z
0
2
+1
x4 + 1
dx.
n 2 | ! .
Z
$ . % 1 +dzzn , C | !,
C
&'( !( arg z = 0, arg z = 2=n &*'(
! !+.
4.146.
1
xn
1 + x2n
Z
0
dx n 2 .
, . - & 4.145
4.146 & / 0!/( R xn .
Z
ikn1 n + k 2!
1
d
4.147. 1,
=
n
n
k
2i
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C
k | ! , C | &&, & ( ( *'& (Z , ( h h 0 .
1
d
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k | !
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4.149 4.152, !& ( 3 . ! 3.17 , ! .
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4.150.
4.151.
1
Z
x cos x dx
6
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1
1 x sin x dx
Z
1 x + 4x + 20
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2
0
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2
1
Z
1
x sin x dx
.
x2 2x + 10
.
dx a b | + .
4.152.
1x sin ax
Z
0
x2 + b2
4.
dx a b |
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.
4.153. f z = eimz F z , m 0, F z
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2 $ $( ( % $ % , ' # x1 ,
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3 F z ! 0, z ! 1 Im z 0.
+ ,
,
m
m
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1
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o
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e
x cos x dx
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dx
.
4.155.
.
x
x2 5x + 6
i
4.156.
11
Z
1
sin x dx
x2 + 4 x 1
1
. 4.157.
1 cos tx
Z
1
dx.
1 + x3
4.158.
1 cos tx
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1 1
x4
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.
Z1 2
Z1
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4.159.
dx
.
4.160.
.
4.161.
.
x2 + b2 x
x x2 + b2
x x2 + b2 2
0
1cos 2ax
4.162.
4.163.
0
Z
0
1sin
Z
2
x2
0
x
x2
cos 2bx
0
dx.
dx.
7 , Z .2iz 0 , $
' # e z2 1 dz , C
' C , '. 15.
4.164.
Z
C
6
e3iz
1sin
Z
0
x3
3eiz + 2
z3
3
x
dx.
7 , . 0
, $ ' #
dz , ' C , '. 15.
. IV.
84
.
4.165 4.168 , , xp 0 x 0 !
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1
xp1 cos ax dx a 0, 0 p 1$%
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1 0
2$ xp1 sin ax dx a 0,
Z
0
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C
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z p1eaz dz , ( ! C !( . 16.
4.166.
1
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0
4.168.
1
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1sin xp
Z
0
xp
0
dx p 12 .
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:::, n , ( - - . . . !", p | ( . ,
p+1
p+1
zlim
!10z f z $1=0:
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3 (, :
1$ p + ,
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n
0
pi
= sin p e
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k=1
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n
0
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k=1
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' ( . 6) z pf z $ dz
C
Z
z p Ln z f z $ dz , C | ( !, 7-. . 17.
C
4.170.
1
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0
dx
xp x + 1
0 p 1$.
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sin a
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85
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4.172 4.174 !( .
4.172.
1 xp dx
Z
1+x
2
0
4.174.
1 p 1$. 4.173.
1
Z
0
xp dx
x2 + 2x cos + 1
1
Z
0
xp dx
1 + x2 2
1 p 3$.
1 p 1, $.
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p+1
lim 0z p+1 f z $1 = zlim
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z !0
3 (, ) ) , :
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1
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n
m
X
X
0
p
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k res 0f z $1z=k
Z
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k=1
k=1
x 0%
2$ p | + ,
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n
m
X
X
0
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k=1
k=1
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4.176 4.178 .
Z1
Z1 2
4.176. x4 dx . 4.177. x4 dx .
x 1
x 1
0
0
86
. IV.
4.178.
.
1xp dx
Z
0
x
1
p
1
4.179.
0.
1 epx dx
Z
1
ex
1
0
p
1.
4.1804.186 .
Z1 1 p
p
x
4.180.
Z
x
1
dx
1 + x3
0
z1 p
C
z p
1
dz ,
1 + z 3
p
1
2.
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C |
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R ! 1.
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4.181.
x1 p
1 + x2
0
p 2!.
1
xp
dx
1
" . # ,
lim
R!1
z1 p
Z
1+z
CR
z p
1
2
dz = 2iepi
CR | % & ' -
&
4.182.
Z1
x1 p
0
4.183.
4.184.
Z1
0
Z1
0
4.185.
4.186.
x
Z1
1
1
x
1
' jz j = R.
xp dx
1 + x2
1
x
x+a
p
x
1 + x
x + 1
1 p 1, a 0!.
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x + a2
p 1
1
1 + x2
Z1
0
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1 p 2!.
dx
p
3
2
x
1
xp
x
1 p 1, a 0!.
dx 1 p 2!.
.
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0
x
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1
x
a
p
1
2
,
2
4.
87
1 a 1 ! .
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p
x
1 x
4.188.
dx, 0
bx
b | ! *! ! b 6= 0, b 6= 1.
0
Z1
4.189.
0
p 1,
p1dx
xn n = 2, 3, :::.
. !
Z
C
pn 1dz zn , C | ! ,
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4.190.
Z
C
ln z dz
1 ln x dx
Z
0
x2 + a2
.
. !*!!( !
, ! C . 19.
z 2 + a2
4.191.
1 ln
Z
0
4.194.
Z1
0
2
x dx
.
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4.192.
x dx 2 .
ln
1
x
0
x
Z
0
0
1+x
&
% . 20.
Z1
4.195. ln 1 x
1
Z
dx
.
x+a
Z
C
ln
pxlnx2x+dxa2 2 . 4.193.
1
pxlnxx+dx12 .
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z
z
1
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,
1 + z2
C | ! , -
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88
.
4.196. |
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h f z i
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z
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O
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k
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6 arg 2 .
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a
a
:::
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f z
z
z
z
i
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4.198.
1
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0
1
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0
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x + 2
C
# . 21.
+ # , 4.19704.199 & 0
| .
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n
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a
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2
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a
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4.
89
4.200. |
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C
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C
a
h
h
bi
z
+
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z
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C
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4.201.
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a
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sh x
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i
R
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90
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e dz
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C
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0
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0
.
ch x
dx
C
. 4.204.
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0
sh x
.
a .
.
4.206.
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0
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h ! 1.
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f z dz ,
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4.
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1
2i
4.208. 1
4.209.
1
2i
4.211. 1
4.212.
Z
1
2i
1
2i
C1
C1
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2
z n+1
Z
C1
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2i
2
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.
z an+1
C1
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Z
az
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C1
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1
2i
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C1
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z 2 z 2 + 1
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.
1
2i
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c
bz
4.214. 3 , 4z 41 z =
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1
2i
ez dz
z +1
=
1
+ 1
) .
tz = ez ln t .
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1
2i
4.210.
ezt dz
z
91
Z
C1
tz dz
.
z z + 1:::z + n
sin z
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.
1 ez dz
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2i z +1
1+ 1 !.
Z
4.215.
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Re
1 Z
2i
1
C1
ezt dz
z +1
=
t
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:
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ezt dz
1 Z p
1 Z p
4.216. 1+
,
2+
.
.
4.217.
2i
2i
2i z 1 + z
1+z
z+i
4.218.
1
2i
Z
C1
Z
4.220. 1
2i
C1
4.221.
C1
C1
ezt dz
p .
z + 1 z + 2
ezt 1
z
e az 2
4.219.
dz
-
a
1 Z
ezt dz
-a 0+.
2i z 1 e az
C1
1 1e az = 1 + e az + e
1
2i
Z
C1
C1
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.
0+.
.
2az
+
. $! !
:::
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.
Z
etz
p
x z
dz x 0.
.
C1 , . p26.
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1
e sh r z
4.223.
p dz a r
2 i
rz sh a z
C
0.
Z
ln 1 + z ezt
4.224. 1
dz .
2 i
z
4.222.
1
2 i
C1
z
1
4.225.
4.226.
.
1
2 i
4.227.
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Z
ebz
z
C1
CZ1
ezt ln zz + 11 dz .
C
Z1 Z zat=z
dt e z dz a 0.
1
2 i
0
1
C1
2
.
1
dz eaz ch x dx a 0, b | !
Z
0
. #! , "
C1
4.228. %& !'
1
X
J z =
k=0
Z
( )
1k
z
k! k + + 1 2
euz
z
dz = 0 u 0.
* !
+2k
! ! | , " 4.214- 2
Z z =4
1 21 i e +1 d = 2z J z
2
1
2 i
Z
C1
e z =4
+1
2
d =
2
z
J z Re 1.
. .!' ( ) / ez2 =4
! 0
" 4.214 4.215.
4.229. 2, " Re z 0
Z
J z = 21 eiz sin i d
3 | ,
. 27, !" / , " !
-
4.
n
,
Z
1
Jn z =
cosz sin
93
| "
n d:
0
. " / 4.23014.232
3 .
4.230.
1
Z
0
e zt Jn tdt
,
( +, 2
Re x 0,
n
| " .
. . ( +! ! +2, "
! ( .
1
Z
4.231. 1
0
J0 at cos bt dt
+ " .
4.232.
1
Z
0
px
cos bx sinpt
2
x2
a2
a2
2
dx
1
Z
0
J0 at sin bt dt
b
| -
t jbj.
. . !, "
r
p
sin ut
t
t 1 Z
=
J1=2 ut =
u
2u
2 2 i
C1
!. " 4.228,
a
!
5 ! "
ezu t = 4z
dz
z 3=2
2 2
(
.
( 3
4.233. 6
" 'z ! C1
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'z ! 0 z ! 1 Re z 6 . 87
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1
zt
f t =
e 'z dz:
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C1
9 , tlim
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4.234. 6
" 'z ! C1 " " 7+/ " 'z ! 0 z ! 1 Re z 6 .
3
,
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94
.
,
1
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t -
Z
ezt 'z dz
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zt
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f t
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t ! 1 ) f t =
1
2 i
Z
C1
ezt dz
z 2 z + a3
Re a 0:
t ! 1 )
4.236. +!
p ,
f t =
p
1
2 i
Z
C1
2
zezt z +2az
p2
z !i z + 2az
dz ! 0 a 0
z 2 + 2az 0
z 0.
/ . 0 C1 , ,
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t ! 1.
1 cn
2
zn
X
f z
n=0
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h
f z
k
X
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zn
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2 " , ,
4 . $ fqn z g |
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n z
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0
lim
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2
1
X
n=0
n
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h
1
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k
k
X
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X
n=0
i
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95
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4.237. - ,
0
Z1 xt
e
x1 x2!3 + x4!5 + 1n x22nn+1! +
1 + t2
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z
x
dt
:::
:::
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/ . 0
n+1 2n+2
1
2
4
n 2n + 1 t
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1
+
+
1
2
1+t
1 + t2
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4.238. - ,
0
Z1 xt
e
x1 x12 + x23 + 1n xnn+1! +
t
t
t
:::
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x
dt
x
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0
Z1 xt
n 1!
e
2
1
1
+
+
+
+
+
t
x x2 x3
xn
x
:::
dt
x
:::
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4.240. - , ( )
1 et2 dt
Z
t
1
x
p
1
X
n=0
2n!
22n n!
1
x2n+1
1
#
t
0
x
x
dt
1
1 X
n=1
2
+ n 1
xn
x
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Z t
t e
1
x
0
0 # # # .
4.242 . - ,
Zz
z t
z z 2 z3 + 3 z5 4 z7 +
+ ! , ! Re
e
2
2
0
p
dt
2
e
2
1
1
2
13
2
135
2
2
:::
z
0 Re
0. 3 Re = 0, # ! (
.
4.243. 4( & '
Z zt
= 21i ez2 +z!dz2
0
z
z
p
f t
4(
!
f t
C1
.
t
:
96
. IV.
.
. C1 , . 28.
Z1 xt p
e x2 + x!2dx
0
4.237. ! t C1
"
, " " " # !, z2 +1 !2 .
4.244. ,
1
1 4n+1
1 Z p ezt dz sin t
2 X
n 4n! p
p
$
1&
:
2
2i
4
2n!
z z + 1
2 t
n=0
C1
4.245. '(
)*
Z
ezt dz
f $t& = 21i
$z 3=2 0 z 0&:
z 1 + z 3=2
C
1
! "- ) f $t& t.
.
p3 1
i
C1 , . 29, z1 = 2 + 2 , z2 =
p
= 21 i 2 3 . !
*.
x 5.
t "* ( C1
.
" #
1 2#
3 4 4.24664.247, ( 2#, ( 74 jz j 1 ( 4 (.
4.246. z 9 2z 6 + z 2 8z 2 = 0.
5.
4.247.
4.249.
.
z z 3 + 3z 2 z + 8 = 0.
2 5
( (
,
( ( -
97
z 7 5z 4 + z 2 2 = 0.
- C ( (
4.248.
jak z k j ja0 + a1 z + ::: + ak 1 z k 1 + ak+1 z k+1 + ::: + an z n j
a0 + a1 z + ::: + an z n k ' ( C ,
z = 0 ( 7 , ',
( C .
4.250. 8 ' ( z 4 5z + 1 = 0 - : (
jzj 1: ( & 1 jzj 2?
4.251. 8 ' ( z 4 8z + 10 = 0 - : (
jzj 1: ( & 1 jzj 3?
4.252. 8 ' ( jzj 1 (
z n + 0 z 2 + 1 z + 2 = 0
j0 j j1 j + j2 j + 1 n | ?
4.253. 8 ' ( jzj 1 ( z = ' z ,
jzj 6 1 & ' z ( ( ( ( j' z j 1?
4.254. 8 ' ( jzj 1 (
ez 4z n + 1 = 0 n | ?
4.255. 8 ' ( jzj R ( ez = az n
n | , jaj eR =Rn ?
4.256. , ( z = e z 1 ( (' (!' ' ( !' .
4.257 . , , ! ! 0, 1
1
1
) n ( & fn z = 1 + +
+ ::: +
z 2! z
n! z n
- ( jzj .
4.258. , 1,
Pn z = 1 + 2z + 3z 2 + ::: + nz n 1
) n ' ( jzj .
2
. ,
(
) 4.257.
&
'z
G
4.259. = &
(
& .
2
(:
1
1 , -2- (
'
(
= 1 (
( &
( :
2
1 , -2- (
'
(
= 1 (
' &
( :
3
( 1 2 ( (!, (
= 1
(
= ,
1 ( 1
1 (
2.
C
'z
j' z j
'z
j' z j
'z
C
C
'z
jj 6
7 ..
.
G
G
'z
'z
jj
G
G
98
. IV.
.
Pn z = z n + a1 z n 1 + ::: + an
.
, z ,
! Pn z k, k | #
, n, k 6 n.
, $ Pn z n + k=2 .
' . Pn z
Pn z = z n 1 + az1 + ::: + aznn
# z R, Re z 0,
* R.
4.261. +
z 6 + z 5 + 6z 4 + 5z 3 + 8z 2 + 4z + 1
.
4.262. + 0
z 4 + 2z 3 + 3z 2 + z + 2 = 0
.
4.263. 2
2z 4 3z 3 + 3z 2 z + 1 = 0?
4.264. 4 0
z 4 + z 3 + 4z 2 + 2z + 3 = 0?
4.265. , 0
z 2n + z 2n 1 + 2 = 0
| 5 , 6= 0, 6= 06 n |
, !
, n,
n . 7 n , n 1,
0, n + 1, 0.
' . 8 ! arg z 2n + z 2n 1 + 2 ,
z 5 # * .
7 $99# 5
Pn z = z n + a1 z n 1 + ::: + an 1 z + an
0 5 5 , , 0 , 5 Pn z , 5
, , !
0 , 5 5
$99# .
, Pn i = 0 | 5 , . . , 0 5 4.260.
j
j
j j
5. .
99
0 5 . : 0
, , 5 Pn z
0. : , 0 !
, , 4.260.
4 4.266;4.268 , , 5 ! P z , !
, 06 0
$ m.
4.266. P z = z 3 + z 2 + z + .
4.267. P z = z 3 + z 2 + z + 1.
4.268. P z = z 3 + + z 2 + z + .
4.269. f z = Pn z + Qm z e z ,
06 Pn z
Qm z | 5 5, n m f z 6 N | Pn z
. , 0 , 5 9#0 f z
, , 5 w = QPm zz e z N
n
w = 1, 0 z
Pn z , z
$ Pn z 0
5 .
4 4.270;4.272, 0 4.269, $99# a, b . . a, b,
0 5 ! 9# , 0, 0, a b | 5 .
4.270. z + a + be z . 4.271. z 2 + az + be z .
4.272. z 2 + az + be z .
4.273. 2 ! 5 8* , 9#0
w = f z z0
f z = w0 + ck z z0 k + ::: ck 6= 0 k 1
r 0 ! 0,
w 6= w0 jw w0 j 0
k jz z0 j r, 5 .
4.274. 0 5 !
, ,
0 9#0 0 .
4.275. 0 5 !
, 0
9# # 0. ,
$ # 0 5 5 5 w = f z , 0 ! .
7
100
. IV.
.
4.276. ,
4.273 k = 1, . .
f z0 = 0, f z "
# "$ $ " % z0 "# jw w0 j .
& . ' " "# jw w0 j % z =
= f 1w.
4.277. ,
4.273 k 1,
w = f z ("# "% % " z0 k- )$ "* " w0 .
0
6
4.278. ' +" " " ), ) 4.2764.277, $, * z0 +" )
" ) +% f z .
4.279. ,
" # f z " ( $
A
A
1
2
f z = A0 + z + z2 + ::: + Azkk + :::
" " ( $ # ()
" ("# )$ "*,
A1 6= 0, k- )$,
A1 = A2 = ::: = Ak 1 = 0 Ak 6= 0.
/("0 "
4.280. 1 F z = z a wf z , +"
f z z = a. 1 " $ '2 , , +"
jwj 0 "* K " z = a,
" F z
+" $. 1
# ,
f a 6= 0, +" %0 )(" w %( "$ " z = a #
F z .
4.281. 1 z = z w | , +" +"
jwj "
z a wf z = 0, f z
z = a f a 6= 0. , $ 4z ,
$ z = a, +" jwj
" #
z = 4a + 1 wn dn f4a5f a6n g:
1 wf 0 z
n! dan
X
n=1
& . 7
( " C "# "* K ,
" " z a wf z = 0
"
. 4.280,
z = 1 Z
1 wf 0 z 2i a wf d:
C
' # +) *" % % " + + w
)$ .
5.
.
101
X
+ ) ) 1 $ % , * ,
n dn1
0
n
' - z = a + wn da
n1 f af a g.
n=1
+ & , ,
* ' ( z =
= z w ) / *.
. + ' (
z 1 wf 0 z 1
$ % , * .
4.283. # ) $ $ ) w &
* ' ( z w, $ *
w = 2z + z 2 ) *
z 0 = 0, ) * z 0 = 2.
4.284. # ) $ $ ) w ' ( z = z w,
$ *
w = 2 zz2 a , ) * z 0 = a.
4.285. 34 )
$ ) $ - Pn z , $ ,& $ ), * ' ( p
zt t2 . 3.108 4.282.
!
1
1
n
, Pn z = n n dzd n z 2 1n .
1
2
+
1
2
!
. 6 )4 4.284 $ 2 ' - z
' (
z = p
aw w2 = z2 az .
4.286. 7 ( ) z = z w $ w = 0
w = zeaz . # ) $ $ ) w:
1 z w9 2 ebzw.
4.287. # $ $ ) w ' ( & z = z w, $ & w = 0
: $
z a = w sin z a 6= 0 2 ::::
4.288 . ;$ 4 $ $ % , * ) z w , a = =2.
4.289. < &, ,
% - . +
f z 'z | ' ( , a, C |
( a r ), 4 4
jf z + 'z j r:
| ) ' ( ) ) ) z a f z 'z = 0,
m n dm+n1
0
m
n
= a + m n da
m+n1 f af a 'a g
$ m n, m = n = 0.
1
1
2
X
!
1
+
!
2
+1
V
.
,
x
X
1. ! "#$ %& '()%
5.1 5.10 .
1
1 zn 2
1 h + in
X
X
5.1.
z n + 2n1zn : 5.2.
+
:
5.3.
:
n
!
n=0
5.4.
5.8.
1
X
n=1
1
X
n=1
z z
n
n=0
ez ln n : 5.5.
z
1
n
n:
z
5.9.
1 sin nz
X
1
X
n=1
n=0
n
n
: 5.6.
:
1+ n
z
2
z
n
z
1
X
1
X
n=0
5.10.
z
n=1
1n
n=1
n
1
X
n
n=0
z
2n
+1
:
4 + z 4 + z 2 :::4 + z n
:
+n
: 5.7.
z
2n
n
1
1
n
X
X
5.11 . , ! an , 1 n n
1
n=1
n=1
X
"#, $! jz j 6= 1& ! '!
an ,
n=1
1 nn
1
X
X
#$!
an z n n
1
a
z
z
a
n=1
z
z
n=1
! )$ #$.
5.12. 1* +' , !,! z ##
# ,#!$ .
1
X
2* , , jz j 1
'.n* 1
z
n
n=1
z
n
=
1 an zn
X
n&
n=1
1
z
1
z 2
z
, $!
'.n* | ! ! # !, ! 0 n, ! , n.
1 X
!. , ! ! ! 0!!
'.n* = m, $! n ,!$! ! ! !!! m, " 1 m.
5.13. +' 2# 3" .z * =
1
X
1
1
X
ez ln n . - n=1
n=1
* 4! ! z = 2 n
z =
1.
103
# !$ .
5.14 5.17 # .jz j 6= 1*.
5.14.
5.15.
1
X
1
1
1 + zn
1 + zn
n=1
1
X
n=1
z
n
1
z n 1
!! .1 z *.
5.16.
1
X
n=1
n
Y
z
2n
2k
z
n+1
1
.
. 1 !. 1' !
: 5.17.
1 + z
1
X
n=1
z
z
2n
1
2n
1
k=0
5.18. ,!'!:
:
1
X
1* !
fn .z * '!! E
n=1
! , "$ " 0 #8!
! N = N ."*, ! n N , ! z 2 E "+p
nX
$ # $ p , !!
fk .z * "&
1
X
k=n+1
2* !
jfn .z *j '!! E
n=1
!#! ! ) '! '!!
1
X
fn .z *.
n=1
5.19. 9 '!, ! !
,!! :
n
o
n n o
n
o
1* 1 +1 n & 2* 1 + 2n & 3* sin .
5.20. : $ ,!! !,!
2# 3 ffn.z *g ! $! #
'!! E , ! , ) ,!! ! )$ '! !,! ! ,!! '! E , . !.
,!! ! zn , ,!'8 '!# E 8 ! z0 ,
lim fn.zn * = f .z0 *:
n!1
z
z
nz
z
n
5.21 5.25 '!, ! !
.
1
1
X
X
1
n+ 1 :
5.21.
e nz :
z
5.22.
2
n
n=1
n
z
n=0
. V.
104
5.23.
X1
n=1
e
z ln n :
5.24.
X1
X1
.
sin nz
n2
n=1
zn
n2
n=1
5.26. ,
5.25.
:
X1
sin nz
n
n=1
:
-
6 1.
1
, 1 ?
n 1
1
5.27. ,
!z+n
n=1
" , = 0, 1 2
&, '
"!.
1 n
z
5.28. ,
n
X
jzj
( 1 0),
X1
n=1
jzj
z
X
zn
n
. * " ,
:::
n=1
' & , 1
zn
n
n=1
X
( 1 0)
& + .
, . - + , . /
" .
1
X
z
5.29. 1) 0
"! j j 0, jarg j 6
1 + z 2 n
z
z
n=0
6 4 (' ! " "! , , , = 0 / j1 + 2j = 1).
=
z
z
z
,
" .
, . 3 , & "! " 1 .
n
X
1 z
2) , ' & "
1 + z 2 n
n=0
"!, "! (. .
).
5.30. ,
1
X
j n ( )j
f
n=1
z
", 1 /! " ,
X
j n0 ( )j.
' & "
G
f
n=1
z
2.
1 )
x 2.
1
105
X
0
n z ,
n
n | '
n=1
n | & , !+
nlim
1
2
!1 n = 1
& .
5.31. ,
0 = 0 +
+ 0, Re Re 0 , & jarg (
2.
0 )j 6
5 . , " 6"
a e
a
:::
z
iy
x
z
z
q
X
n
a e
n=p
=
n z
q
X
n
a e
n=p
(
j
e
az
e
bz
j=
Zb
z
e
a
z
=
n z0 e n z z0
= + )
6 jzj ( ax
a b z
zt dt
z
x
iy
e
x
e
bx ):
5.32. ,
"!
= 0 , "! Re
Re 0 .
7 , 5.31 5.32, ,
"! ( + ) Re
1), "! "!
c ( c
( + ) | Re
a
( a
1),
" "!
Re = a , " "! ' . : c a !
.
. 5.33;5.39 " ( c ) "
"! ( a ) .
1
1
n
X
X
1
n3 zn2
z ln ln n
5.33.
5.34.
z
z
z
z
z
x
x
z
x
x
z
x
x
x
x
x
e
n=0
5.35.
5.37.
1
1
X
1
n=1
p n
n
1
n=2
X
e
(
:
n=2
e
z ln ln n :
1)n e z ln n :
5.36.
5.38.
e
n
1
X
1 n=2
X
1
n2
n=1
:
( 1)n
e
e
z ln n :
.,
z ln ln n :
5.39.
1
X
en e zn2 :
e
n=0
, 2, . IV, x 2 .
. V.
106
ln n
lim
!1 = 0,
5.40. ,
n
5.41. ,
.
n
ln n
lim
n!1 n
ln jan j
c = xa = nlim
!1 n
x
c 6 l.
= , a
l
x
x
5.425.46
! "! .
5.42.
X1 # 1%n z
n=
X1 zn
e
ln n
5.43.
:
X1
1
5.44.
1
n=1
n
e
:
5.45.
X1
n=1
1n
n2
n=1
1
n3
e
z ln n
e
zn
:
5.46.
:
:
-
X1
p
1 n
n
n=1
zn
e
:
& . '. " 3.64.
5.47, 5.48 + ,
! , ! .
5.47. -"
n " + "
lim ln n = 0
lim ln jan j =
1
n!1 n
n!1 jn j
,
6 arg n 6 , . + / = + , / ! 1 2
cos sin
0,
, / ! 1 2 cos sin
0
1
5.48. ! ,/
n n z . -"
n=1
ln jank j
# % = klim
#
%
= lim
!0 # %
!1 jnk j
f k g | ! , , 6
6 arg nk 6 + #
"4 " / !! f mg, mlim
arg
" ! 5 # % = 1%.
n
m = ,
!1
ln n
,
lim
= 0, . + "n!1 n
. , = + / ! +. " +
" + cos sin # % 0, / ,
54 / .
k
z
x
x
iy
'
'
k
:
'
y
'
X
:
k '
k '
x
'
'
'
'
k '
n
G
x
y
a e
n
k
z
'
y
'
x
'
k '
iy
'
k '
G
x 3.
,
'
5.49. ": !" | ! / " # ",/
",/%, +4 / "+ ", # % | 7" ,
! ! / ! , ! C
f
z
z
z
3. ,
107
/ .
, ! 54
Z "" 8
7" , ! # % = # % ,
C
7" ! ! Z /
0 # % = z0 # %
D
C
F z
f
z
d
z
F
9
Z
f
C
#
z
%
z
f
z
d :
C
d
. ,/, . .
!, -
7" ,, ! - . , 2
" 5 . " "+ ", !
/
: "+4 ! , , " . Z
5.50. : / # %
C
f
z d
C
5 ! : + /- . 0 6= 1 "
. , ., +.
0 "4 # % ,
Z
# %
E
C
"
"
f
z
d
"
C
5
/ " " ,
54 / - 0 54 / ;" "
" . , .
5.51. '7" ,/ / / ,
" , 0 = 1. <
" : " "8 . ,.
5.52. ,
j # %j 6 j # %j Z Z
5
j # %j , # % z
E
C
C
C
f
E
z
z
z
d
f
C
z
d
C
5 .
5.53. -" # % | 7" ,
!
! , ! , ! 54 / . , , ! 54 "" , + , , , " , 5 , 7" + # %,
":+
,
,.
,
. ,/
Z
# %= # %
f
E
z
z
z
D
C
f
z
z
F z
f
C
z
d
" .
#. . / "/ !. . %, 7"
#
%
0 # % = Z @f
@z
D
D
F z
F
z
d
C
108
. V.
.
.
D
5.545.61 , .
Z1
Z1
zt2
5.54. !" # = z 1 t
" z 1 = z 1 ln t # 5.55.
z
5.56.
Z1
sin t
0
5.59.
5.61.
t
tz
dt
ezt
t
1
zt
t
dt
dt
" = 0#
c
6
" =0
c
6
e
cos t
tz
0
1
+1
c i
cZ i
t
Z1
5.57.
dt:
+1
cZ i
c i
e
0
+ 1 zt
cZ i
c
z
&
5.58.
dt:
5.60.
:
:
t
=
e
t ln z
Z1
0
e
tz
t
dt:
Z1
sin tz
dt:
t
0
" = 0#
dt
c
:
6
#
:
'
&
e
e
0
"#
"1#
f t dt
( ) " # *0 + ,- .
1, .
5.62. / . ,0 :
1# "1# = 0 ,
Re
Re 0 , .
jarg " 0 #j 6
23
2# "1# - , = 0 ,
- , Re Re 03
ln f t
3# tlim
!1 t = , "1# - , Re
Re
+ " 0# " . ' , 0
- , , = 1#3
4# lim ln ftt = , "1# - ,
t!1
Re
.
& , ( 5.62, , - - , ' " - 0 , # , Re
c
Re
, a |
a 3 c ,
' .
f t
a
a
z
z
z
z
z
z
=
z
z
j
z
z
j
z
"
z
"
j
j
z
z x
z x
x
x
3. ,
109
Z1
zt " # ,
5.635.69 c a
0
" # | ( ).
2
2
5.63. " # = 1 5.64. " # = t
5.65. " # = t
5.66. " # =
8 t2
0 6 ln ln 3 ln ln 2 6 ln ln "2 + 1#
" =2 3 #
=:
t2 ln ln "2 + 1# 6 ln ln "2 + 2# " = 1 2 #
5.67. " # =
8 e =2
0 6 ln ln 3 ln ln 2 6 ln ln "2 + 1#
" =2 3 #
=:
e =2 ln ln "2 + 1# 6 ln ln "2 + 2# " = 1 2 #
5.68. " # =
8 e
0 6 ln ln 3 ln ln 2 6 ln ln "2 + 1#
" =2 3 #
=:
e ln ln"2 + 1# 6 ln ln "2 + 2# " = 1 2 #
t
ln "2 1# 6 ln 2 " = 1 2 #
5.69. " # = t
ln 2 6 ln "2 + 1# " = 1 2 #
x
x
e
f t
dt
f t
f t
:
f t
e
:
f t
e
:
f t
e
t
e
k
k
t
t
k
k
k
:::
k
::: :
f t
e
t
t
t
e
k
k
t
t
k
k
k
:::
k
::: :
f t
e
t
e
t
t
k
e
f t
Z1
0
e
t
k
e
k
t
k
t
t
k
k
k
k
:::
k
::: :
k
:::
k
::: :
5.705.73 . . '
zt " #
) .
f t dt
5.70. f "t# = 1:
t
k
1.
5.71. f "t# = 0
0 6 6 1, " # = 1
t
f t
=t
2
5.72. f "t# = 0 0 6 t 6 1, f "t# = 1=t t 1.
5.73. f "t# = 0 t = 0, f "t# = 1 0 t 6 1,
" # ,0 -:
" # + 1 "2 1#2 + 1 6 "2 #2
"# =
" # 1 "2 #2 + 1 6 "2 + 1#2
t
1
f t
f t
k
f t
k
t
k
f t
t
k
" =12 #
k
::: :
VI
.
x
1. !"# $
6.1 6.7 .
1 1 1
1
Y
Y
1
6.1.
1
=
.
6.2.
1+
= 2.
2
n
2
nn + 2
n=2
n=1
1 n2 4 1
1
1
Y
Y
2
6.3.
=
.
6.4.
1
= 3.
2
n 1 4
nn + 1
n=3
n=2
1 n3 1 2
1 h 1n+1 i
Y
Y
6.5.
=
.
6.6.
1+
= 1.
3
n +1
n=2
6.7.
.
1
Y
n=1
3
e1=n
1 + 1=n
= eC , eC = nlim
!1
6.8. ,
1
Y
n=1
n=1
n
X
1
k=1
k
cos 2n = sin .
.
k
Y
n
n=1
ln n |
sin ' = 2k sin 2'k
cos 2'n :
6.9.
! "# 6.8, ,
p2 p 2 p q p2 p ::: = 2 :
2
2+
2
2+
6.10 . & #!! '
2+
2
2
=
1
Y
n=1
2n
2n
2n 1 2n + 1
.
, () , arg pn 6 ,
1
Y
( pn * * 6.11. ,
#
#
1
X
n=1
ln pn .
n=1
1.
111
6.12. !', '! $' )' ' 1# ', $' $'!, :
13 0 6 arg pn 25 23 arg pn 6 + 2 62 03.
6.13. %,1 $! /$7# ',' /
Y
''!
61 + an 3 . . /$7# ',' !
1
1
n=1
X
X
ln 61 + an 3 / ', ' , / !
an '$!
n=1
n=1
/$7.
1
1
Y
Y
6.14. *''! pn ' qn !!. 8$ 'n=1
, ''#: n=1
1
1
1
1 pn
Y
Y
Y
Y
13 6pn + qn 35 23 6pn qn 35 33 pn qn 5 43
.
q
n=1
n=1
n=1
n=1
n
6.15 6.19 '$ ', ' /$77
', ''!.
1 h 1n+1 i
1
Y
Y
n
1 + n . 6.16.
6.15.
n1 .
1h
Y
n=1
6.17.
n=1
i
1n+1
np
1+
n=1
6p 03. 6.18.
1
Y
1
Y
1 + n1p 6p 03.
n=1
1
X
6.19. cos zn , $' ', !
jzn j2 '!.
n=1
n=1
6.20. %, ' ''
1
Y
n
61 + z 2 3 = 1 1 z
n=0
'' '! /$7.
6.21 6.29 #' /$' ',' ''#.
1
1 zn
1 z2
Y
Y
Y
6.21. 61 z n 3. 6.22.
1 + 2n . 6.23.
1 n2 .
1h
Y
n=1
6.24.
6.26.
6.29.
1
Y
n=2
n=1
1
Y
1 1
cos
z
.
n
1
n
n
6.27.
n=1
i
z n .
1
Y
n=1
6.25.
sinz=n
.
z=n
1h
Y
n=1
n=1
1 + 1 + n1
6.28.
61 + cn z 3, $' ', !
n=1
1
X
n=1
1
Y
n=1
n2 i
zn .
1 + nz ez=n .
jcn j '!.
112
. VI.
.
6.30. ,
1h
Y
n= 1
1+
1
n+1 i
nz
nz = z ln n
Re 1=2 Re z 1.
6.31. ffn z g | !" #,
G,
$ % !" , $ &
, ' G. , jfn z j 6
1
X
6 n z 2 G, 1$ n z , n Y
n=1
, !" F z = (1 + fn z ) # n=1
G.
* 6.32-6.35 / / # &$$!" , / ' &
$., $ , (2, &. VII, x 4) (3, &. VII, x 1).
6.32. ,
1z + 1 =
1 n n + 1 z n + 1 z
Y
n+1
= ez ln n
n=1
z
+n
n
n
# , $ # z , / "/$ "/$ $,
!" , # , $ z = 1 2 :::
6.33. ! $ 2#
1z = nlim
!1
n! n
z z
+ 1
z
z
+ 2:::
z
+ n
nz = ez ln n
, :
1 1z + 1 = z 1z 4
2 1m + 1 = m!, m | .
6.34. ,
1 n + + n + 1 + 1
Y
= + + 1
= 1 2 ::::
+ n + n
n=1
6
6.35.
1
1
Cz Y 1 + z e z=n
=
e
z + 1
n
n=1
C | .
" # . $ # # 6.7.
2.
113
6.36 . 9
/
p1 p2 ::: pn ::: |
1 X
p1 = 2 p2 = 3, p3 = 5 :::, s =
ns ns =
= es ln n | -!" : $,
Re s 1 $. 5.23. , :
1
.Y
1 z = 1
1 pn s 4
n=1
-
n=1
2 !" s $ # Re s 1.
9 $ . ; -!" ' 6 . <$., $ , $& ! : = $ 6 >. ?.
= -!" : $. | @.: ;A, 1953.
6.37 .
/
x 2.
1
X
,
, . n=1
1
n
p
, & fpng |
. !""
6.38. 9 f z | $ $ ! !" /$ $ a1 a2 ::: an :::,
$ 0 ja1 j 6 ja2 j 6 :::, nlim
!1 an = 1.
D $ An / !" f z an
n = 1 2 :::. $, ' $/ Cm , ' ' $ $:
1 Cm an 4
2 E/# Cm E Cm+1 4
3 $ $ Cm
$ & Rm &
m ! 14
4 6 / Lm Cm Rm & /$, . . Lm = ORm 4
5 zmax
2Cm jf z j = oRm % , , / ,
!" f z & Cm .
,
% $ $ E !" f z #6
f z = f 0 +
1
X
n=1
An
1
z
n
a
+
1
n
a
$ $ $ # # $# , E'# an , $ $$/ &
&$/, ' $, /$ $E Cm
Cm+1 m = 1 2 :::.
8 ..
.
114
. VI.
.
.
Z
1
2 i
Cm
f d
z
m ! 1.
. ., , 1, . VIII, . 4#.
X
X
$ 6.39(6.46 ).
1 2z
1 1n 2z
1
1
1
6.39. ctg z = +
.
6.40.
=
+
.
z
z 2 n2 2
sin z
z
z 2 n2 2
6.41.
6.42.
X1
tg z =
n=1
1
cos z
6.45. z z
e
6.46.
h
1
2z
2n
n=1 z 2
X
n=1 z 2 +
=
h
1
X
n=
z2
1n 2n
= 1 z2 +
1
1
sin2 z
i2
1
2
1
X
=
6.43. th z =
n=1
1
z
h
n=1
.
1
2n
1
i2
.
2
2z
2n
1
1
i2
. 6.44.
1
sh z
= 1z +
2
X
n=1
2z 2
z + 4n2
2
1
n 2
2
1
X
n=1
1n 2z
.
z 2 + n2 2
.
.
6.47. f ,z - | /0 /0 0 a1 a2 ::: an :::, 0 ja1 j 6 ja2 j 6 :::, nlim
!1 an = 1. 2 , fCn g, 03 0
0 1-(4- 6.38,
0
f z
max
= o,Rm -:
z 2C f z
m
2 ,
f ,z - = f ,0-ezf 0=f 0
0
1
n=1
)
1 az ez=an :
Y
$ 6.48(6.54
6.48. sin z = z
1
1 nz2
Y
n=1
2
2
.
n
).
2.
1
Y n
6.49. cos z =
n=1
6.50. sh z = z
6.51. ch z =
6.52. ez
1
1
Y n
n=0
2
1+
h
1
2z
2n + 1
Y
2
n=1
i2 o
.
1 + 4nz2
2
+bz=2
cos z = z
2
1
Y
n=1
.
2
ebz = ,a b-ze a
6.54. ch z
115
1 + nz2 z2 .
n=1
1
i2 o
2z
.
2n + 1
Y
1 = zez=
6.53. eaz
h
1
Y h
n=1
1 + 4nz4
2 2i
1 + a4n2b2z .
4
4
.
6.55. f ,z - | 0 /0, 3 0
3 a1 a2 ::: am , 3
z = 0 1 2 ::: 2 ,
fCn g, 03 0 , Z
lim
f ,z - ctg z dz = 0
,1n!1
1
X
m Cn
X
f ,n- = res f ,z - ctg z #z=ak .
n= 1
k=1
6.56. 2 , 0
Z
,1- nlim
!1
1
X
n= 1
Cn
f z dz
sin z
,1-n f ,n- =
m
X
k=1
res
-
= 0,
h
f z i
:
sin z z=ak
$ 6.57(6.64 0, 0 a , 0 .
6.57.
6.60.
6.63.
8
1
X
n= 1
1
X
n=0
1
X
1
.
a + n 2
1
n2 + a2
1
2k
n=1 n
6.58.
. 6.61
1
X
n=0
1
X
n= 1
1n
n2 + a2
1n
.
a + n2
. 6.62.
,k | -.
6.59.
1
X
n=0
1
X
n=0
1n
1
.
2n + 12
.
2n + 13
. VI.
116
.
, z = 0 | f z ,
6.55
# # $,
# # $ # #
n 1 +1,
nh = 0. i&
# res ctgz2k z
z =0
' # # ( 3.102.
.
6.64.
# -
Z
Cn
zeibz dz
a z 2 sin z
2
6.65. . ,
1
X
n=1 n
n
2
p
3
n
4 2
3
=
p3=2
Z
0
1
X
1 n na2sin bn
n2
b .
. , Cn | 6.56 .
x ctg x
p
3 x2 3 4x2
n=1
p
dx + 16 ctg /2 3 0:
. , # -
Z
Cn
z 2
z ctg z dz
p
3 4z2 3 ,
- Cn | , - #1$ . 30 ##
' .
x 3.
1
& f z | M r = jmax
jf z j. 2 =
zj=r
ln ln M r
= rlim
!1 ln r # $ . 3 0
ln M r
1, = rlim
!1 r # . 3 = 0, f z # $ 4
= 1, f z # $
4 0 1, f z # $
.
1
.: . .
! .| #.: $ ! , 1956)
* +2, . VII, x 1-.
3.
117
6.66. . 1 # ( .
1 3 6= 1 6= 1 | ## f z , ' - " 0 ( R" ,
r R # # ##
M r er +" M r e+"r :
5 ( ( # frn g frn0 g,
6 1 ' , 6
"
0
M rn ern
M rn0 e"rn :
2 3 - - k
M r
0 rlim
!1 rk 1
0
M r
f z | k , 1# rlim
!1 rk .
. , # ## 7 8 91
X
# - f z = cn z n .
n=0
3 3 f z | ,
lim ln M r = 1:
r!1 ln r
, 6 6.67;6.79 6 $ n | .
n
6.67. c0 z n + c1 z n1 + ::: + cn . 6.68. eaz a 0 . 6.69. z n e3z .
3
2
6.70. z 2 e2z e3z . 6.71. e5z 3e2z . p6.72. e2iz . 6.73. sin z .
6.74. ch z . 6.75. ez cos z . 6.76. cos z .
1 zn
X
6.77 .
m | .
2m n!
n=0
6.78. ee .
z
6.79.
Z1
0
ezt2 dt.
6.80. = f z 0 6 6 1 .
. , P z f z + Qz , - P z Qz | '
- , ( .
6.81. = f1 z f2 z , ## # 1 2 , 1 6= 2 . 2 (
$ f1 z f2 z f1 z + f2z ?
6.82. = f1 z f2 z - - (
, ## # 1 2 , 1 6= 2 .
2 ( $:
1 f1 z f2 z 4 2 f1 z + f2 z ?
118
. VI.
.
6.83.
f1 z f2 z . :
1 f1 z f2 z 2 f1 z + f2 z ?
1
2
Y
6.84 . f z =
1 z , 0 n = 1 2 ::: , lim n =
2
n
n=1
1.
n
n!1 n
n =
0, nlim
' , ( f z | *+
!1 n
, ( , *, ( 6 6 .
1
2k
Y
6.85. f z =
1 z , 0 n = 1 2 ::: , 2k | 2
n
n=1
n
( . ' * :
n = 0, f z | , -
1 nlim
!1 n
., ( k
+
1
X
n =1
1
2 nlim
1, f z | , 2
!1 n
n=1 n
- , ( k
+ .
/ . 0 * 1 - ( .
6.86 . ' , (
* .
21 ( 6.8766.94, ** - :
* * f z =
1
X
= cn z n, ,
n=0
**
n ln n e 1= = lim n1= p
n jc j :
= nlim
n
!1 ln 1=cn
n!1
6.87. ' , (
1
X
Azn
f z = n
A 0 0
+ 1
j
j
n=0
= 1= = A1= .
/ . 0 * 9 +
n p
h
1 i
: n + 1 = n
2
n
1
+
O
e
n :
0 (; 6.8866.94
1 ; .
1 n
X
X ln n n=a n
z
6.88. f z =
.
6.89. f z =
z a0 .
n
n
n=1
n=1
3.
6.90. f z =
6.92. f
6.94.
1
X
n=a
1
n
n ln n
1 zn
X
z =
n=2
nn1+a
n=1
z J z =
1
X
n=0
z
a 0.
a 0.
6.91. f z =
6.93. f z =
1n z 2n
n! n + + 1
1
X
1 ch pn
X
n=0
n=0
119
en2 z n .
n
n! z .
1) J z |
9## -o .
:# | f z 0 1,
ln jf rei' j
h ' = rlim
!1 r ( # # f z .
+ 8 6.9526.101 # h ' # # -& .
6.95. ez . 6.96. ez + z 2 . 6.97. sin zp. 6.98. cos z .
n
sin z
6.99. ch z . 6.100. ez . 6.101. p .
z
6.102. < f z # h ', P z |
. = # h ' :
1 f z + P z ) 2 f z P z ?
6.103. P z | ( . ?#
f z - , #
h ' -, # f z + P z $ .
VII
.
x
1. !"
1
2 i
Z
' d
z
C
C | 1 ' |
, C , !, "# $,
! ! !, ,
. ' ' 1
,
| .
() z
F z , !
$ " , $ * !
C . + ,
n z = n!
Z
' d
:
z n+1
1
C
+ # ' C )
0 6 1 , ' 2 Lip , . .
j'1 '2 j k j1 2 j
! 1 2 $ C , a k | . 0 ,
! 0 , * F
2 i
=
F 0
1
2 i
Z
' d
0
C
!
().
1
! $ # ! "! "
Z
' '0
'0
1
1
b 0
F 0 =
d +
2
'0 +
Ln
2 i
0
2
2 i
a 0
C
1
.
" . #
( ) .| 2- .|
!
,
. .
, , !
,
.: %. %. &
'
,.: &--., 1950.| &. III. ,! !
.
C
1.
121
! a b | C , . 4 !
# Ln " , ! ! a = b
! ! ,
=
F 0
1
2 i
Z
'
'0
1
d +
'0 :
0
2
2
0
C
5 " !# ! F 0 F 0 # !
() F z z ! 0
C
C , 67
1
1
+
F 0 = F 0 +
'0
F 0 = F 0
'0
3
2
2
+
= 1 9F + 0 + F 0 :
F 0
2
F
+
0 F 0 = '0 :
5 C "7
F + | # ! F + z ,
" # D+ , F | F
" # D 2 . 6., ,
x 3: 93, . III, x 3:.
7.1. < #, !
4
"!,
z , 92, . III,
|Z # '
() =z =
d $ "#
z
C
' = '+ + ' , '+ ' |
! ! , ! 7 C ,
=+ z = '+ z = z = ' z + ' 1:
C
1
2 i
+ ! . 5 ! '
'+ $
,
() " *
() )
" .
7.2. + # C | . > F +z F z,
#
() | n | # ! :
1
1 ' = an @ 2 ' =
a C @
an
3 ' =
7.3. >
1
an
a C .
z F z , :
1 ' |
! ! , !
F
+
2 34
!
.
,
|
,
122
. VII.
.
D+ , ak ,
2 ' | ,
D , ak ,
ak ! z = 1.
7.4. $ F + z F z ,
ln 23
3
+
i 2
+
4
i
' = 4 3 2 4 + 2 4
C | ! j j = 3=2.
7.5. $ F + z
F z , ' = ctg
C | ! j j = 5.
2
7.6. $ F + z F z , ' = 2
+ 1 C | ! !, .
) . )
*+ , Z1
'
1
! F z = 2i z d !
,
7.7. $
' = a20 +
1
, .
F + z F z , ! F
1 C , C | ! j j = R,
an cos n + bn sin n | ,-
X
n=1
. ! ! = 'Re i .
7.8. 1 )! C | ! j j = =2 f | , j j 6 =2. $ ,
Z
Z
I1 z = 21i f ctg z d I2 z = 21i sinf d z
C
C
,, z , ,
z + k k | C .
2 0 + ! , . 1, ,
C | ! j j = .
7.9. )! C | 11 12, ,
' 1. $ F z C , ! F
F C . 3 !, , F i, F 0 F 0.
7.10. )! C | ! j j = R, 0 arg R ' 1. $ F z C , ! F F C . 3 !,
, F 0, F iR F iR. $ F 0 0.
1.
123
7.11. )! C | ! j j = R, arg 0 R ' 1. $ F z C , ! F C , F 0 F 0 0.
7.12. )! ! *+ ' = 1= n . $ F z C , C | :
1 ! r jz j R
2 Im = ,
3 jIm z j
4 ! j j = R, 0 arg R
5 ! j j = R, arg 0 R.
3 . 4 5 ! F C ! F 0.
3 , 7.1387.18 F z C , , C | , - a b, ' | .
7.13. ' 1. 7.14. ' = .
7.15. 1 ' = n
7.16. ' =
1
z0
2 ' =
z0 2= C .
X1 c
n=0
n
n
| .
1
z0 n z0 2= C . 3 !,
: . 3 ! !
'
= ' 'z + 'z :
7.17. ' =
z
7.18. $ F + z , F z
z
z
, F z0 .
! F ,
C | ! j j = R, ' | , :
1 ' = ln = ln R + i' ' 6
2 ' = Ln = ln R + i' 0 6 ' 2.
: . 0 ! , -
jzj = R 1R 02 10 R2 |
.
) . ; !
,
! ! , ! !
.
7.19. $ F + z , F z ! F C , ' = ln 1 , C :
1 ! j j = R R 1
. VII.
124
.
2 Im = 1, .
7.20. F z F C , ' =
= ln 1 , C | ! = R R 1, 0 6 arg 6
R.
p 0 6 arg p
7.21. f + z F z , ' =
C | ! j j = 1.
# $ 7.22&7.24 F z , C | ,
a b ! , ' | ) *, , +, a
b !, C .
a
7.22. ' = Ln
Ln 1 = 0.
b
j j
r
'1 = +1.
. . / , !+, , ! ' z 1 .
7.23. ' =
7.24. ' =
a
b
b
'
a b1
=1 .
=1
7.25. F z , C , a ! D+ , b | D ' = Ln ab
Ln 1 = 0 | , , +, a b +, C 0 .
. . 0 C 0 a, z ! D+ , 0 b, z ! D .
7.26. / 0 1:
1
2
Z1
0
Z1
0
t
t
1
1
t
t
t
h
z
= sin 1
= sin 1
dt
h
dt
t
i
z
z
1
1
z
2= 20 134
cos
i
2 0 1.
. Z. /
d )
1
5 6 2i
,
| ) *, 1
z
1
C
20 13 1 1,
C | *
: j j R R 1
20 13. # ) ,
) 7 $ * .
1.
125
7.27. #
Z1 r
2
1
1+t t
1
7.28. :
Z1
0
Z1
z 2= 20 134 2 ln 1 t
t dt
t z
1 ln 1 t
1 x 1:
tt +3
dt
x
1
0
t
dt
t
7.29. 8
F z t0
= 21i
Z
C
d
t0
z
= +i
2 0 1.
06
1
C | , +, a b, t0 | 9
t0 | , +,
t0 1. : t0 * C ,
, $ C 4 !
t0 , t0 b C .
/ +, ! .
1 # a
ei
F z a =
z a + F1 z z 2= C
2i sin
= ctg2i t a + F1 t t 2 C
F1 z | ) *, a.
. . 8
Z
1
d
ei
F1 z =
z a
2i
a
z
2i sin
F t a
C
, ) 7 $ * , , F1 z ) * a.
2 # b
ei
F z b =
z b + F2 z z 2= C
2i sin
= ctg2i t b + F2 t t 2 C
F2 z | ) *, b.
3 # t0 C
F z t0 = z t0 + F3 z C
F z t0 = F4 z C
1
F t t0 = t t0 + F5 t
t 2 C
2
F3 z , F4 z F5 z t0 .
F t b
. VII.
126
7.30.
.
1
2 i
Z
C
ln 1 d z
z = R z = R, C | j j = R
R 1 ,
! " R .
$ . %&. 7.20.
*& . +
" &.: M y ! .. /. %
.| 0.: 1&, 1962.| 4. I5 4 ! 1. 6.
.| 0.: 1&, 1963.| 4. I.
x 2.
,
,
/
D u
D u v
=
ZZ
G
u2x + u2y dx dy
,
=
ZZ
G
uxvx + uy vy dx dy
| 9 " & .
7.31. 6 9 " 6! 9 " & " " : &:
ZZ
1
1 D u =
u2r + u2' dx dy z = x + iy = rei' 5
r
G
2 D u D u v : &! " G5
3 D2 u v 6 D u D v & & u=v = const 5
4 :< f z = u + iv
G,
ZZ
2
D u = D v = D f =
jf z j dx dy:
0
G
= & & D f ?
7.32. *
: & " 4
ZZ
Z
@u
v ?u dx dy + D u v = v
ds
@n
G
C
n | &5 ? | @ , 9
" & ! :<" u, u1 , u2 v | :<, u :
2.
1
=
D u
Z
u
C
,
@u
ds =
@n
Z
u dv
C
Z
& 2
C
@u
ds =
@n
Z
dv
C
127
= 0&
3 1 = 2 , 1 2 &
1
2
4 @u
= @u
, 1 2 = const .
@n
@n
7.33. , * * j j
j j 6 ,
u
u
C
u
C
u
u
G
u
G
u
z
z
u
0 = 21
7 .
Z
u d'
jzj=r
Z2
i
u Re
R
4-
R
= R1 2
d
0
Z
C
ZZ
i'
u re
jzjR
@v
ds
@n
r dr d':
= 0 , *
& . * 43 4 r
4 4 ! 0, 3 4 4
4 ! .
7.34.
r
r
R
ln
ZZ
C
j
= + ,
i
1
n
zj
Z
d
d
ln j 1 zj j j
d
C
|
Z
@
@n
C
ln j 1 zj j j
d
C
.
0 :
8
1
ln
j j
= : j1zj
= i &
ln
j
j
6
0
R8
1
ln
ZZ
j j
1
1
j
z
2 R2
ln j zj
= : 1 j 1 h jzj 2 i
ln R + 2 1 R
j j 6 &
j jR
Z
Z
@
1
3
arg "
ln
j j=
@n j z j
1
1
2
Z2
ln j 1 zj
z
R
d
Re
z
R
d
z
R
d
d
d
C
z
z
R
n
C
4 3 " &
Za
@
Im 0
4 @
ln j 1 zj =
|
Im 0 0
a
, 4
*
0.
"
*
,
, = + , = +
*
C
d
'
z
'
z
a a
G
g z
z
x
iy
'
z
'
a
g x y
i
128
. VII.
.
x, y , ,
G, z = ,
g z = ln
1
j + :
, . .
g z g z ., , "2, . VI, x 1% .
7.35. & ' ( ) ( * , , g z .
7.36. - w = f z . (' .( G jwj 1 , f = 0 2 G .
) ' 0
g z = ln jf z j
1
g +ih
f z = e
2
( h z | , . g z .
7.37. -' 0 1 ' ( 7.36, *
g z ( ( *:
1 ( jz j R6 2 ( Im z 06
3 ( 0 Im z 1.
7.38. ) ' ( .( n | , r | . jz j = r :
1 u z ' z = ,
Z
lim
u r @g@nz ds = 2 u z 6
r !0
r
z
j
2 u z (
' z = ,
lim g z @u@nz ds = 06
r !0
r
Z
3 u z G (
C ,
Z
u = 21 u z @g@nz ds z 2 G :
C
7 ' . 8
Z
@g g @u ds = Z u @g g @u ds
u @n
@n
@n
@n
r
C
0 ( * ( r ! 0.
3. , ,
x 3.
,
129
,
9 ( * u = u R ( . = Rei 0 6 2 ,
Z2
u z = u r ' = 21 u R R2 2RrRcos r ' + r2 d
1
0
( jz j R z = rei' , u ' , u :
lim u z = u
r !
2
2
z ! . &
f z = u + iv, jz j R, ( :
Z2
1
f z = 2 u + zz d + iv 0
0
v 0 | ' ( * .
7.39. ) ' ( .( :
Z2
2
2
1 21 R2 2RrRcos r ' + r2 dr = 16
0
2 u r ' u R 0 =
Z2
2
2
= 21 "u R u R 0 % R2 2RrRcos r ' + r2 d 6
0
3 ju R u R 0 j " ( j 0 j ,
Z
1
R2 r 2
j
u
R
u
R
0 j
2
R2 2Rr cos ' + r2 d "6
j j
4 j j j' j =2,
R 2Rr cos ' + r 4Rr sin 4 6
5 j' j =2 . 3 ,
ju r ' u R j " + M R2A r
0
0
0
2
2
2
0
2
2
0
Z2
( A = 4 Rr sin2 4 , M = ju R u R 0 j d .
0
9 ..
.
. VII.
130
7.40. ,
.
= Rei ,
z
= rei' ,
+ z = R r + i2Rr sin '
z
R2 2Rr cos ' + r2
, :
1 r n
= 0 +
n cos + n sin
R
2
2
:
u z
X
u
= 0 +
v z
1
X
v
n=1
= 0 +
f z
!
n
a
= 1
1
X
f
Z2
u R
0
c
n
a
n=1
r
R
n
n
c z
n=1
cos
n'
b
n
f
b
= an Rnibn = R1 n
n'
n cos n' + an sin n'
n
n d
b
0 = 0 + 0
u
= 1
Z2
iv
Z2
u R
sin
n d
0
u R e
in
d:
0
7.41. " , ! # $ % u z !
1
ZZ
X
2
2
2
2
"&
=
D u
jzjR
x + y
u
u
dx dy
=
n=1
+ n
n an
b
'$$% n n # .
' ) ! )
) .
* . # ,
,
1
ZZ
X
r 2n 2
2x + 2y
=
n + 2n
r =
R
a
b
r '
r R
D
u
u
jzjr
u
dx dy
n
a
b
:
n=1
7.42. " , # $ %
1
X
sin n!
=
n2
n=1
! "& !
! j j
$ % ! ) ! "& = 1.
7.43. + ! , ,
"& ! j j
: # $ % , !
) j j , ! )
! j j = . - 1 .
* . + 1 = 2 ..
u R
z
u z
u R
D u
z
z
R
u z
R
u
u
z
R =z
R
R
3. , ,
131
7.44. , jz j R $ /% .
x 3 $% f1 z = u1 z + iv1 z ,
" ) , :
f1 z
= f
u1 z
v1 z
n
R2
z
= f 0
= u 0 +
= v 0 +
1
X
n=1
1
X
n=1
1
X
c
n=1
R n
an
r
n
zn
c n
= R2n c.n
cos n' + bn sin n'
R n
r
o
bn cos n' + an sin n'
" an , bn, cn , 7.40. '
Re f1 = Re f = u , Im f1 = Im f .
* . 3 $ /% jz j R z =
= R2 =z.1 , = R2 = . j j = R.
3 & 7.4557.50 # $% f z jz j R f1 z jz j
R , $ # /%,
u | $% .
7.45. 1 u = Re 8' + 9: 2 u = Im 8' + 9:
" ' z | $% , jz j 6 R, a z | jz j R.
1
7.46. u = Re n . 7.47. u = Re n .
7.48. u = Re ln
R 1 .
1
r
r
7.49. u = Re
1,
0: R 1 .
1
1
7.50. u = Re ln .
7.51. , $ /%
f z
= i1
Z
j j=R
u d
z
f 0 :
* . 3 + zz = 2 z 1 .
7.52. $ , "
"
$ /% & # Im z 0, . e. " $% u z $%
f z = u z + iv z
u t 1 t 1 .
* . 3 $ ) ".
7.53. 3 $ /% 0 Im z 1.
* . 3 $ )
.
9
. VII.
132
.
! z G
z G , G
, 1
0 |
. " ! z G
# .
$ 7.54(7.57 G | z 1 ! z 1 2 | = 1 2 : w = e , 1 6 6 2 .
7.54. + + , ,
j j
i
! z 1 2 =
1
2
Z2
1
1
1 r2
2r cos ' + r2
d
, ! 0 1 2 = 2 1 =2.
d! z 1
7.55. -
. d
i'
z = re | . .
/ . 0 ,
d!
= 21 jjww zzjj
d
w | , 1 w . 1 z .
7.56. 2 w , 1 w
. 1 z . + | 1 2 , z | , . w , w . . 0 ,
z 2! z 1 2 .
7.57. - ! z 1 2 1 2 . + 3, , , . 41
4 . 7.54, . 1 1 2 0 | . 4
.
7.58. 0 . . Im z 0 . 4
! z a b a b, 1 b a 1, 5
3 ?
7.59. - 4 0 arg z .
7.60. 0 . jz j R, Im z 0 7 . # 8, # 3
.
7.61. - 4 . # 8
jz j R, Im z 0.
7.62. - 4
# 8 jz j R, 0 arg z 2.
0
0
0
0
0
0
3.
,
,
133
7.63. " + "%$ 3 !r
jzj
R.
5 #$ 7.6388.1 (! G " 3% 4,
& n %$, "#$ 4 = 1, 2, :::, n . '($#
$# 3 ! : +
( G; ! n | : + ( G.
G Z 4 % "
df z :
!, " u z # G, #
f z = u z + iv z
Z
@u
#! 4 i @n ds.
7.643 .
7.65. !, # = g + ih ( G g z | = ( G, h z | 3 g " # #! 4 2i! z ,
"# ! z | " 4 ! ( G. n
!,
!
1.
X
=1
. =
Z
# u = 21 u z
G
u # #. # 7.38 .
@g z
ds
@n
7.66. 5% ! ! z " +, " + G
+ u z , +&+ % c 4 = 1 2 ::: n .
7.67. !, # v z , 3
u z , # " G, #% p #! 4 #Z
z
+ # p = u z @!@n
ds.
7.68. )! !~ z | 3 ! z " p | # w z = ! z + i!~ z #! 4 .
1 !, p = p , = 1 2 ::: n .
. 5! ! #
Z
1
@!
p =
!
ds:
2
@n
3 7.64 7.70 . e : . ! "#, % &'( )% *# . | ,.: - # , 1953.
134
. VII.
2 ,
Xp
n
=1
.
= 1 = 1 2 ::: n.
7.69. c = 1 2 ::: n |
.
1 , p | , 7.68,
Xp
n
! "#
=1
c c
0, # # #
X
% & , & c #' ( .
n
) . * &# "+ ! z =
c ! z Z
=1
@!
# "# D! =
! @n ds #. 7.32. #
( , # n | !!
# .
n 1
Xp
2 , ! "#
=1
c c
!!! -
' , . . ' ! / #
fc g, 0 ! c1 = c2 = ::: = cn 1 = 0.
Xp
7.70. , #
n 1
=1
A = B = 1 2 ::: n 1
A | # % ! 0(/ B .
! 1#, , ! 0( &# G "+ uz , (2 &!, , #' ( ! A1 , A2 , :::, An 1 , ( &# ! "+!
u1 z = uz +
(
G.
XA !
n 1
=1
~ z
5 6 7 8 7 VIII
.
.
x 1. !"#$%" &'%(% )" "
X1
f z = z n ' ! : n=0
z = a jaj 1. / !/ a 1 ' ! ' "+0 f z ?
1 zn
8.2. <## & ! f z =
' ! : n
n=1
z = 1=2. * ( , 0 (
# (# ' "+! f z ?
1
zn
8.3. , "+! f z =
1n+1
#' (
n
8.1. 9+!
X
X
n=1
' (= %0 ( # !
2
3
1 z
ln 2 =
1 z2 1 z2 :::
2
X1
2 2
3 2
8.4. < !
f1 z =
n=1
zn
n
f2 z = i
+
X1
1n
z 2n
n=1
n
#0 (2 ( / #. # #,
"+ f1 z f2 z !!0! # '#
& &.
8.5. , "+, ! #
az
1 a2 z 2
+
2
1
z
1 z 3
!!0! # '# & &.
1 + az + a2 z 2 + :::
8.6. !
1
11
z
:::
f z = a0 + a1 z + ::: + an z n + :::
# / # R = 1. ! # # z =
. VIII.
136
= 1 +Z Z ,
= 1 +Z Z = = 0 + 1 + + n n +
, :
1 1 2, = 1 $ % & , = 1 2'
2 21
1, = = 1 z z o % & , (
j j 1 ) 1 z z = '
3 = 1, . 2 % & Re 1 2'
4
1, % &
, ( ) 1 z z = .
1 n
X
2
8.7. - , $ =
n=0
% & , j j 1
j j = 1 $ $ &$ . . % &$,
$ . .
/ k . 0
= pk2 + 4 + + 2k +
+ 2 , , $ = 2 1 | ! 1 ! 1 0
1.
3 8.8, 8.9 % &, . . . , . . . 1
X n!
8.8. =
.
f z
f
F Z
c
c Z
:::
c Z
:::
=
f z
z
=
f z
F Z
F
f z
z
f z
z
=
f z
f z
z
z
z
f z
f z
f z
f t
t
f z
z
z
:::
z
k
t
z
n=0
/
n!. 6 | . &.
= n! .
1 zn
X
8.9. =
n .
p
2pi=q
re
q
n
q
,
r
!
f z
n=0
2
0 . 3 8.788.9 . .
$ . ) .
6 1 . 9%%&
X n
1 2
=
n
a z
f z
n=0
n n :::
1.
137
$ k+1 1 + k ,
0, &
& % & .
1
X
1 1 j j
8.10. - ,
n
1 zn
n=0 1 z
1 j j 1 % &, . 5.1485.17.
8.11. 0 | . &. % &
1 1 zn 1 zn
X
=
. - , .
n 1 + zn
n=1 1 + z
= 12 ; + < + 12 ; <
j j 1 % & , j j 1 |
% & .
8.12. 1 - , | $ &
1
X
1
,
1
n z e in j j
2
n=1
j j 1 % &, $ . j j = 1 $ &$.
/ . - ,
! 2im - .
0 . 0 .
$ $ ..
0 | , , 1 X
$ .$ .. 6
n
n=1
, 1 2 n $ . $ , $ $ 1$ X
, = ancn z n=1
% & , $ , $
$ , $ 9 $ $
. = . > , . 2, . XVI.
2 - , | $ & ,
. 1 & % & .
8.13. - ,
1
X
1
z
z ln n
nz + n =
n
n
f z
z
+1
z
f z
' z
1
S z
1
z
f z
z
' z
S z
f z
' z
f z
z
' z
z
2
z
z
z
e
L
c
a a ::: a :::
L
L
F z
L
n=1
Re 1,
$ &$.
z
n
e
Re = 1 $
z
. VIII.
138
8.14. ,
f z =
X1 e m z
!
m=0
Re z 0 Re z = 0 .
8.15. ,
f z , Re z 0 ! f z =
X1 ane z, an = 1n
n
n=1
+1
,
2k k = 1 2 :::, % % Re z 1.
1
2k
& . ' f z f z =
1 e ze e 2kz
k=1
, ( ( 1 e ze 2k Me 2k ,
M | (.
* . * +, ,
, ! , ! % (+! .
8.16 . - f s, Re s 0
2k 1 = 2k, 2k = 2k + e
X
Z1
, . f s = e st et sin et dt, %
0
Re s 1.
8.17. / 0
Z1
Re z 0 1z = e t tz 1 dt tz 1 = ez 1 ln t .
0
* 2 ,
,
1z %
+ 0 1 2 :::
::: n :::, + n 1n =n!.
8.18. *, 1 % %
1
Z1
n
X
1
, + 1z =
+
e ttz 1 dt.
n! z + n
n=0
Z1
1
& . ' e ttz 1 dt
e t
0
% .
8.19. 4 3.26 (+ , 0 x 1
Z1
Z1
0
0
tx 1 cos t dt = 1x cos x
2
tx 1 sin t dt = 1x sin x2 :
4 ! (! z ( + +
+?
1.
139
8.20. , 1z % % ( , , +
Z
1z = 2 sini z w wz 1 dw wz 1 = ez 1 ln w
C
C % , (! 8 .
8.21. * z | 9 . 5.13
z =
1
X
n=1
Re z 1:
1
nz
, Re z 1
z =
Z1
1
z
0
wz
ew
1
1
dw,
%
z ,
z = 1; + ! (
z z = 1.
& Z . % w z 1
dw, C | 8.20.
ew 1
C
8.22 . *
f z % f z =
1
X
=
an z n, , ! R = 1. =( n=0
1
X
an z n
'z 'z =
' z | ,
n=0 n!
3.150.
$ %,
f z , |
. -
z 1
Z1
e ' ztdt = f z :
j j
t
0
$ % ,
Z1
e ' zt dt '
0
f z ( % G,
t
-
' ( :
(
,
,
'
*
| +
( %,
'
, j j
1, ,
+- +-
* - +- ,
| , , %
.
z
G
G
fz
. VIII.
140
8.23.
: 1
1
1
1 z n 2 z 2n 3 z 4n.
X
n= 0
8.24.
X
X
n=0
n=0
-
! F "z = 21 i
Z
C
' d
z
C|
% & '(% ' '" | )'*, ( C .
,' & : -.( & )'*% F + "z F "z " . . 127 .(
-' % -& 2 C , .
-, -.( f "z 0 .
8.25. ,' & , - )'* '" -'% 1 % -' % & '% Z C , -' C .( ! 21 i ' d
.
z
C
1 ' & . 2 & ) 3*' ( & -% !.
8.26. | . (% % & '(% ', % & 1 , 2 . '-(
-' z1 z2 ". 31, G+ |
. , G | .
. , Z ,
F "z = ' d
z
'" = a, 2 1 ,
'" = b, 2 2 "a b |
' '( (.
4 % )'* F + "z F "z
-' )'* F "z . G+ :
-& 1 . -& 2 .
8.27. G | & . , -
' ! 5, '"z | )'*, -' & '% . G + + 5.
,' & , - )'*
Z
'"z 21 i ' z d
-' Z! ' , )'*
'"z + 21 i ' z d
1
- , C .
,
'
2.
.
141
| ' 5. 7. ' 5 ! - % '.
x 2.
.
2
2& -' z = a ' k 1 )'* w"z
"k | - , k 2 '& , -
w"z , ' ' -' z = a
X1 c "z a
1
X1
w= c z
w=
n=k
n
1
n
n=k
"a 6= 1
"a = 1:
8 ! '- - '9))* cn * ( ' - , -' z = a " z = 1 &(
" . . .. : - -' &(
" . -' & - ' .
4 % % -'% z - ' )'* w"z
. - & -( . -'
* ( -' , ( -' .(
-' & - ' . 4 % )'* w"z z - '( ' -' & ' '.
: ' % '% ' w & -% )'*%
' t:
1
w=
cn tn
X
n= N
1=k "a 6= 1
t = "z z 1a=k
"a = 1:
" . . . -'
z = a z = 1, '( ' ' - . w"z ' - -' . 0 jz aj
r " R jz j 1 .'-& - . ; '
w"z ' . . . & -% -'% )'*% ' t = Ln "z a, Re t
" t = Ln z , Re t . 3 , -
2 " #$
$. %1, . VIII&' ( ). ). * +, .| ..: (,0, 1951'
3
4. 5 $06. | ..: 7*, 1955.
. VIII.
142
z ,
. . .,
.
8.28. , z w z
p z - , :
1 w = z 2 9 z $ 2 w = sin z + z 2 + 4 Ln z $
3 w = sin z + z 2 + 4 2 Ln z .
)
w0 z ?
8.29. +,p , -.:
1 w = z z , 2 w = z 2 Ln z w 0 = 0,
z = 0 0
,
, 0 .
8.3028.38 -. w z
t
,
z - $ ,
.
8.30.
8.31.
8.32.
8.33.
8.34.
w
w
w
w
w
p12 z = 1
pp 1 2 = 1
=
p p 1 =1
= 1+
p p 1 =1
= 3 1+
pp
p
=
=
1+
z
z
8.35. w = e1=
8.37. w
z
p
= ctg 3
z
z
= 2.
z
z
z
z
z
z
z
a
p
z
z
b
z
= 1.
= 2 z = 1.
z = 2
z = 1.
z = 1 a 6= b .
8.36. w =
= 0.
z z
= 5
sin pz
pz
2
z
8.38. w = sin z
= 0.
= 0.
z
= 0.
8.3928.45
, z - ,
, ,
-.,
, 0 z - .
8.39. sin p
1
z
8.42.
1
1
1 . 8.40. sin 1 + pz . 8.41. sin 1p .
1+ z
p p
1p . 8.43. 3 p
sin z
z
8.44. tg i Ln z . 8.45. tg
+ 2 z.
i
4
Ln z .
4 -. w = f z , , -. z w
, ,
-. z w f 0 z , f z f z , . 5 6 z0 6= 1
. . . k 1 -. z w ,
-
2.
z0
) 2 % &
X
f z
1
f z
=
w0
X
1
f z
-
+
z0
=
=
n=k
n=k
1,
cn z
cn z
.
# (
z0
ck = 0
6
n
z0
6
n
ck = 0
" )
X
1
f z
=
n=k
cn z
=
+
w0
X
n=k
:
)" #' (
6
n
ck = 0
1
f z
cn z
6
n
ck = 0
:
8.468.52 ' z w ,
% &.
z
8.46. w = z 1 z . 8.47. w = z 3 3z . 8.48. w =
8.49. w =
8.51.
8.52.
1
w
=
Pn z
w
=
R z
z
# $
n-
|
w
1 + z 2
2
z
z a
. 8.50. w = 2
z
143
0
1
a
-
.
1 .
.
& ' % & .
8.538.61 (' ), *( #
% & w z , ' # ( ( ' R
w - '. ' z - ' , ((.* #
# R.
8.53.
w
=
1+
8.54.
w
=
z
z
8.57.
w
=
1
2
z
z n
. 5# '
n
a n
1
.
8.55. w =
b
2
'"
z
+
1
z
8.56.
.
n + 1 . 6 ) $
n
z
w
n
=
"
1
! 1.
z
z 2
.
-
(, ' " % & w (, ("', ( # ( ( ((.
(# $ ".
8.58. w =
z
.
1 + z n 2
8.61.
Tn z
w
7"8( .
=
=
8.59. w = z
1
2n1
zn
.
n
cos n arccos z
n
8.60. w =
1 Tn z
1
z
+
zn
.
n
| #"
8.628.65 % &, " "#.
8.62. w = e1=z . 8.63. w = etz1=z t | # .
. VIII.
144
8.64. w = cos z + sin z .
8.65. w =
.
8.66 8.74
w- .
8.66. w = cos z . 8.67. w = sin z . 8.68. w = tg z .
8.69. w = ctg z . 8.70. w = ch z . 8.71. w = sh z .
8.72. w = th z . 8.73. w = cth z . 8.74. w = z + ez .
8.75. ! " w-
# ! $" # w = R%z &,
$" #! z %w&,z :
1& w = R%e &* 2& w = R%sin z &.
8.76 8.81 "
$" #! % z - &.
p
p
8.76. 1& sw = %z a&%z b&* 2& w = %z a&%z b&%z c&*
n
Y
3& w = 3 %z ak & % " -
k=1
- n&.
p
p
8.77. 1& w = 3 z a* 2& w = 3 %z a&%s
z b&*
n
Y
p
3& w = 3 %z a&%z b&%z c&* 4& w = 3 %z ak &, n 3.
sin z
z
k=1
p
8.78. w = n %z a&%z b&%z c& n 3.
r
p
p
z a
8.79. w = 3
+ z c. 8.80. w = p3 z 1.
z
b2
p
m = sin z.
8.82 8.87 ./ -. $" #! w%z & z %w&.
8.82. w2 + z 2 = 1.
8.83. w2 = z 2 . 0 . 1 1
z
8.56.
8.84. w3 + z 3 3wz = 0. 0 . 2
z = 1 +3t t3 w = 1 3+t t3 :
8.81.
3
w2 = 1 z z . 0 . 2
3
z = 1 +t t2 w = 1 +t t2 :
8.85.
2.
8.86.
= t tt
2
1
2 +1
.
w2 = z 2 11 + zz .
.
0 . 4 / z =
t2 1
,
t2 + 1
zn
.
1 + z n 2
145
w=
0 . 5 " - - n.
6 z = a | . ! ! f %z & $" # w%z &,
f %z & z = a / ! f %z &, " ! z = a jz aj r r ! 0. 8 . . . .
! . 6 $" #
a | "9 . , ,
# -, . %.
/ " 4.76&. 8 # -$
, / . . / . /" " "" %. -":
: " . . . ; # ! $$#
"!.| <.: : , 1951.| : . I, x 7&.
8.88 8.99 . $" #!, /
! . # -$ p
.
4
6
8.88. w =
z .
p n
8.89. w = m z %n m | " &.
p
p
p
8.90. w = e1= z . 8.91. w = sinp z . 8.92. w = tg 1 pz .
z
1+ z
n
z
i
8.93. w = e
n | . 8.94. w = z = ei z .
8.95. w = sin Ln z . 8.96. w = 1 Ln 1 . 8.97. w = z + Ln z .
z
1 z
1
1
8.98. w = Arcsin z . 8.99. w = + Arctg z .
z
z
8.1008.103
! " #.
8.100. w = z a a | " "
.
8.101. w = Ln $z a z b %.
8.102. w = Ln $z a z b z c %. 8.103. w = Ln sin z .
8.104. &! = 'z | ' ( ' "' ! " ', Rz' | z - " +
w = f | ' "' ! " '
, + ' G . "" , ' Rz' "- !" - #:
8.87.
wn =
1
Ln
Ln
2
10 ..
.
. VIII.
146
1 wz = f 'z , 2 wz = f z + 'z , 3 wz = f z 'z ,
?
, ! , ", # ' | #%
% ", a f | .
( )* 8.105.9.2 !0 , ) )0* " wz
)0 , | 1
2 * % , # ) , !0 ! *p n m | 0 .
p
8.105. w = z 2 ! z 2 .
p6 4
p
8.106. w = z ! 6 z 4 .
pn m
8.107. w = z
! pn zm .
p
p
8.108. w = n ez . 8.109. w = n sin z . 8.110. w = Ln z i .
8.111. w = Ln ez . 8.112. w = Ln z 1=z .
8.113. w = Ln ez 1. 3 ! ! * .
8.114. w = Ln sin z . 3 ! ! * .
8.115. w = Ln tg z . 3 ! ! * .
8.116. w = Arccos cos z ! cos Arccos z .
8.117. w = Arctg tg z ! tg Arctg z .
8.118. 1 w = z r1 r2 r1 , r2 | 0 . 4 !
z r2 r1 . , ! , r1 = 23 r2 = 32 1
2 w = z r1 z r2 1 3 w = z r1 + z r2 .
pp
p p
8.119 w = n z m 1 z . 8.120 w = n m z 1.
pz
8.121. w = p
4 z + 1.
p
8.122. w = n Ln z . 3 ! ! * .
8.123. w = Ln Ln z . 3 ! ! * .
8.124. w = Ln z 1i .
p
8.125. w = n Arcsin z . 3 ! ! * .
8.126. w = Arcsin Ln z . 3 ! ! * .
p
8.127. w = Ln n z 1 . 3 ! ! * .
pz
pz
1
p
8.128. w = Ln
.
8.129. w = Arcsin
.
1+z
1+ z
8.130.
8.131.
w = Ln
z | "! .
p
w = z + Ln z . 3 ! ! * .
2.
.
147
w = Ln z + Ln z . 8.133. w = Arcsin z + Arccos z .
w = Arctg z + Arcctg z . 8.135. w = Arth z Arcth z .
3 ! ! * w = Ln z i
! 2! 0* )" w " . . !.
" z = 0, 0 :
1 r ! 0, ' = const1 2 r ! 0, ' 1
3 r = const, ' ! 11 4 r ! 0, ' ! 1.
8.137. 3 z | ) ! # jz j 1, # 2 ) 2 jz j = 1.
(0 , * )* a )0 )0 , * :
p
1 w = z + n z a1 2 w = z Ln z a1
3 w = a + z n1 4 w = a + ez .
8.138. (0 !
":
p
n
1 w = a, 2 w = Ln a,
# = 1 z , | ) ) 8.137.
8.139. 8 !
! 0* * ", 0* !! :
1 w = z Ln z i , 2 w = z Ln z 1i ,
# z | ) ) 8.137. 9", ! , %
! . . !.
: ) . ( ) ! ; ) 8.136.
8.140. 3 f z | .
3
p !0 ! * ":
1 w = f z 1 2 w = Ln f z 1
3 w = f z | .
s
8.132.
8.134.
8.136.
8.141.
3 ! ! * w =
1
X
2n
1
X
z n! :
n=1
3 f z = z + 2n +z 12+n2 + 2 . 3 !0 ! p* ": n=1
1 w = f z 1 2 w = Ln f z 1 3 w = Ln f z2 + Ln f 21z .
: ) . 3 ! ) , f z ! # jz j 1 2 jz j = 1 ! " ! " # ".
8.142.
10
IX
1. !"#$% 1
P
w- , Ak k = 1 2 ::: n | # , $ %& $ '& P , k |
n
X
k = n 2 . ) * % w = f z , $ +k=1
, % ' ++ Im z 0 P , &% % . / ..%01 *
x
w = f z = C
n
Zz Y
0
k=1
z ak k 1 dz + C1
1
& 1 a1 a2 ::: an 1 |
x, +,
# A1 , A2 , :::, An P 3 C C1 |
% .
. 1 '&% &$ , & + ak | & ' # Ak % C C1 .
4 n ak $ , & '
% ' $
& . & ' n 3
' % ' C C1 n + 1 & '
& % + &
. 1. *
& &+, ' $ . 6 li Ai Ai+1
i = 1 2 ::: n 1
li =
6
aZi+1
ai
jf xj dx = jC j
0
ln " An A1
ln = jC j
n
Z1 Y
an k=1
jx ak j
aZi+1 n
Y
ai k=1
k 1 dx +
jx ak j
k
n
Za1 Y
1
k=1
1 dx:
jx ak j
#
k 1 dx
:
1
.: 2, . VIII. x 7, 3. . II, x 3 .,
. !" # $ %.| '.: (), 1963 -. %
/ $ 0 $ # $ %
1 2 / . .
1.
149
n 3
n
Zz Y
n 3 ak . t =
!z ak "k 1 dz 0 k=1
#$
% P
t- , # . & ' #
w = Ct + C1 , ) P P .
*#$
% $ % zn
Z Y
dz
F !z " = C
!z ak "k 1
+ C1
!2"
2
2
, n 3
0
0
0
0
k=1
z
b z
b
b | , -) # w- . ak | , -) Ak % ! % 1 a1 a2 ::: ak 1".
n
X
k |
% k = 2 k
k = n + 2 .
0
0
0
0
k=1
9.1. /%,
% | #
# , , an = 1, #$-)
Zz nY1
f !z " = C
!z ak "k 1 dz + C1 .
0
k=1
0
. % #
= 1z , ak 6= 0,
= z 1 a , a 6= ak , ak = 0 !k = 1 2 :::
::: n 1".
9.2. /%,
%
Ak $ 1,
!1" , k % $
-) ,
.
0
. &% Ak = 1. 3 k 1, % % P , P Ak Ak , Ak Ak $
0
0
00
0
00
150
. IX.
Ak 1 Ak Ak+1 Ak . 32, 1,
1 P Ak ! 1,
Ak ! 1.
k 1, Ak Ak P .
. 32, 2 , !
, 1.
9.3. "
k , # 1, !% !
0
0
00
0
00
&'' ", %! . 33.
) % . * , ' %
% 9.2.
9.4. 1 , %, !
' ' jz j 1
'' P , , ! -# .
f z = C
n
Zz Y
0
k=1
z ak k 1 dz + C1
' ak = ei' '1 '2 ::: 'n | jz j = 1,
-# Ak , ! , k | ' '' P .
2 , %, . , ! -# ' jz j 1
1.
'
'' P ,
z = 0 w = 1,
f z = C
1
Zz nY
1
k=1
151
, -
z ak k 1 dz
z2
0
| ' P .
9.5. 3 %' ! # 4 56 . 1, . . ,
'' P ! . z = z w %
w- .
) % . 7'' , % P -!
% , ! %
% - w- .
8 % 9.659.8 ! % -- Im z 0
% ! P , w- , %
P
. " ! . z w, ' G
! % w- ! B ; ' . . 38.
9.6. 1 P | 0 v h< w1 1 ! z 0 1 2 <
2 P | 0 v h< w1 1 ! z 1 1.
9.7. P | 0 u , v 0< w0 i1 ! z 1 1 1.
9.8. 1 P | ' ' '
,
3
i!
< w 0 ! ! + p ! z 0 1 1<
6
'
ak = ei'k '1 '2 ::: 'k ,
0
0
0
0
k
0
3
2 P | ' ! ' <
w0 ! ! + i! ! z 0 1 1<
3 P | ' <
p
w 0 ! ! + 1 +2i 3 ! z 0 1 1:
9.9. 3 !
Zz
w, .
wz = t 1 1 t 1 dt arg w x = 0
0
0
0 x 1
! -- Im z 0, :
1 0 6 2, 0 6 2< :
2
w- z - : A
wA B ::: ! za b :::
! a, B ! b, :::
-
. IX.
152
+ 1 + = 1
+ 1,
+ = 2 = = 3=2
2 1 6 6 2, 1 6 6 0, + 1 :
a = 1, = 0 + = 1 = 2
= 2, = 1=2 = 2, = 1.
9.10. 1 w,
! " #$
w %z =
Zz
0
ptt t
1
i
h
p
p p
dt = z %z 1 + %1 2 ln % z + z 1 i2
& ' (( . ) : 0,
0 1=2, = 1=2, 1=2 1 1.
2 w,
! " #$ r
h
i
Zz
p
p
w%z = 21i pt dtt = i1 ln % z + z 1 z z 1
t t 1
1
& ' (( . ) : 0,
0 1 1.
9.11. * + ' (( Im z 0
w- , + !
. 34, +
:
1 w%A = 0 B = 1 C = 1 ! z %0 1 1
2 w%A = 0 B = 1 C = 1 ! z %0 1 1
3 w%A = 0 B = 1 C = 1 ! z %0 1 1
4 w%A = 0 B = 1 C = 1 ! z %0 1 1
-
1.
153
5 wA = 0 B = ia C = 1 ! z 0 1 1.
9.12.
Im z 0
w- , . 35 0 1:
1 wA B = 0 C = 1 ! z 0 1 1
2 wA B = 0 C = 1 ! z 0 1 1.
" # , $% | ' # = p=q, # () $ # *+ ,'.
9.13.
, . 36 %$ AC | .
wA = ai B = 1 C = 0 ! z 0 1 1:
/ . 0 + (
.) = a=w % ) # + # % # 9.12, 2.
9.14.
Im z 0
w- , . 37,
wA = h B = 1 C = h D = 1 ! z 1 0 1 1:
" + . $
.) % ' ++ .
9.15.
Im z 0 + w $ + a 3 A 4 d . 38.
154
. IX.
wA = 0 B = d C = d 1 + e D = dei ! z 0 1 1 1:
.
9.16. !" ! Im z 0
!
i
w- ! , "
% . 39. ( % a, b a 0, b
0 | % " * + | " %
%
! , ! " :
1 wA = 0 B = 1 C = H + ih D = 1 ! z 0 1 a 1.
2 wB = 1 C = H + ih D = 1 C = H + ih B = 1 !
! z1 a 1 a 1. .
. ( ! ! %
! / 0
3 wA = 0 B = 1 C = 1 D = h3 ih2 h1 E = 1 !
! z0 1 a 1 b.
9.17. 1 1 2" / wz , *" " jz j 1
" ! n-"! / ! + w = 1, "!, w0 = 0,
w 0 0.
2 " jz j 1 + n-"!
"!, w0 = 1, wx 0 0 x 1. ! c 1
! wz = c 1 =z + :::
9.18. 1 1 2" / wz , *" " jz j 1
" ! % / ! + w = 10 "! : w0 = 0, w0 0 0.
2 " jz j 1 + %0 w0 = 1,
wx 0 0 x 10 ! c 1 ! wz = c 1 =z + :::
0
0
0
9.19. 1 ! ,
" 2" /
Zz
n
dt
wz = 11 +tnt 2=n+
w0 0 0 1 1
0
% " jz j 1.
n
2
1.
155
9.20. !" ! Im z 0
!% n-"! w- ! / !
+ w = 10 : wi = 0, w0 = 1.
9.21. (" ! P w- ! + 3 %", * n , % *
w = 0 . 40. ("
Ak | + % P ! k | " * "!%
n
X
k = 2 , Bk | + % P
k=1
/ % A1 , B1 , A2 , B2 , ::: ! %
!! P , lk = Ak Bk | ! % %. 6, 2" / w = f z , f 0 = 1, * %
" jz j 1 P ,
n
Y
f z = Cz z ak k
k=1
C | ! , ak = = ei'k | " jz j = 1, " * Ak . 7 bk = eik , " * + Bk , ! "
n
X
k
z
ak
k=1
z = 0:
3
1
8 ! % C , ak , bk ?
.
. ( ! / "
2" / f z "
% !, :" 2" /
'z = f 0 z =f z
! z - ! .
( . (!" 2 "! " +
!. II, ,
2.128, 20 2.129, 10 2.1310 2.137.
= " + "
%
! %, * *" 2 "!".
9.22. + " ! . 41
+ " jz j 1 "! w1 = 1, w0 1 0.
9.23. 6
, 2" / w = f z , *
!" ! Im z 0 + %
9.21,
n
Y
C
f z = z z z z z ak k
0
0
k=1
z0 | !" ! , * 1.
. IX.
156
9.24. P w- 0 1,
n 1 , w =
= 0 B i = 1 2 ::: n 1,
m 1 , Ds
s = 1 2 ::: m 1
1, ,
!" !# # . 42. Ak k =
= 1 2 ::: n | '( P , k | # (, Cj
j = 1 2 ::: m | '( P 1,
j | # (, A1 B1 A2 :::An C1 D1 :::Cm | !() *( P .
+ , , *" w = f z , ! # " ## Im z 0 P ,
n
m
.Y
Y
f z = C z ak k
z cj j 4
i
j =1
k=1
ak , cj | x, # ' Ak , Cj . - bi , ds
x, # ' Bi ,
Ds , " "#" " "
n
X
k
k=1 z ak
m
X
j
= 0:
z
cj
j =1
5
/ ( C , ak , bi , cj , ds ? 1 ",
ak , bi , cj , ds 1?
2 . + ,
n 1
m 1
n
f z X k
=
f z k=1 z ak
0
m
X
Y
j
i=1
= Y
n
z
c
j
j =1
k=1
z bi
z ak
Y
s=1
m
Y
j =1
z ds
z cj
:
9.25. + , , 4 9.24 "
) P *), ! ) 0 1 . 43.
5 ' ( " ,
( " , # "#.
6 9.2689.30 ##
Im z 0 w- , ( # -
1.
157
( " ,
a 0, b 0.
9.26. 9 , # . 44,
( a b
w A1 = 0 B = hei2 A2 C = 1 ! z 1 b 1
1:
9.27. 9 , # . 45,
w A1 = 0 B = ih A2 C = 1 D = iH C2 = 1 !
! z a 0 a 1=a
1 1=a:
9.28. 9 , # . 46,
w A1 = 0 B1 = he i A2 B2 = hei A3 C = 1 !
! z 1 b 0 b 1
9.29. 9 , !:#
1:
. 47,
w A1 = 0 B = hei2
A2 C1 = 1 D = Hei2 C2 = 1 !
! z a 0 b 1=b 1 1=a:
9.30. 1 9 , #
. 48, 1,
w 1 = 1 w 1 = 1=
2 , # . 48, 2 ( ! =n=
. IX.
158
=2n ,
-
w1 = 1 wA1 An ! z 1 1:
9.31. P w- ! " H ,
Bi
1
1 D s = 1 2 ::: m 1 . 49.
, w = f z , ! "!" Im z 0 " P , % &
i = 1 2 ::: n
s
f z =
X h ln z
n
X
ak
Xm l ln z
j
cj + C
6
j =1
k =1
)& ak , cj | x, +% Ak Cj " P , hk | % & %, & % " ,
lj | ! . - bi i = 1 2 ::: n 1 ds s = 1 2 ::: m 1
x, +% Bi Ds , " %
n
m l
hk
j
7
z
a
z
cj = 0:
k
j =1
k=1
k
X
/ ! & " !% 0 C , ak , bi , cj , ds ? 2 % , "
& !% ak , bi , cj , ds & 1?
3 . 4& & & 9.25. 6 & f z , & , " ! & & %"0 / "78 .
9.32. 9 " P ! "!" Im w 0
) "0% %, & % " 1 Bi
0
1.
159
i = 1 2 ::: n 1 Ds s = 1 2 ::: m . 50.
& , ' (! w = f z , ) ! *
Im z 0 P ,
n
m
f z = hk ln z ak lj ln z cj + Az + B
X
X
j =1
k=1
8
ak , cj | x, " Ak Cj
P , hk , lj | !! ) ,
. < A0 *
m
X
1, A 0 Im B = lj .
j =1
0 bi i = 1 2 ::: n ds s = 1 2 ::: m x,
" Bi , Ds , !! ! ! !
n
hk m lj + A = 0:
9
k=1 z ak j =1 z cj
2 A, B , ak , bi , cj , ds ? ,
A0 * a0 6= 1,
n
m
f z = hk ln z ak lj ln z cj +
j =1
k=1
+ L H ln z a0 + z Aa + B 10
0
X
X
X
X
H=
Xn hk
k=1
L=
Xm lj
j =1
A 0 Im B = L:
= ak cj 1, ' 10 .
5 . > ! ' 2 ''!9:(.
160
. IX.
w a0 , ak , cj ,
f z .
9.33. " P | w- $ $$ ,
1 $ Bi
i = 1 2 ::: n . 51 . $,
w = f z , )
)) Im z 0 P ,
f z =
n 1
= Az 2 + Bz + C + hk ln z ak
X
k=1
ak |
x, ) , Ak P , hk |
$$ . " A0 1, A 0 B | - . . bi i = 1 2 ::: n , ) ,
Bi ) $ - f z .
/ A, B , C , ak , bi ?
1 $ . 2. $ $ 9.32.
9.34. " P w- $ $$ , 1 . 52 . $, 0
w = f z , ) )) Im z 0 -
1.
161
P ,
n 1
m 1
f z = hk ln z ak lj ln z cj + z Aa0 + z C c0 + B 11
j =0
k=0
X
X
A 0, C 0, h0 = Im Dm B1 , l0 = Re D1 Bn , ) bi ds , Bi Ds )
$ , $ 9.32.
/ A, B , C , ak , bi , cj , ds ? " $,
a0 = 1,
n 1
m 1l
h
j
k
f z =
ln z ak
ln z cj z C c + Az + B 12
0
j =0
k=1
A 0 C 0. 5 ak k 6= 0 cj j 6= 0 1,
12 ) .
1 $ . 2. $ $ 9.32.
9.35. 6$ )) Im z 0
X
X
w- , $ . 53 $ $
) $ 7 - a b a 0,
b 0 :
1 A1 B A2 C ! 1 b 1 1 7
2 A1 B1 A2 B2 A3 C ! 1 b 0 b 1 1 7
11 ..
.
. IX.
162
3 E O D ! 1 1 1 4 E O D ! 1 1 1
5 E O C ! 1 1 1 6 A0 B A1 ! 1 0 1
7 A1 O C ! 1 0 1
8 A0 B1 A1 B2 ! 1 1 + a 1 0 d = Re B2 B1
9 A B C D ! 1 1 a 1 d = Re D B h = Im B D.
x 2.
,
3
uz k =
Zz
0
p
1
dt
t2 1 k2 t2
1
!" # 1 " t = 0, %#&
1- .
'( k %#& # )*( & , +
0 k 1.
,( %#"&"() () z = sin ' " &# t =
= sin "# - " #"
usin ' k = F ' k =
/0 !"
Z'
0
p
1
d
:
k2 sin2
2
z = sn u k
3
"", # "1 0%+" 1, ' = am u # & ) "% 
" %#& , sn- . %
" & , + sn 0 k = 0. 2 !") sn u k &# %
# " !""
p
p
cn u k = 1 sn 2 u k dn u k = 1 k2 sn 2 u k
4
%#( &#&# cn- " dn - . 4#" )
5& &#" (" cn 0 k = dn 0 k = 1. 6&" 01"(&" # %"" ( k, "* & sn u, cn u, dn u.
3
, !
,
! ,
, :
# $. &. '
.| ).: $ , 1970.
/
'. 0., 1 23. $. 4 . 0. 2.| ).:
5 , 1963. 7 8., ' ! #. 1 9
! . 0. 3. '
,
:
) .|
).: $ , 1967. ;
".
4
! =>
! !
> !
: 4 8., 4 0.
.| ).: $ , 1968. @ A., ' ! 5., : 9 5.
.| ).: $ , 1964.
2. ! "#
163
% 174 & , +
d sn u = cn u dn u d cn u = sn u dn u
du
du
5
d dn u = k 2 sn u cn u:
du
,+" !"" uz k " z = 1 ' = =2 , . . "
Z1
p
t
1
0
dt
2 1
k2 t2
= K k
%#&
p 1- . 4"+" k = 1 k2 %#& .
K k = K " K k = K %#5&
.
4 %+1 +& "& % 5& & 58" %#(
&*" 0%#" 1 -""+& "1 "#
# :
K k1 = kK + iK K k1 = k K + iK
6
ik = kK :
K ik
=
k
K
K
k
k
,("( :, +
0
0
0
0
0
0
0
0
0
0
0
1Z=k
p
1
t
1
Z
0
2
p
dt
1 1
t
kt
2 2
dt
=K
kt
1 + 2 1 + 2 2
0
&# k2 t2 + k 2 2 = 1
0
= K 0 &# t = tg :
9.36. ; % , + 0%( #1) & &" Im z 0
" 0:"" & (8 5 ( -""+& " 1-
Zz
dt
u= p
2
2 2
0
1
t
1
kt
# & ( " & #*"(" K , K + iK 0 , &#&# 58"(" + ( 1, 1=k.
': - 0:" "!" &"(("", % ,
+ 0 0 !" z = sn u # ""+& & "(" 4K " 2iK .
<&&( &#&#" (: & & (" u " ', u =
= F ' k.
= % " . '"(" "!" &#&#" "!. /&0
#"(" 0" "%(" ( du
, z 0dz
)&#" 5 & .
11
164
. IX.
9.37.
Im z 0 w- a, a + ib a 0, b 0 .
9.38. ! " z - # $ %!
u=
Z1
z
p
dt
t2
1
k 02 + k 2 t2
:
'"# % # ( ,
z - !
1 1* +1 1 i1 ik =k* +ik =k i1 :
' , ! " z = cn n k , " , !
" "! " 4K , 2K + 2iK .
9.39. ! " z - # $ %!
0
0
0
u=
Z1
z
p
dt
t2 t2
1
k 02
:
'"# % # ( , z - ! 1 k *, +k 1 . ' , ! . ( " z = dn u k , " , !
" "! " 2K 4iK .
9.40. " z - # $ %! 1- "
0
0
0
u z k1 =
Zz
0
p
dt
t2 1
1
k12 t2
" k1 ! k, 1=k, k , 1=k , ik=k , ik =k.
/ . 0 . 6 .
2 Zz r
Z'q
2 2
v z k = 11 kt2t dt = 1 k2 sin2 d = E ' k
0
0
z = sin ' 4
.
0
0
0
7
0
' z = 1 ' = =2 !
2-
r
Z1
1
E k = E =
1
0
k 2 t2
t2
2-
dt:
0 " ! E k = E , !
0
0
8
2. ! "#
1
165
E
= k1 + E k 2 K i E k2 K *
k
E 1 = 1 + E k2 K i E k 2 K *
k
k
ik 1
E
= E E ik = 1 E :
0
0
0
0
0
0
9
0
0
0
k0
k0
k
0
k
' 4 %! 4 1- 2- ", !$
" 4 " k k , 4
EK + E K KK = =2:
10
7 z = sn u , " $ v z k , "
0
0
0
0
Zu
E u k E u = v sn u k = dn 2 d
11
0
4#$ %! 2- " . ( %! 1- ". 8 . ( E u . ( Z u , " .
12
Z u = E u KE u:
9.41. " z - # $ %! 2- "
v z k1 =
Zz
0
r
1
1
k2 t2
t2
dt
" k1 ! k, 1=k, k , 1=k , ik=k , ik =k.
/ . 0 . 9 , !
0
1Z=kr
1
1
1
k2 t2
t2
dt = i K
0
0
0
0
E
0
k2 t2 + k 2 2 = 1.
9.42 .
Im z 0
Im w 0 #
Re w = a, 0 6 Im w 6 h %: w0 =
= 0, w1 = 0, w 1 0.
' . (
% ) *+ ,
z
Z
b2 z 2
wz = C p
dz = C1 -k 2 b2 1uz k + v z k .
2
2 2
0
0
0
1
z 1
k z
/ C1 = C=k2, w1 = a, wb = a + ih,
C1 , k b.
. IX.
166
9.43. ,
w = Z u +
dn u cn u
sn u
0
K , 0 K , u = + i, h !"
i
i
i
w
h 2K ,
2K
i
h 2K u = + iK , 0
0
" dw
= 0. $ %
du
, ,
jj K , jj K j , ! . 54.
$ . ( )
) ! |
| . %" : D a rw i n. Some conformal transformations involving elliptic fonctions,,
The Philosophical Magazine. Ser. 7.| 1950.| V. 41, 0 312.
3 z
Z
dt
wz k =
p
=
2
2
2 2
0
0
1 + t
1 t 1
Z'
=
0
kt
d'
p
2
1 + sin ' 1
!
.
k2 sin2 '
= 4' k 13
3- -
5 " " z = sn u
Zu
wsn u k = 1 + d
:
sn
0
6 41 k = w1 k = 4 =2 k !
2
3- .
14
9.44. 7" z - 89 % 3- 13
0 k 1. : ,
:
1 1 1; 2 1 k2; 3 k2 0; 4 0 1.
: = 1 = k2.
< w- , ! 9 ! ! " " z . 6 98
u- , z = sn u.
2. ! "#
3 Z1
w= p
z
dz
4z 3 g2 z
g3
=
1
Z
z
p
4z
dz
e1 z e2 z
e3
167
15
= = g23 27g32 6= 0 % e1, e2 , e3
! !
1- ,
z = w
16
! - - .
?
0
!@ % @
" 2!, 2!0
Im !! 6= 0 . (
9
02 = 4 3 g2 g3 =
= 4 e1 e2 e3 : 17
A w , . 55, 98 9
! 0 = 0 !,
! + !0 , !0 :
e1 = ! e2 = ! + !0
18
e3 = !0 :
3 17 ,
e1 + e2 + e3 = 0
e1e2 + e2 e3 + e3 e1 = 41 g2 19
e1 e2 e3 = 14 g3 :
9.45 . 3 z - 89 % 1- 6") 15 8 !@ g2, g3
= 0. : g3 0, g3 0, g3 = 0. 7" ! w.
< . : @ "
Im z 0 89 .
9.46 . 3 z - 89 % 1- 15 8 !@ g2 ,
g3 = 0. : , , " g2 = 0. 7" ! w.
< . C = 0, e1 , e2 , e3 !, " . $ e2 | " , e1 = + i , e3 = i 0. :
. IX.
168
z e2 j = je1 e2 j, Im z 0 j
.
9.47.
Im w 0
ABC !" :
1% &A = 0 B = ! 0 C = !&1
p + i%% ! &1 1 0%(
a 3
2% A = 0 B = a 0 C = 2 e =3 ! &1 1 1%(
3% A = 0 B = a 0 C =
p
a 3
2
i
ei
=6
! &1 1 0%.
2.
169
. 9.45 ! " g3 =
= 0 9.46 ! " g2 = 0.
9.48. $! % & 1'15, ) % z -
!
% . 56, &
!+ , 1 jwj 2
! = 2 . . 36.
1
- 9.49'9.51 & %" !+ jtj 1 ! % & .
9.49. .!+ jRe uj K , jIm uj K 0 k 1. / "
) & ) .
9.50. ! 0 jz 1j + jz + 1j = 2a a 1 3a 14, 31 a4.
9.51. ! 0 jz 1j + jz + 1j = 2a a 1. / "
) 5 ! & ) .
9.52. + !+ jtj 1 &
& 1'3, ) % z - !
% . 57.
0
7 0 - 5! ,"
& & )
- " ! !+ 0 , +&% &%, ) & ) 8 !- ) %- " %- !- , - !+
!+ , ., , +!: : ;. <. .
+ 0 .| >.: / ! , 1966.
X
x
1. 1
w z = ' x y + i x y
$
% ' $
.
' = const |
. ) * V
w z
V = V ei = Vx + iVy = w z
1
, | -
, = const |
0
V = w z = arg w z
V = grad ':
j
0
0
j
2
, * C | $ , -
*
C *
- , - $ $ .
Z
Z
Z
/ = Vs ds = Vx dx + Vy dy = d'
3
C
C
C
$ V C .
Z
Z
Z
Q = Vn ds =
Vy dx + Vx dy = d
4
C
C
C
n | * C , -
* $ V
C . Z - V - AB
- Vn ds n $* .
AB
23 $ 3 4,
/ + iQ =
1
Z
C
w z dz:
0
. 3, . III
.
5
1.
5 w z
,
0
X
/ + iQ = 2i res w z :
C
171
$
0
5 a | 6 w z , w z a
z = c n n + ::: + p 1 + + iQ ln z a + c0 + c1 z a + :::
0
w
z
a
2
z
a
+ iQ
2i
,
ln z a / Q | * $
2i
a * Q
/,
p 1
$ a% Q / 2 ,
| 2 z
a
p, $ a% p p | % - |
p , 7c a
k
, * $ $
6
z ak
a 2k .
) ,
1
n
w z = cn z + ::: + p z + + iQ ln z + c0 + c 1 + :::
+ iQ
2
2i
z
ln z 1
* Q
2i
p
/,
z | p 2
1 - p,
* $ $ ck z k | 2k .
8 , $ V = 0, * , w0 z = 0, $ 6
% :
: * $
$ . 5
n 1, :
6
-$ =2n. 8
1.
10.1;10.14
* : * $
,
* V , $ , * *
* , $
*
1.
10.1. w = cz c = + i .
10.2. w = zn , n = 2 3.
10.3. w = 2+iiQ ln z. < *,
Q = 0.
10.4. w = 2+iiQ ln zz
Q
,
/=0
a
.
b
a
Q
2
= 0,
! . ! = 0,
" #
0, # $ %, #
aQ.
.
. X.
172
10.5.
w = 1z .
10.8.
ia.
z
2 i.
10.6. 1 w = z + R 2 w = z
2
R2
. 10.7. w = z12 .
z
w = ln z 2 a2 a 0.
-
10.9. w = ln z 2 a2 a 0. 10.10. w = Q ln z
z +a
2
1 .
1
10.11. w = ln 1 + 2 . 10.12. w = ln z 2
2
10.13.
10.14.
2
2
w = az +
z
Q
2 ln z
a 0 Q 0.
1 .
z
z
w = az + 2i ln z a 0 0.
jz j R,
2
w = a z + Rz + 2i ln z a 0 0:
" # $ : 4aR, = 4aR, 4aR.
10.16. ' (
# )( * wz ( , + , # fak Qk k g k = 1 2 ::: n
#-., + $- V1 = V ei .
10.17. / )
+ , -.( :
1 # 2 # 3 # # # ?
10.18. ' ( + +# , #$ , . a 1, $-. )
2# ) 3 c1 6= 0 c 1 6= 0:
1 = + c1 z a + ::: 2 = + c z 1 + :::
3 = zc 1a + c0 + ::: 4 = c1 z + c0 + :::
10.15.
10.19. ' (
:
+ +#
n- )
= + cn z an + ::: cn 6= 0
= + cznn + ::: c n 6= 0:
10.20. 4 + , #
* $
## + # ( )( $,( $
3 * ( , #
, | , #$ , | . , , #$ ( , - . ' ( + +#
## .
1.
173
6 # . 7 + 10.20 $ )(, $ 2 # )# #, 8 +$ ( #. + 10.22910.30.
+ * ## $ , # ( , $,, $ 3 * ( ) ## )# 3( .
: ) + ) ,
$+ ) (, 3 *
# - .
10.21. ; z - + )# # , ( (.
1 '( # $ ,, )
$ jz j = R ( , , ,
:
) 3( $
) (
) (, .
2 7 3
( $ ,, ) $ jz j = R 3 * ( (.
10.22. '( # ) *) ( ( $ Im z 0 + )# # V1 :
1 < V1 = V .
2 7 a V1 = 0.
3 a Q V1 = 0.
4 4 a p V1 = 0.
5 7 fak Qk k g k = 1 2 ::: n, a p
V1 = V . > # + 1?
6 7 0 Q 0 p V1 = 0. @ + # # ## p? 7 , +# ,
6= 0?
10.23. 7 $, jz j R , # -. :
1 a 2 a p.
10.24. '( $ +# ( $, jz j R, :
1 # - fak Qk g k = 1 2 ::: n, ) $ $,
2 # . 1 # - fa0k Q0k g
k = 1 2 ::: m, ) $ jz j = R. 7
$ ( # ) *) (.
10.25. 7 jz j R , # -. :
1 a , V1 = 0 * $ *- 1 = 0
,
174
. X.
2 a p, V1 = 0 1 = 0
3 V1 = V e 1 = 0
4 V1 = V ei
z = R.
! " . " " #""! 10.25 " & " " #"" ( " & ! , & #
( ., " , )3, . III, . 49+.
, #""!"- 10.26-10.29 # , ! #"" / & " & !
/- ! "- " " .
10.26. , &" z 1, Im z 0, - ia , a 0.
10.27. , 0 arg z =3, ! aei=6 Q , a 0.
10.28. , "" Re z 0, Im z 0,
!
1 Q.
10.29. 1 , "" Re z 0 Im z 0, c ! 1 Q i Q.
2 , "" Re z 0, Im z 0, ! 1 + i Q
0 Q.
10.30. ! ( z - , # , ! - ( Im z 0 - ! f ak Qk k g k = 1 2 ::: n a p, x 1 " ( ( V1 = V ei . , " " !
# ?
10.31. ! ( z - , # , !
jzj R - ! f ak Qk k g k = 1 2 :::
::: n a p,
jzj = R 1 " ( ( V1 = V ei . , " " ! # ?
10.32. , # ( &" D, " ! ( C ,
!
( " C , 3 - !
f ak Qk k g k = 1 2 ::: n. , " " ! # ?
10.33. , &" D, " ! ( C "3 ( & ! " ! , !
( " C ,
3 - ! f ak Qk k g k = 1 2 ::: n #""
V1 = V ei . , " " ! # ?
, #""!"- 10.34410.41 " "" & " " ! / " ! /- / ".
5""! 6" 3 7 &" " 6
", - ( .
10.34. & " " ! " C #"" (
( V1 = V ei . 8" &" 3 /( " w z !" = 0?
i
j j
j j
1.
+ yb2 = 1:
1 #"" ( V1 , & #
2 #"" ( V1 ( .
10.36. & " " jxj 6 C , y = 0:
1 #"" ( V1 , & #
2 #"" ( V1 ( , ( #
, !&/ # " ! ( -" " .
10.37. & " 7 : 3 #"" ( V1 ( , ( 3 "" : 4;"/ " /( 7 ! ( -".
, #""!"- 10.38410.41 & " #"" /- .
10.38. ""&/ y 2 = 2px # # .
2
x2
10.39. "( &/ 2 y2 = 1 # # ,
a
b
V1 = 0.
10.40. /- 1 x 1, y = .
10.41. /- 1 jxj 1, y = 0.
, #""!"- 10.42410.46 " "" ! !
V z + ! = V z ! ( /- "- " ""-. =
1- ! ( ( / / 7 &"# " ( / /, #" !
#" "# 7 /7 ( / /- & (.
, #""!"- 10.42, 10.43 " & , - "! " 1 " / " 1 ! - ! ( #"" / / "" .
Q
10.42. 1 w = ln sin z 2 w =
ln sin z .
2
2i
p
10.43. w = ctg z 0 6 arg p 6 .
2
2
10.44. , ( ( z - S : 0 x ! , ! , &"#" - ! a Q , a 2 S , 3
#"" / V x + i1 = iV , V x i1 = iV1 . , " " ! # ?
- "! " 1 " / ,
= 0 Q = 0.
> " # " . ! #" #" #""! 10.42.
10.35. & " 1 "
3
.
2.121 .
x2
a2
175
2
. X.
176
10.45.
z - S : 0 x !, , a p , a 2 S ,
V x i1 = iV . " # $ % & .
10.46.
z - S , ' C1 , C2 , , C1 , C2 , & # , S & V1 V2
&# # (1 (2 & S . ) &
' .
* & , ' w z
$ + % .
+ %
& + + %, 2! 2! , Im !! 6= 0
- , Im !! 0 . . $' ,
f z + 2m! + 2n! = f z , ' m n | & % & .
" ' - z0 , z0 + 2! z0 + 2! z0 + 2! + 2!
z0 | & .
1 f z | $ + %,
& 2 :
1 f z
'
2 & + % f z # , 4 &# ' ,
3 f z = a ' ' ' a, '
' $
& $ + %
4 4 # #
+ % f z , 4 &# ' , , . .
X
X
k k = 2! + 2 ! | % & :
- & % + %
Y
2
2
z = z
1 1 ez=+z =2
6
' ( = 2n! + 2m! (, & . 8 % z .
- & + + %
X 1
1 + z
z = zz = 1z +
+
7
2
z
' (, & .
8 % z .
0
0
0
0
0
0
0
0
0
0
0
0
1.
177
5 7 "z $ 2! 2! ". . 167$ "z $ "z $ = "z $.
8
9 "z + 2!$ "z $: = "z $ "z + 2!$ = 0
"z + 2!$ "z $ 2
, !,
"z + 2! $ "z $ 2
!
| . ( & "z $,
! , = "!$ = "! $.
7 , , ! !
!
! = i=2:
;'
= 1
= 3 + = 2
! = !1 ! = !3 ! + ! = !2 :
5 k "z $ &
k z z !k
"k = 1 2 3$:
"8$
k "z $ = e
!
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
k
=
"z $ = k zz :
0
k
"9$
k
5 k "z $ 7 "z $
?' sn z , cn z , dn z &* :
p
"z $ ek = kzz
"10$
p
"11$
sn u = e1 e3 zz cn u = 1 zz dn u = 2 zz
3
3
p
! u = z e1 e3 ek = "!k $ ". . 167$.
3
( * "z $ "z $ 0 &'& &
&.
) f "z $ ! & bk Ak "k = 1 2 ::: n$,
f "z $ =
n
X
k=1
Ak "z bk $ + C:
"12$
) f "z $ ! ak & bk
"k = 1 2 ::: n$, 0 / , !
12 ..
.
. X.
178
,
=
X X
f z
b1
=
n
n
k=1
ak
1
X
1
=
v
i
n=1
2
#
v
z
z
an
bn
13
.
= 1
#
2
1 n n1=2 2n1iv =
q
e
=2
+ 12
v
4
#
a1 z a2 :::
b1 z b2 :::
-
#
bk
k=2
z
z
C
v
X
1
2
1 n n+1=2 sin 2 + 1
q
n=0
n
1
3 v = q1=4 eiv #1 v + 2 + 2
= iq1=4eiv #1 v + 2
#
14
15
= i , = 0 .
- "- " #"
q
e
! =!
z
2
= 2 1 1 z =2!1 ##10 v0
! e
16
1
i z
v
2
= 1 z =2!1 ##0+1 v0
i
e
i
i+1
=1 2 3
17
=2 1 .
&"' - ( )* +( ," , , -)',. &) ")( 16 , -) 13 ". -
z
! v
f z
=
z
#1
C
z
#1
a1
2!
#1
z
a2
2!
::: #1
z
an
2!
18
b1
z b2
z bn :
#1
::: #1
2!
2!
2!
&) ")" 11 , 16 17 , ". - . ) ( 0 + sn , cn , dn * - .
1 )(#" --), * | () *),
0 | * "", . . -)))"" - ) -") ".
z
z
z
!
!
= 2M
+
)
"-) " -)" -* * "
-)" 3
| "" -) -)))"" -.
4", *, )*:
1 = 0 ) Im = Im 0 ) )" 3 - ,"* ) 7 -) )3 )10.47. & , *
M
M
u
!
f u
u
Cu
1.
179
" +., ')" ( =
3
2
+2 =
3 - ,"* ) 7 -) ), ) " +. =
.
10.48. & , * *, -)" "-) "
-)" k
= 1 2 3 , *" * 10.47
- = 0 -"' -) , .
8 . 1-) ")" 8 9 .
10.49. & ,.* *,
-)" "-) " -
u
)" 0k 2u!
=
1234
". . 165 , k
2!
*" * 10.47, 2 -"' )(, -+(.
10.50. & , * *, -)" "-) " -)"
". . 165 , -"' )(, -+( * * 10.47, 1 .
8 . : -) #
K 2 = 1 3 2 KE = e e1 e + !2
t
f u
!
f u
f u
t
u
k
f u
C
u
#
#
k
Z u
E u
e
e
!
2
K
3
:
10.51. ;( "-) ( -)
-* * " -)" 3 , 3 -)))""
-.
1, " )*
+ 7))-* (
) Im = Im 0 , Re =
+ ) )"
7 -)" )" ) + 3 - ,"* ) 7 -) ).
1 *, 10.52<10.54 ) -* *, -)" " "-) " -)"
.
10.52. sn . 10.53. cn . 10.54. dn .
10.55. ;( "-) ( -)
-*
* " ,* " 3 = , 3 = -)))"" -. 4", *, )* = 0, = ,
= + 0 = 0.
;(
, )'( )
+2 =
=
.
1 *, 10.56<10.58 ) *, -)" " "-) " -)"
.
10.56. 1 ln sn 3 2 ln cn 3 3 ln dn . 10.57.
.
u
10.58. ln k
= 2! = 1 2 3 4 .
:) - "-) -)
( +) +* *) " +. ) :
j j 1 = 1 | ") 3 ) )" f u
M
N
f u
u
!
u
!
f u
u
u
u
f u
Q
Q
!
!
!
!
f u
f u
f u
f u
u
#
v
u
v
k
u
u
f z
D
R
12
t
=
D
R
!
. X.
180
1
t = e ui=! 0, 2! , 2! + ! 0 , ! 0 u-
, ## $ % %
= !! = i ln 1 .
0
'
# 10.18. )
, # $
*. # 10.20+, %,
-*u+ 2! , 2! 0 . f *z + = -/u*z + *.
0. 1. 2 ,
#
. 169+.
10.59. 5, %, :
1+ R: r1 jz j r2 , 7
%$ ,.
2+ # , , # , D 4 + 7 %$ .
3+ $ , $ , 7 %$ $ V1 = 0.
4+ # , D, ,
, 7 %$ V1 = 0.
10.60. : R: jz j 1, # *. p+ * a 1+ #
% %. 1 t = f *z +
$ , .
; # . < # # 10.18, 1+ 10.51.
10.61. : # , D, ,
, # %
% 1 V1 = V ei .
10.62. : R: jz j 1, # , $ z = 1, % # . :
t = f *z +. >, ,
, .
; # . ? %, f *z +
f *z + = c2 2 + c1 + c0 + c1 *z 1+ + :::
z
1
z
1
% , # c2 c1 # %.
10.63. 1+ < R: jz j 1, , # $ *a. 7+ * a 1+ % 71 * jz j = 1+ 72 * jz j = +. @ # # 7, 71 , 72 ? >,
4
D
, .
, ,
2.
181
, 72 = 0 72 = 71 .
1 , &
t = f +z ,,
2
it=
Z =pe
t = f +z ,, Z = e4 it=
4
=
0
s = Z Z0 +Z0 = e
, 0 | - , . : u
C & .
2, < - D, -
, , & & 71 , 72
V1 = V ei .
x 2.
: C E = Ex +
+ iEy = Eei % - - w+z , = u + iv ,
&-
/ v &
+ !,, u |
. 0
v = const |
, u = const |
. : C
E = grad v = iw0+z, E = jw0 +z,j = 2 arg w0 +z,
@v @u
@v
@u
Ex =
=
Ey = = :
@x @y
@y
@x
< % % C , C % % %, &%
& , ,
+. . &- ,.
E a | w0 +z , w a
w+z , =
z
cn
pi
+ 2qi ln
+ ::: +
an
z a
z
1
a
+ c0 + c1 +z a, + :::
1
2qi ln
a - &-
z a
& = 2q , &- +a/ 2q , +
& - , z - a,
% q ,/ pi=+z a, a p, &- +a/ p, +p |
/ pF ,/ &
& ck =+z a,k +k = 2 ::: n, a
2k .
2, 1
c
w+z , = cn z n + ::: + piz + 2qi ln z + c0 + 1 + :::
z
2qi ln z 1 - &- & = 2q , piz | p.
. X.
182
w = u + iv E = iw z |
! V = w z , # :
0
0
u
u = const
v
! " #$
%
& '
I
! "$
&" I
|
+' a, )$
# " a, 2q$
q = )=4
|
.
|
|
/ p
0 " '
v = const
vI
v1
2
du
( ) =
Vs ds
N=
En ds
|
/ p= 2i$
" " , 3 !
% & ' 10.64-10.71 & w z / ,
! , ' / 1 ,
! ' ' ' q | . 0 1
& 10.1-10.14.
1
10.64. w = cz c = + i . 10.65. w = 2qi ln .
z
10.66. w = 2qi ln z
. 10.67. w = 2qi ln z 2 z 2 a 0 .
a
z
b
p = jpje . 10.69. w = z
1
10.70. w = piz + 2qi ln
p 0 q 0 .
10.68. w =
10.71.
::: an .
pi
i
z
X
w = piz + 2i q ln
n
R
2
z
.
z
k=1
k
1
z
k
a
p 0 qk 0 a1 a2 :::
10.72. 3 & & & a4 2q a4 p :
1 / !4
2 ! & .
2.
183
10.73. 5&, , / & & a4 2q & & / D,
w = 2qi ln 1 + c
f z a | , / ! # / D , f a a = 0, c | .
6 & ! v z 7 / D . & 7.36 .
% & ' 10.74-10.80 & & & 10.73
, ' , / &' & &
&' / '.
10.74. % ' Im z 0, & z0 4 2q .
10.75. 1 % jz j R, & z0 4 2q 4
2 1 jz j R, & z0 4 2q .
2
2
10.76. % 1 2 + 2 = 1 & 14 2q .
10.77. % 1 & jxj R, y = 0, & 14 2q .
10.78. % 1 jxj d, jy j d, & 14 2q .
10.79. % jxj a, jy j b, & 04 2q .
10.80. % 0 x 2a, 0 y 2b, & z0 4 2q .
% & ' 10.81-10.85 , / & & .
10.81. % jz j R, a4 p .
10.82. % 1 jz j R, a4 p .
10.83. % 1 & jxj R, y = 0, 14 p .
2
2
10.84. % 1 2 + 2 = 1, 14 p .
10.85. % jxj a, jy j b, 04 p pi = ei .
10.86. ;&, , / & a4 p & & / D,
w = f z , f z / !
/ D 1 & & , f a = 1,
f z a
, a 6= 1, piz , a = 1.
3 f z , & t z , / ! # / D:
1 , a 6= 1, t a =
= 0, t a 04
2 1 jtj R, a = 1, t 1 = 1
t 1 = 1.
f z a
x
y
a
b
x
y
a
b
pi
z
0
0
a
. X.
184
10.87.
, f k 2 k g = 1 2
.
! | #!$ % . . 127 , ' $ ) ! ' * * ! )1 )2 )n
! , , | ! ) * ) +
, +- . . | #!$, ' ), #!
% !
Z
z
= 21 @g@n
D
a
q
k
::: n
:::
a p
g z
D
n
D
u z
D
u z
h
Z
= 21
u z
u
@
@n
ln
u
1
j
ds
ln j 1 zj @u@n
zj
i
ds:
. , ! !- ! #!$ ' , * #! 1 . ! #!$ % 1 ,
1 = ln j j + + jz1j
= zlim
!10 1 ln j j1
! ' , , -2' - ' , .
10.88. 3 !-2 ! , | ! :
Z
a
1 = ln jz 1 aj 21 @g@n
ln j 1 zj , 6= 1
D
u z
u
g z
g z
z
o
g z
D
:
z
z
e
n
g z a
ds
2 1 = 21
g z
Z
@g 1
ln j 1 zj ds,
@n
, ! !- !
Z
1
3 21 @g@n
ln j 1 zj = ,
ds
2
z
a
D
2
z = D
, ! !- !
Z
a
4 21 @g@n
= 1, 6= 1
ds
a
a
D
D
-
=12 .
D
7 . . 1 #!$ % = ' ' #!$ ' . . 2 ' '
#!$
g z
g z
D
2.
185
ln j j + 1 1 , *
1 = 1 . . 3 | ,, * #!$
ln j j 1 5 .
8!$
1
0 = 2 ln
jz aj
z
g
g
z
g
z
g z
g
v
z
q
' 2 .
+ - 0 ' # $ !* * : 2 1 2 .
! ! ) ! ! , ! . 190,
).
Z
; '
= ln j 1 zj
a
z
v
a
q
z
q
q
v z
ds
! ! $
, $ *
) * - 2 , . . !2 2 2 2 .
8!$ | - ' -! ), = 1, ' ' #!-
Z
= 2 ln j j + jz1j
2 =
ds
ds
v z
z
z
v z
q
z
o
q
ds
, $ ! ! 1 2 .
; '
Z
@
@n
ln j 1 zj
1 =
v z
v
q
z
)
ds
-
, ' ), ! , ) ' | . . | !' ,! , !2
,
| !' , !'
n
z
z
n
d' z
ds
=
v1 z
Z
cosj zjz
5 .:
.
" , 1941.| . V, x 2.
ds
=
Z
z
d' z :
.| .:
-
. X.
186
,
Z
1
2 z
@
1
z
ln
ds =
@n
zj
0 z
j
. 7.34.
!"# $ gz a % D & '" ) ' , % * a+ 1 ' "* % D. , 10.88, 1 ' *, a 6= 1 ) . a
# ' &# % = 21 @g@n
. /
), '. 4 10.88, 1. a = 1 *0 1+ 1 ) * .# ' &# %
1 @g 1
=
0 * 1 % ' #
2
@n
' * '" . 10.88, 2. )1 1 *
% *#
a+ 1.
1 10.89310.92 0 "# a+ 1 ' & % a #.0 '"
' vz a
% 0, *1 .
10.89. | 0 & &, Im a 0.
10.90. 1 | & jz j = R, jaj R+
2 | & jz j = R, jaj R &, ,
0 a = 1.
10.91. | 0 &0 jxj 6 R, y = 0, a = 1.
x2
y2
10.92. | ) ' 2 + 2 = 1, a = 1.
10.93. 6 0 7"# $
% D, .0 z = 1, 8& : 0 ' & ' " % D , #.
D ' *0 '" 6 .
: . 6. 10.88, 3 4.
1 10.94310.96 8& '% <%)
*1 %0 D.
10.94. D | 8& jz j R.
6
""
. (" .
, !"
,
#
" ! !"
! , . . 185
E
E
2.
187
10.95. D | 8& jX j 6 R, y = 0.
10.96. D | 8& ) '
x2
a2
2
+ yb2 = 1.
1 10.97310.100 0 . . 190 *1 .
2
x2
10.97. jz j 6 R. 10.98. jxj 6 R, y = 0. 10.99. 2 + y2 6 1.
a
b
10.100. jz 2 a2 j 6 a2 a 0.
10.101. /& ' '* 77" & 0 & 7" ' .
=
&, 0 & & ' >8
Z
'
1
d 7 '" 0
2i z
' &# ' , & | 7 '" ' ' &# 21 d'
.
ds
10.102. = &, 7"# v z , # # 10 ' ' Im z 0, ' &
7 '" 0 :
Z1
1
@
v z =
ln 1 dt:
v t
@n jt z j
1
? vz 1, ' & 7 '" ' :
Z1
@v t
1
v z = v 1
ln jt 1 zj dt:
@n
1
10 ' ' Im z 0 0 ' *0
'" ) ' , '" vz ' 0 &0 * ' * .
,'& '" &# 7"# '.&# 1 #.1 0 &0 . . 132:
1 ' 1 a, 0 a 1+
2 ' a b, 0 1 1 a, b 1+
3 '1 '2 ::: 'n 1 1 a1 a1 a2 :::
::: an 1 an 0 an 1 & a1 a2 ::: an +
4 '0 an 1, '1 '2 ::: 'n 1 1 a1 a1 a2 . .
: . '. 1 ' & & 7* % ' + &*1 | ' & & '' "
' & & &0 7 0 @"
' ' , . . 129.
10.103.
. X.
188
10.104. wz v z
d = v2 v2 v1 , v2 :
1 r1 jzj r2 #
2 D.
10.105. &, D |
( , ) * , + v1 v2 ,
wz = i v2ln v1 ln tz + c + iv1 vz = v2ln v1 ln jtz j + v1
tz + ( D 1 jtj | D v1 ( jtj = 1# c | .
10.106.
v1 v2 :
1 - ( jz aj = R a R v1 |
( #
2 - ( jzj = r1 v1 jz aj = r2
a r1 + r2 #
3 , ( jzj = R v1 jz aj = r 0 a R r#
2
2
4 * xa2 + yb2 1 , )
+ * v1 #
5 - 1 jxj 1=k, y = 0 0 k 1# v1 #
6 - jxj 1, y = # v1 .
10.107. 3 D | 4, ) n 4k k = 1 2 ::: n, !k z |
4k . . 132. 5 D ,
- 4 .
& ) ( .
1 5 D ,
n
! z = 21
k
Z
@!k
ln j 1 zj ds
@n
! z = 1 21
n
Z
k = 1 2 ::: n 1
@!n
ln j 1 zj ds:
@n
5 D ( ,
Z
1
k ln
! z = ! 1 21 @!@n
j zj ds k = 1 2 ::: n:
k
k
2.
189
2" * , * +. 1", ,
+ ) , +( 1 +
, ", 0
+ , , 6= .
. - +. 1" . " + ( + *, * * ) , + * , + ) " =
k . /
= 21 @!@n
* 1 "
1 " + ( . +
* * ) .
/*
)
, * + " = 1 2
", . .
z
D
D
k
n
D
i
i
k
!k z
D
k
D
::: !n
!
z :::
z
pik
!k z
i
pik
= 21
Z
i
i k
::: n
@!k
ds =
@n
21
Z
!i
@!k
ds
@n
* (
* * * 7.6727.70.
10.108. 3 *
", ) *
" , + * + + * :
1"
1 j j 5
2" + , .(, )(( 6 .
10.109. | 10.107 " | *
+ 6 + , +( + *
=1 2
" * + " .
' ( ) .
1" " =
".
pik
!k z
k
pik
z
D
D
k
k
X
v z
::: n
k
n
v z
k
=1
k !k z
2" %
Z | & ,
" = 21 @v@n ln j 1 zj 5
D
n
v z
)
D
ds
n
) 1, Z
" = 1" 21
v z
v
@v
ln j 1 zj ds:
@n
' , + * * . *
=
=12
" + *
, * .
3" /* * ) *
Z
2 = 21 @v@n =
k
::: n
D
i
qi
ds
i
X
k
n
k
=1
k
pik
k
190
Xq
n
i
4
1
2
. X.
=1
= 0.
. .
ZZ
D
i
Xp :
7.68, 1.
grad v 2 dx dy =
n
ik
ik
i
k
=1
. . 7.69.
5 wz | !" #$
1 @v
$ ( ) * +% =
=
2
@n
%, #
1
2
# #& jw j.
0
, #& D | ( & % ) (% % * #& ) $" -, # %." + (!/ # ( -1 ::: -n . . #(0# 1 ! # * +% * # D
+0 0./ /
* #"
! (% #( # a b | ( &! # * # D, A | ( & .
1 2
. 3# * + 0. % 1 $% f z
%# % 0 b 411$# A +%
A + c z b + ::: b 6= 1
f z = z b c 11
Az + + ::: b = 1:
8
:
z
2 2
# * !( #&
#&
# , +!
, (!/ %.
#,
$# (
#. 7 $% f z %# % a, 0 b 411$# A +%
8 A
+ c z b + ::: b 6= 1
f z = : z b c 1
Az + z + ::: b = 1:
1
3 2
$# (
#. 7 $% f z
%# % (% f a = 0, f 0 a = 1 # -k ,
/ %.) ( + #&.
4 2
$# (
#. 9
# (, / %./ ( (#00 ( (:00 ) 0
+ #, # * + %# % # #&0 * ( " *% ( # .
., : < <. =. < # % # % 1 $"
) ) .:| =.: 2 , 1966.| < . V.
10.110. # % ( #! 1 $0, 1 # * + 0.0 * #& D #& &! , ) -
2.
191
! !k z ) !/ # ( -k k = 1 2 ::: n 7 , "#
#$ 4 # # # )
%, ( ) ( * # D a> p, ) ) $ - * #
.
10.111. # % ( # " 1 $0 < * # D, "# #$ 4 # # # )
% ( 4# " * #, * ( ) # ! % a> 2q a 2 D 0.)
! #$ ! k
) !/ # / -k k = 1 2 ::: n. ? ) * #& D
(% , #& 1 10.73.
10.112. 3 #& / # 4 # # # )
%, ) ( ) (% " * # D ! #$ w =
= f z , ) f z | 1 $%, # * + 0. % 4# * #& #&
, &! " #(# & " .
10.113. 3 #& / # 4 # # # )
%, 1
) ( * # D ! #$ w = 2qi ln
f z ,
1 $% f z # * + # * #& D :
1 #&
$# ) (! )
$# (
#>
2 )
$# ) (! ) $# (
#>
3 ) ( &$
$# ) (! ) $# (
#.
? ( / %/ "# # ( # %+ # %
) ! #!.
10.114. , # #& / +% 4 ( #$ &!/ (!/ " 4 # # # / ":
1 * ( ) ( * " ( (% " * # 1>
2 * ( ) ( ) " ( (% " * # D # !
% .
? * / %/ #$ ! ) !/ # / # %!.
10.115. 1 ?! #& #$ v z 4 # # # )
%,
* ( ) ( ) (% " * # D % * +% 2qk
X
n
-k
k
=1
q
k
= 0 ) !
! z )
k
) !/
# (. 2
+ # -k #$ # %.
. ? & ( #& % &# # 10.109.
2 ?! #&
!"
#$ v z ,
% a> 2q .
X
n
k
=1
q
k
=
q 6= 0
# % # -
7
., , x 1
. ! : #
$. % & ' , (
) ( * '.:| .: ,-, 1953.
. X.
192
.
vz + 2qgz a
. 1.
10.116. v z r1 jzj r2,
! ! " # 2q1 2q2 , $ $
q1 + q2 = q 6= 0 ' $$!
a( 2q.
. )* + #
' * , $ 10.63, " ' .
/ * ! ! " # $! , ! 1 + .
x 3.
/ $ ! $ 1 wz = u + iv u |
, !
. 3 Q = k grad u = kw z k | 411 , , ! !
. / Z$ Z C
Z
@u
Qn ds = k
ds = k
dv
@n
C
C
C
n | , C , " # . 5 1 uZ $,
C # ik w z dz . 6 " $ a
h
i
C
c 1
q
w z = ::: +
+
c0 + c1 z a + ::: +
ln z 1 a
z
a
2
k
q
ln z 1 a $ a a( q " $ 2k
q, $ zc 1a | $ a.
7 *' $ # 4 $ :
0
0
u
u
v
= const
w z
= u + iv
w z
= u + iv
Q=
V
$%
! "
#
"
&
k grad u =
= kw0 z
&
= grad u = w0 z
iw z
E=
=
v
+ iu
grad u =
= w0 z
! "
#
"
v |
#
2.
v = const
*
*
a, q
$
193
q
a,
k
$
+
%
q
2k
a,
- .
- "
- "
%- , - ! %
% ,
,
.
iwz ,
4 % % 1 "
0
,
0
.23
0
10.117. 91 #
$ $ ! $ ! " . ! " , , $ 4 " $ (
$ .
3 $ 10.118;10.121 ! ! " ! $ , $, $
" .
10.118. 3 Im
0( $ ( .
10.119. 3 j j
( $ ( .
10.120. 3 j j
,
0 $ ( 0.
10.121. 3 j j
, jj
$ 0( .
10.122. 1 < * 1 +
" .
2 9$ 1* + " , ! 4 " , , $
$ ( $! =k = 1 2
! $ k . > ' *
$ k $! .
10.123. !
1 j j 2 , , $
$ ( $! # ! $: 1 | # j j = 1 2 | # j j = 2 .
. 9 . $! $ 10.63 10.116.
D
a q
C
z
z
R
x
a q
a q
a
x
y
a
ih q
y
b
q
g z a
D
D
D
a q
k
r
u
u
!
r
z
z
r
a q
u
z
z
r
13 *.$.
.
h
::: n
XI
,
x 5 . I
! ! "#. $ %
& '
:
dz) = dx i dy = dz d' = @'
dz + @'
dz):
@z
@z
1. ! " #
# p
,p 1. p 6= 1 ,0 6 ., #
Ox.
11.1. 0& , ' & ,
h ! & & p 2
x2 2 xy + y2 = ph2
x
= p cos2 + 1 sin2
=
p
p 1p cos sin = p sin2 + 1p cos2
jz + z)j =
= p
1 2i
p+1
e
= 2ph
p+1
:
, , 11.1 &2 ! & & . 4 #
2 = 1.
. 5 ,
j j 1. &
= 0, p = 1, = = 1, = 0.
11.2. 6& % #
& & :
r
1 + jj
2
+
1. p =
=
+
+ 2
1 jj
2
2. 2 = arg + , 6 arg
2
.7
2 ! -
p + 1p = + 7
1.
8
2
tg 2 =
p
3.
+ 2 + 4
:tg =
8 j j2
2
: j j2
j j
1
2
jj
= 1+2
cos 2
195
2
j j2
+
j j
j j j j2
1
6 6 p j j 6 12 p 1p
p
4
5
1
= 2
sin 2
1
= 1
2
cos 2
+
,
.
!! "
#$ ":
6
jp2 p1 j 6 j2 1 j j2r 1 j j2 1 j
8
j2 1j 6 jp2 p1j p2 1 p1 1 j2 1j
p
p
+
+
j2 1 j 6 jp2 p1j 1
prp
1 1
p2 p p 1 p
j2 1 j
r
:j2 1 j 6 jp2 p1j p2 1 p1 1 j2 1 j
p
p
p
j 2 1 j j2 1 j
j 2 1 j 6 j 2 j j 1 j
jp p j p
p2 p1
j2 1 j
6
+
+
2
1+
+2
2
+
8
2
2
2
1
1 2
1
+
p2 + p1
) .
sin
1
sin
1
sin
+2
1
sin
+
2 1
7
1
1
1 sin
.
" !"" 6*8 $ + !-
"" "
jr2
cos
2 r1 1 j 6 jr2 e 2 r1 ei 1 j 6 jr2 r1 j
pr r j j jja ib j ja ib jj 6 ja a j jb b j
2 1
2
1
2
2
1
1
2
1
2
1
i
cos
+2
sin
+
,
+
+
- .! $! .! "
"
,
w fz
=
=
u x y
+
iv x y
.
+-
u a1 x b1 y c1 v a2 x b2y c2 :
a b
a1 b1
2
2
=
/ . # 0
+
+
+
=
.! 1 =
+
+
1
. 2 1 = 0,
.! -
" "+! + .
11.3.
1 655
" 4"
.!
w Az B z C:
A B C
=
8+! #559+
13
,
+
55+
.! 4.
$!" " "
7+
:! #559+ $! .! -
196
. XI.
1 , = A 2 jB j2 .
2 6= 0,
z = A1 w + B1 w + C1 :
!" #
$%%& "
$%%& " 1 , 1 = jA1 j2 jB1 j2 = 1=.
3 6= 0,
) *")
$ " $ ". + " * , ) , = B=A, 0 = A=B , 0 -
.
. -
# ") ) w = Az + C w = B z + C .
4 0,
) ) 01 0,
* ) 0 .
5 = 0, $%%& " a1 , b1 , a2 , b2 " -,
* 0 0
j
j
w = 2jAje +=2 jz j cos ' + 2 + C
#0 ' = arg z , = arg A, = arg B . 3 # * , *"
i
$ #
.
" - ) p , * ) $ ,
*") # *. 0 11.3, 3.
w = ux y + ivx y
* J 0 " - ) pz , z 1 z , z ,
z * ) z %% #
du = ux dx + uy dy dv = vx dx + vy dy dw = wz dz + wz dz:
4 * * * #
du2 + dv2 = d 2
*", $
dx2 2 dx dy + dy2 = p dh2
dh | * , , 0 # , ,
jdz + d zj = p 2+p 1 dh:
11.4. 6 , 0 " 0 # %% #
*- * 7
=
1:
= b2 + b2 = jj
a2 + a2 a b + a b
1
2
1 1
2 2
1
2
1.
197
11.5. 6 , 0 ) " 0%% & * #
"* * 0 " 7:
2
1 u2 + v2 = u u + v v = u2 + u2 = pddh2 = J1 1
x y
x y
x
x
y
y
1
2
2
2
2
2 ux + uy + vx + vy = J p + p 1
.
dw jw j + jw j
3 p = max dw
min
dz = jjwz j jwz jj 1
dz
z
z
w
p
1
z
4 = w = p + 1 e2i 1
5
r
z
J
p
6 ppJ .
6 dw
dz
.0 " 0%% & *
w =
= u + iv "* * "
pz , z z , z , z
z , 0 *" $ " $ * ) * *" #.
11.6. 6 ,
% * 0 * ,
ux + uy = vy ux + uy = vx
,
* , % * , wz = wz :
3 . 3 - jdz + dzj = const jdwj = const.
4 * . 3 " 0 " - . 9 "
.
11.7. 4 ,
0 " 0%% & *
w = u + iv "* *,
0 *" # *" $ " ) * pz , z z , z , z z , 0
* ,
ux + uy = vy
ux + uy = vx
,
* , % * , wz = wz = = pp + 11 e2i :
4 * . - *
pux + quy = vy qux + puy = vx
0
198
. XI.
p = 1=, q = = ,
p q -
. . ! . "#$ $%
& ", 1965 .
11.8. + , , - wz = z wz ,
$ ,& w,
, wz = z w.z , /
$ ,& & z .
11.9. /,
$! w = u + iv ! / $ , $
/ 1 / p, , $ / 1 / p1 , 1 $! /,2
, p, 3 p1 , 1 ,
& , , 3 1 , 1 , 1 . + ,
$! w = u + iv , , ,&
ux + + 1 uy = vy
1 ux + uy = 1 vx
2 ! ! , ,
wz = q1 wz + q2 w
.z
p1 p2 1 e2i q = p p21 1 e2i1 :
q1 =
2
pp + 1 p + p
pp + 1 p + p
1
1
1
1
4 . 4, jdz + dz.j = const , , ,2 jdw + 1 dwj
. = const , / , 1 .
11.10. = f z | ,
$! & f :
fz = f fz
w = g | , $! & g :
g = g g :
+ , F z = g7f z 8 = g f , $! &
+ g f z =fz :
F = f
1 + g
f f z =fz
+ ! 9
f F f 1 = fz = f z g f 1 :
g = z
F
f z
f z
fz 1 f 1 g
f z 1 f F
f z
. : , / F , ,
g j
sup ln 11 + jjm
gj
! , , ! f F . ; ,
F ! g f 1 .
1.
199
pw=z , w=z | p, , $! w = f z . ,
pz=w = pw=z z=w = w=z 2 + arg fz
! ,
$! w7z t 8
11.11.
pw=t 6 pw=z pz=t :
, , / $!&
u=f x v=y
!-!
u = x
v=f y
!-!
= r
= f ' , !-!
p = max f f1
$! = f r , = ' !<!
p = max rff rff :
11.13. ,
$! jz j R
$, , % z = a jaj R , ,2% ,!/ ! jz j = R. # p.
11.14. ,
$! & /
x 0 x tg y x tg + h
2
u 0 0 v h
$ ! ,
/ !
$ , & . # p.
11.15 . ,
$! , ,
% , /
20 . 58 , .
# p.
11.16. @,& ,
2
@ 2 u + C @ 2 u = F x y u @u @u
A @@xu2 + 2B @x@y
@y2
@x @y
2
1 AC B 0 ,
$! = z = + i $ , ,
@ 2 u + @ 2 u = F u @u @u :
1
@2 @2
@ @
11.12.
0
0
0
0
. XI.
200
, z
@
@
@
@
A @x + B @y @
pAC B 2 = @y
B @x + C @y
@
p
= @x
2
AC B
, , , !
1
2
C=B=A=p
AC B
, A 0.
x 2.
$% w = u + iv, '( '
wz + Aw + B w) = F
1
+ A, B F | % z ,
.
- .+ +
, ( 1, ! 1 .
11.17. /, 0
ux vy + au + bv = f
uy + vx + cu + dv = g
+ a, b, c, d, f g | %
x y,
1: wz + Aw + B w) = F .
- A, B F
. % .
11.18. /, wz q2 z w)z + Aw + B w) = F
(' 2 +"
w = az ! + bz )!
2
1. 5 (' 2
, + .
11.19. /,
wz q1 z wz q2 z w)z + Aw + B w) = F
(' ( !z q1 !z + A ! + B w) = F . 5 (' , + .
0
1
0
0
0
. : . . ! " .| $.: % " , 1959.| *. III.
2.
201
6 . /
' ' w)z q1 z w)z q2 z wz + Aw) + Bw =
= F , ' !)z , a . % az bz .
11.20. /,
wz q1 z wz q2 z w)z + Aw + B w) = F
jq1 z j + jq2 z j 1 ('
z
' , ' z ! z = q1 z ,
w q2 w) + Aw + B w) = F:
5 q1 q2 +
z .
11.21. , . %
1 vy = ux + + 1 uy + au + bv + f
1 vx = 1 ux + uy + cu + dv + g
. 2 07 +, 0
wz q1 z wz q2 z w)z + Aw + B w) = F
jq1 z j + jq2 z j 1, 1 0, jjq1 z j jq2 z jj 1,
1 0.
6 . 8. 11.9. 8 1 0
1 0 w = u + iv w) = u iv.
11.22. , wz |
%
! wz q2 z w)z = 0, + q2 z |
% z jq2 z j 6= 1,
wz = 'z1 +jqq2 zz'j2z
+ 'z |
'
-
2
%.
11.23. , wz |
% ! wz q2 )z w)z = 0, + q2 )z | % z jq2 )z j 6= 1,
Z
wz = 'z + q2 )z ')z dz)
+ 'z | % z .
6 . 8 , wz % z % z).
. XI.
202
x 3.
G | ,
C.
-
11.24 2 .
ZZ
@f
1
@z
f x y
2
ZG
f d
,
:
Z
= 21i
#
g x y
G
dx dy
+
C
3 =
f z
f d
CZZ
&= 2
g d
1
2i
Z
C
i
G
f d
z
@f
@g
dx dy #
@ z @z
ZZ
1
@f d d
@ z
= +
i
.
G
) . * , + @f@ 1 z = @@ f z ,
. 1 , , - j j ! 0 .
11.25. /, +
, , , + -
+ &
/ /
, / @f
= @g
.
@ z
@z
G
f
z
g
G
11.26.
,
f dz
f
g
G
C
0
ZZ
C
0
&
0
f g dx dy
G
G
f g d
2
G
+ ,
11.27.
g dz
0, +
| , +
,
ZZ
Z
&
=i & &
f g dx dy
,
/
= 2i
Z
&
f dg :
C
0, + , -
, , / +
= , Z
&
=i
f z
0
G
C
I
0
f C
f df
2
C
, # - , -
, , = .
S
G
0
G
f z
C
0
I
S
2 11.24 11.27 . . : .
! "#, % &'( )% # .| .:
+ # , 1953.
3.
jj
11.28.3 . G |
$ =
4 '!
'
e2i
ln 1=
z
$ =
$ = 1
ZZ
F z
%
& %
G
1
= +
203
R
e
i
i
' d d
z
z
=
i'
re
:
$ 2
z
G
@F @F @F @F
,
,
,
z 6= 0.
@x @y @z @ z
% @F
"&
@ z
,
% &
%& & .
11.29. 3 , ' $ %& ',
1 Z
.Z Z
@f
=
lim
@ z
2i
f z
f d
C
dd
G
& & .
0 . & # . 1 11.24.
G
z
11.30. ' $ %& & .
3 , '
ZZ
$ = 1 ' dzd
'
G
F z
G
&& &! @F
= $ , & @ z
11.29.
11.31. 3 , & & 11.30 " # &
@f
= $
@ z
1 % &
& & &
&
! @F
@ z
G
' z
' z
G
$ = 21i
Z
f z
C
0 .
f d
z
1
ZZ
G
' d d
z
$
:
& # . 3 11.24.
3
11.28 11.31 .: 1 2. 3. 4''5&
".:| .: 6 , 1959.
I
1.1. 1
1
2 i 3 1 + 3i 4 8.
5
1.2. 1 3
arg z 2 2
i
2
p
p
p
3 2 4 2 3 5 29 arctg 5
4
2
p
p 4
7 29 arctg 5 8 29 arctg 5
2
2
p2 2
b
b
p
arctg
6 29
9 jbj
jbj = sgn 4
2
b
2 b
10 a + b , arctg %& a 0, arctg + %& a 0 b 0, arctg
a
a
%& a 0 b 0.
2k
1.3. z = cos 'k + i sin 'k , ' 'k =
, k = 0 1 ::: n 1 z = 0.
p p
2
1
2
1.4. 1 1
4
2
6
2
p
i 3
i
+
n
p2 p 2 p2
3+
3 2
2
p
p2
2
2
i
5 1 i
p2
p
i 3
i
1 + i
i
3
p2
1 + i
2
5
2
p2
2
1 i
i
1 i
pp22 + 1 pp22 1
7 2 +
i
p 2h 2k + 3=4
2k + 3=4
8 2 cos
+ sin
= 0 1 2
p5 h 2k +31 arctg 3=4 3 2k + 1 arctg 3=4 i
i
i
i
9
5 cos
1.15. zk = z1
5
k
+ i sin
2k
2k
cos
+ i sin
n
n
1.16. z3 = z2 + z2
k
5
=0 1 2
::: n
k
1.
=0
:::
4.
cos 2n sin 2n . 1.17. 4 = 1 + 3 2 .
z3 z1
1.18.
z2 z1
.
z z z z
1.19. 1 2 3 4 = 1 3 : 2 3
z1 z4 z2 z4
.
z1
i
z
z
z z z z
4
sgn
b
n
:
sgn
sgn
b=1
b= 1
b 0
b 0:
z
z
I
205
1.20. 0 . 1& ,) & *2, %& ,+, &+ & , ,,
&,&, - %& %&, ,) zk & cos + i sin . 4' , Re zk 0 Re 1=zk 0 %& 5*, k.
1.23. 6& &' & R 7&, z = z0 8' ) &' &) ' ) &'.
1.24. 9% :, - z = 2 *+ %5 5=2.
1.25. 6& + '%&* :, - z = 2
++ %5 3=2.
1.26. 1& , , %&% & &, 52, z1
z2 , %&- 2 & & 8' &.
1.27. 1 1& , x = C %%, &%) %&
2 %%, &%) %& ,+ y = C .
1.28. 1 1 y 0.
1.29. 6& ' &)2 %)5 ++ &+ & &,, *&52,
++ 5 ', & &
' ) ' &+ z0 .
1.30. 1&* y 2 = 2x + 1.
1.31. 1%, '& %& ,+ x + y = 1 &)2 &.
1.32. 1& , , %&- 2 & z1 z2 &+ 5
z2 &), ,&, &+ ) &, 52+
z1 z2 &+ z2 . p
p
1.33. 6& &)+ jz ij = 2 jz + ij = 2, 5 , - *2+ .
1.34. 1 6& *, '& + &, 0 6 x 6 2
++ , , %& .&-, r = '
2 ,) , %& %. 1 % &,
0 2 ++ .
1.35. 1 ;,+ &)+, 52- & ,,+ , , ,, & ,+: C x2 + y 2 = x
2 ,+ &)+, 52- & ++ , , + .
1.36. 1 ;,+ '%&* x2 y 2 = C
2 ,+ '%&* xy = C=2.
1.37. <) | &), 52 ',& , ,, , & + &- z1 z2 % &) .% z1 z2 .
1.38. ;,+ ' &)+ 7, - z1 z2 8
,+ - ) %& ,+- & 7, z1 z2 8- & &) * 5 .
1.39. 1 <) | ',& , , %&
& + &- z = 1 z = 1 % , :, z = 1. 1& 1 ,+ | %& , &,
206
1 ,
! 0 1. = 1
r2 = 2 cos 2'.
2 ! " z1 z2 , z1 , z2 | r
z 2 + az + b = 0. !
r
% , jz2 2 z1 j ,
r
, jz2 2 z1 j . = jz2 2 z1 j %% % z1 +2 z2 .
1p
1p
1.40. 1 jz jmax = ' a2 + 4 + a , jz jmin = ' a2 + 4 a
2
2
p
p
2 jzjmax = 21 ' a2 + 4jbj + a , jzjmin = 12 ' a2 + 4jbj a .
1.41. ) * + r = '.
1.42. !" * r = e' .
2 3 2 4 5 0.
1.43. 1 2
3
x
y
x2 + y 2
1.44. = 2 2
,
= 2 2
,
= 2 2
z = x + iy = 1+i .
x +y +1
x +y +1
x +y +1 p
p
1.45. '1=2 0 1=2 , '1=2 0 1=2 , '0 1=2 1=2 , ' 2=4 2=4 1=2 .
. / 0 , 0 =2 =4 ' * , /1 .
1.46. 2/ * tg ' =2 + =4 3 z = 0. 45/" , 4 " | 7
.
1.47. 1 % 2 8 % =
= 2 arctg r =2.
1.48. 1 9 * / %
2 , * *
' . . 1
3 , * 0 ' . .
% % 8 , 1 .
1.49. z1 z:2 = 1.
1.50. " 180 , *
% *% z- .
1.51. 1 . 8 2 8
3 8 =2 =2 ' |
4 8 =2 j j
5 / 8 6 8.
1.52. )% / %, 1 4 " ' 3% %, 1% , 7*8 / *, %, % *% 1%
d, | / *, /1 ,
% arctg d *% .
1.53.
/ , 1 4 %" .
I
1.56. kz a =
p1 + jjzzj2pa1j + jaj2
207
kz 1 =
p
p1 +1 jzj2 .
z1 = p1 + jz1j2 , ( #& # , + ,%& # & zz
z2
1 + jz2j2
p
jz z1 j = jz z2 j, jz1 j = jz2 j, ,%& # jz z1j = 1 + jz1 j2 ,
z2 = 1. i i=2 i=2 p i=4 p i4 p 3i=4 p 3i=4
1.59. 1, e , e , e
, 2e , 2e , 2e , 2e
.
1.60. i 1k
1.61. e2 , 1 e2 , 3 e3 , 4 2 e3 , 2 4 a, ' , ' 0,
' + , ' 6 0 1, ', j'j , , j'j = 2 sin 2
++
+3
.
2 , + 6 ,
2 , +
n + 1x
nx
nx
n + 1x
sin 2 cos 2
sin 2 sin 2
2
nx
1.62. 1
4 sinsin nx
2
3 sin2
x
x
2
sin
x
x
sin 2
sin 2
sin n +2 1x cos nx
cos n +2 1x sin nx
2
2,
5
, n | &#&
cos x2
cos x2
n | #& .
sin n +2 1 n
sin n +2 1 n
1.63. 1
2
cos + 2
sin + 2 .
sin 2
sin 2
1.67. 1 sin z = sin x + iy = sin x ch y + i cos x sh y ,
p
j
sin
z
j
=
sh 2 y + sin2 x
p
2
2
2 cos z = cos x ch y i sin x sh y, p
j cos zj = sh y + cos x
sin
2
x + i sh 2y
sin2 2x + sh 2 2y
3 tg z = cos 2x + ch 2y , jtg zj = cos
2x p
+ ch 2y
4 sh z = sh x cos y + i ch x sin y, jsh zj = psh 2 x + sin2 y
5 ch z = ch x cos y + i sh x sin y, jp
ch zj = sh 2 x + cos2 y
sin 2y , jth zj = sh 2 2x + sin2 2y .
6 th z = shch22xx++icos
2y
ch 2x + cos 2y
sin 4 i sh 2
8 + 15i
1.68. 1 cos 2 ch 1 i sin 2 sh 1 2 i sh 2 3
2cos2 2 + sh 1 4 17
sin 2 6 40 + 9i .
5 shch44icos
2
41
1.69. Im ez = 0, Im z = k Re ez = 0, Im z = 2k + 1=2
Im cos z = 0, Re z = k Im z = 0 Re cos z = z, Re z = 2k +
+1=2 Im sin z = 0, Re z = 2k + 1=2 Im z = 0 Re sin z = 0,
Re z = k Im tg z = 0, Im z = 0 Re tg z = 0, Re z = k=2
Im ch z = 0, Im z = k Re z = 0 Re ch z = 0, Im z = 2k + 1=2
Im cth z = 0, Im z = k=2 Re cth z = 0, Re z = 0. "8 k | -
k = 0 1 2 :::.
k
k
1.70. 1 Re z = +
4 2 2 Im z = 4 + 2 .
1.57.
208
ln 4 + 2ki, 2k + 1 i i 2 2k + 12 i 2i 3 2k 14 i
h
i
4 21 ln 13 + 2k arctg 32 i, 12 ln 13 + 2k + 1 arctg 32 i.
1.72.
2 Ln z ln z2 . 1, . III, . 19" .
1.73. 1 4 2 2 3 0 4 4 .
p
p
1.74.
p21 cos2k 2 p+ i sin2k 2 p
2 2 cos 2k + 1 2 + i sin 2k + 1 2"
3 e2k cos ln 2 + i sin ln 2 4 e2k 5 e k = 6 1p2i e k =
1.71. 1
2
h
1 2
2 +1 4
i
= k cos ln 5 arctg 4 + i sin ln 5 arctg 4
7 5e
3
3
h
i
=
k
8 5e
cos ln 5 arctg 34 + i sin ln 5 arctg 43 .
() k | + k = 0 1 2 ::: .
1.76.
a a )- ). / ,
)-, /0 1, a /0 .,
2 ) 2 2 1 3 2 2 3 2 , 1 1 1, . III, . 20".
1.80. 1 Im Arccos z = Im Arcsin z = 0, z | )
jzj 6 1 Im Arctg z = 0, z | )
2 Re Arsh z = 0, z | jzj 6 1.
p
5
1.81. 1
3
+
2
k
,
+
2
k
2
2
k
3
2
k
i
ln
2
+
6
p 6
p3
4 2k
h i ln1 2 1 , 2ki + 1i i ln 2 + 1
1
5 2 arctg 2 + 2k + 1 + 4 ln 5
h
i
p
6 ln 5 2 + 2k 12 i 7 14 ln 5 + 12 arctg 2 + k + 12 i.
(-). k | + k = 0 1 2 ::: .
p
p
3 7
3
1.82. 1 z = + 2k i ln 2 1 2 z =
+
2
k
i
ln
p
4
4p
2
p3 1
3 z = + 2k i ln p z = 3 + 2k i ln 3p+ 1
arctg 4 3+2
arctg 4 3+ 2 +1
2
2
2
4
2
4
2
4 z = 2ki 5 z = ln 2 + 2k + 1 i
6 z = 2k + 21 i z = ln 3 + 2k 12 i.
(-). k | + .
2k + 1
3 z = 24k1 ++ 12i
1.83. 1 z = k 1 i 2 z = k 1 + i z =
1+i
z = 24k1 12i . (-). k | + .
1.95. 45 ) / - . 1.96. 45 ) / - .
1.97. 65 ). 1.98. 45 ) / - .
1.99. 45 ) / -
1 ' 6= 2k k = 0 1 2 ::: , 15 )
1 ' = 2k.
1.100. 45 ) / - . 1.101. 65 ).
I
1.102. 45 ) / -
1.104. 45 ) / -
1.105. 1 z = 0 z = 2
209
. 1.103. 65 ).
.
2 z = 0, z = 1=m, z = i=n | m n | -/3 +3
3 2 2 4 2 21. jzj 6 1.
1.107. 2 7 ) lim zn 0 1.
n!1
1.109.
: x = 1, 2 6 y 6 0. 1.110. y = x2 .
1.111. y = x2 .
1.112. ! " ! a ! # $ z = 0.
1.113. % " y = 1=x, & " .
1.114. 1' % ( ! " ! 1 ! # $ z = 0)
2' $ " ! ! 1 ! # $ z = 0, &
.
1.115. 1' * : x = a+t sin t', y = a+1 cos t')
2' +!$
$ +a b'
y = a b cos t.
$ ' +a b', +a = b' # : x = at b sin t,
1.116. 1' ( x = C - ! C 6= 0 u =
= C v =+4C ', C = 0 !" v = 0, u 6 0) ( y = C
- ! C =
6 0 u = v =+4C ' C , C = 0 !" v = 0,
u
0) y = x ! !" u = 0, v
0)
! jz j = R - ! ! jwj = R ) $ arg z =
= | $ arg w = 2) $
- ! x = C ,
y=C C=
6 0 $ arg z = )
2' ( u = C - ! x y = C +
2
2
2
2
2
2
2
2
2
C = 0 | (',
( v = C | xy = C=2
+ C = 0 | ('.
1.117. 1' ( x = C - ! ! u2 + v 2 u=C =
= 0, C = 0 | !" u = 0) ( y = C - ! !
u2 + v 2 + v=C = 0, C = 0 | !" v = 0) ! z =
= R - ! ! w = 1=R) $ arg z = - !
$ arg z = ) ! z 1 = 1 ! u = 1=2)
jj
jj
jj
2' ( u = C - ! ! x2 + y 2 x=C = 0,
C = 0 | !" x = 0) ( y = C - ! !
y
x2 + y 2 + = 0, C = 0 | !" y = 0.
C
1.118. 1 # w = z + 1=z
!
= 0,
u2
v2
+
= 1,
R + 1=R2
R 1=R2
2 6 u 6 2) 3 # w = z 1=z
2!
u = 0,
14 ..
.
jj
! " z = 1
v2
u2
+
= 1,
R 1=R2
R + 1=R2
2 6 v 6 2.
jj
6
! z = R = 1 2 v=
jj
6
! z = R = 1
jj
! " z = 1
-
210
u = C xx2 + y2 + 1 =
1.119.
= C x2 + y2 v = C | yx2 + y2 1 =
= C x2 + y2 . 59.
1.120. , ! "# "#
$ w = 1=4 $ w = 1.
1.121. 1 &$ '# = eC , = C , " = e
2 # y = ex + 2k.
1.122. 1 ) *+ x = C v 2 =
= 4a2 u a2 + 1=4 * ,$ $ w = 1=4, a = C + 1=2,
$ + $' , $ w = 1=4 ! "#
"# a = 0 y = C ,$ #*+ v2 = 4C 2 u + C 2 + 1=4 $ + $'
, $ w = 1=4 "# ' "# /
#
2 x = C $ '# 0# # "# $ w = 1 w = 1 $/ /
# / "
# # $ '# 0#: u a2 + v2 = a2 1 jaj 1
a = cth 2C y = C $ 2 $ '#
$#! $+ w = 1 w = 1, $/ / 2 / 2 2
"# # # $ $ '# : u + v + b = 1 + b
b = ctg 2C
3 x = C
=
C 2 2 = C 2
=e
, x = 0 $ = 0, 0 6 1 4
I
211
y = C = e2 = C2 C2 ,
y = 0 = 0, 1 6 1.
1.123. * y = C
/ # u = x + ex cos C , v = C +
x
+ e sin C , $ *+ x = C | 2 # u = C + eC cos y, v =
= y + eC sin y.
1.124. 1 ) jwj = R
/ $ '# r = cos '=ln R
R 6= 1 # " R = 1 $' arg w = $ '# r = sin '=2k = 0 4 +
"# " k = 0
2 2 * x y = ln R 2xy = + 2k.
1.126. 5"$ f z = z Re z=jz j f 0 = 0.
1.127. 1 2 6 * #*, # # # #. 1.128. 2 6 3 .
1.132. 1 c = 1, b = a f z = 1 aiz 2 a = b = 1 f z = eiz .
1.133. 7 #$! # $ 0 arg z =4, arg z 5=4
f z = z =2 arg z 3=4, 3=2 arg z 7=4 f z = z .
@u
@v @u
1.135. r =
,
= r @v
. 1.138. 0.
@r
@' @'
@r
1.155. 1 6 , u 6= const 2 f u = au + b.
1.156. jf z j | , #$! # 2# $, arg f z ln jf z j | 2# $ .
@ u 1 @u
@ u
1.157. 9u =
+ r @r + r1 @'
u = C ln r + C .
@r
1.158. p = x, q = y p = x y , q = 2xy p = x 3xy , q = 3x y y
p = x 6x y + y , q = 4x y 4xy pn = rn cos n', qn = rn sin n'.
y
1.159. v x y = 2xy + y + C . 1.160. v x y =
x + y + C:
1.161. a v x y = arg z + C v x y = arg z + 2m + C .
1.162. v x y = arg z arg z 1 + 2m + C v x y = arg z
arg z 1 + C vxn y = arg z arg z 1n + 2m + C .
1.163. v x y =
mk k + C
k arg z zk + 2
4
2
2
2
2
2
2
2
4
4
1
2
1
4
2
2
4
1
2
2
2
2
3
3
2
2
3
3
2
2
X
X
vx y =
X
arg z z + 2m
k=1
n
k
k=1
X
k=1
n
k
k
k=1
+C
2
3
3
2
X
n
k=1
k
= 0, , #$-
! vx y ## # .
1.164. 1 ) 2 3 # .
1.165. f z = z + 5 iz i=z + Ci.
1.166. f z = zez + 2i cos z + z iz + Ci.
1.167. f z = 1=2z + iz + 3i + C .
1.168. f z = 2i ln z 2 iz + C . / C | "# "# ##.
1.169. u = C x + C .
1.170. u = C ax + by + C . 1.171. u = C arctg y=x + C .
1.172. u = C xy + C . 1.173. u = C lnq
x + y + C .
p
C
x
1.174. u =
+ C . 1.175. u = C x + x + y + C .
x +y
2
3
2
1
2
1
2
1
2
1
2
14
2
1
2
1
2
1
2
2
2
2
2
2
212
1.176. He . 1.177. f z = ei z 2 ez .
2
2
1.178. f z = ei ez . 1.179. f z = Aez =2 .
1.180. f z = Azez |
.
az , iaz . 1.183.
1.182. + ,
+ ,
1.184. ln + , ln + , ai ln z , a ln z .
A|
iaz + , az + , eiaz , eaz .
,
az iaz e e
ai z a z e
e
1.185. a ln z + , ai ln z + , ea ln z , eai ln z .
1.186. a=z + , ai=z + , ea=z , eai=z a | , | .
p
1.187. w = z 2 : 1 # = 0, k = 2$ 2 # = , k = 1=2$ 3 # = =4, k = 2 2$
4 # = arctg 4=3 , k = 10.
w = z 3 : 1 # = 0, k = 3$ 2 # = 0, k = 3=16$ 3 # = =2, k = 6$
4 # = 2 arctg 4=3 , k = 75.
1.188. 1 * jz j 1=2, jz j 1=2$
2 jz + 1j 1=2, jz + 1j 1=2$
3 jz j 1, jz j 1$
4 Re z 0, Re z 0$
5 Z
jz 1j 1.
Z
Z jz 1j 1,
jf z j2 dx dy, L = jf z j ds.
1.189. S =
l
p G
1.190. 2 e2 1 . 1.191. 2e2 e2 1 .
1.192. +,- D . e 6 jwj 6 e2 . / 0
0 1.189 , , 03 .
4 II
2.1. w = 1 + i 1 z . 2.2. w = 2 + i
2.3. 1 z0 = 1 + 3i, # = 0, k = 2, w + 1
2
z0 = 2 + 2i, # =
,
k = 1, w
z + 1 3i.
3i = 2 z + 1
2i = i z 2 2i $
2
i
3 $
2
3 0 0 $
4
= 1, 0 0 $
w1
a
az1
,
1 a
w1
az1
=a
1 a
w1
az1
$
1 a
a 6= 1,
# = arg a, k = jaj, w
z
a = 1, 0
0
$ a 6= 1,
b
b
b
z0 = 1 a , # = arg a, k = jaj, w 1 a = a z 1 a .
2.4. 1 w = az + b$ 2 w = az + b$ 3 w = i az + b $ 4 w = az + bi.
5 a b | 3 0 a 0.
2.5. 1 w = z + bi w = z + 1 + bi$ 2 w = z + b w = z i + b$
3 w = z + b 1 + i w = z + 1 + b 1 + i .
5- b | 0. 6 6 6
z0 =
5
0 , , 6. 3, 37 37, 037 3.
+, 0, 3 0 3.
II
p
213
w = z h a $ 2 w = z +ha + h + i$ 3 w = 1 b+ k e i=2+arctg k z$
p 2
1+k
e i=2+arctg k z ib1 .
4 w=
b2 b1
2.7. w = ei Rz + w0 .
2.8. 1 * 37 u = 1=a, 37 -0- - $
2 37 v = 1=b, 37
-0- - $
3 b u2 + v 2 + u + v = 0, -7 0
v = u -0- 8 - $
4 0 37 v = ku$
5 0 , 77 0 0 0 0 w0 = 1=z0 8 0 7 , 7 0 0
w = 0 w = w0 $
v3
6 . u2 =
.
v+1
2.9. 1 5 , -7 0 w = h ,
:
C x0 u h1 2 + v h2 2 u h1 = 0
C y0 u h1 2 + v h2 2 v h2 = 0
" z0 = x0 + iy0 , h = h1 + ih2
2
$ % w = h jw hj = 1=R
, ' ( w = h arg w h = .
2.10. 1 *
$ +
z z1
z1 z2: z z = . , % A B , $' " 2.6. 1
2
2
, ' ( z1 z2 . 60, ( z1z2 / 0(
/ . 1 ( 0 ( , ( C | % $ -
214
AB , z1 z2 j = d, 1 2
= cos = rR1 = rR , R = 1 d2 , r1 = 1 d2 , r2 = 1 1 2
2
2 , z1 z2: arg zz zz1 =
2
= . #, $ $
= = ,
% %
3 % $ $ ' . 61 ,
)%% zz zz1 = R j
2
% * arg zz zz1 = '% 0, % 2
$ $% z1!
z2 % 0, | %$
4 $ % $ $ *
. 61
%.
2.11. . % jwj 1, Im w 0.
2.12. . % , w = 0
jwj = 1 jw + 5i=4j = 3=4.
2.13. . % , % %% 'Im w 0 % p $ 1 %% jw 1=2 + i=2j
2=2.
2.14. 1 . % , Re w = 1
jw 1=2j = 1=2
2 $ % ,
jw 1=2j = 1=2 jw 3=4j = 1=4.
2.15. . $$ % , 2 Re w =
= 1=2 jw 3=4j = 2=3.
2.16.
1 w =
+ hi % w = d
d2
1
d
z
d2
+ 1 + hi % w = dz + hi
d2
z
1 + 1 + hi 3 w =
2 w =
d2
d1 z d2
.
z d1 + d2
d1
d1
d2
z
1 +
II
215
2i z + 1
1 + 2iz + 6 3i .
4z 1 5i 2 w =
5 z i
1 + iz + 1 + 3i 2 w = iz + 2 + i 3 w = 1 i 'z + 1
2.18. 1 w =
1 + iz + 3 + i
z+1
2
1 + 3iz + 1 i 2 w = z 1 4i 2 1 i
2.19. 1 w =
1 + iz 1 + i
2z 1 i 4 i
z 3 i 1 + i
3 w = 1 + i 1 z .
2.17. 1 w =
w = izz i1 $e %% $ * .
az + b
2.21. 1 w =
, a, b, c, d | $ % * % ad bc 0
cz + d
az + b
2 w = cz
+ d , a, b, c, d | $ % * % ad bc 0
az + b
3 w = i cz + d , a, b, c, d | $ % * % ad bc 0.
2.22. 1 w = 2='2 z 2 w = 2'2z + 1='z 2.
R z
2.23. w =
$ % $% % u 0, v 0.
R+z
2.24. 1 '2 + i=5 2 9=2 + i.
2.25. 1 jz j = p
2 2 x = 1=2 3 jz i=4j = 1=4 4 xx0 + yy0 = 1=2
5 jz z0 j = jz0 j2 1 ' . . 1
2.20.
%
6 'x2 + y22 'x2 y2 = 0 '%
7 $% * % $ $ 1=z51 , 1=z52 , 1=z53 ,
$% ,
$ z = 0 ' .
=2i
2.27. 1 'x = + 2 arg 'x
2 w ' = e 2b
3 % b 2, $ %% %
p b 2, 5
$
%
,
%
$
j
z
j
2b. '6
p
jz 5j = 2b *$ .
z i
2 w = i zz + 22ii 3 w = ei =2+ zz aa + bibi .
2.28. 1 w =
z+i
0
w = Ri zz + ii + w0 . 2.30. w = zz + 22ii . 2.31. w = 4 z zi2+ 24i .
z z1
2.32. w = ke +argz2 =z1 i=2
, k 0. 7 , $*
z z2
w = 0 $ %% Re w 0, $ $ $ z-% , % $ jzj 1 z1 z2
% $ %% Re w 0 % 2 $
w = 0 $ $ $ jzj 1
)%% % z1 z2 .
z z
jz z j
2.33. w0 = 2 1 , R = 2 1 .
z2 z2
2jIm z2 j
za
a
za
wb
i
= e z a . 2.35. ww
2.34.
wb
a = i z a.
sin ' sin #
2.36. 1 '' = ' + 2 arg 'ei' a = ' + 2 arctg
cos ' cos # ,
a = ei# .
2.29.
216
i
= 1 jaj2 ei , w a = 1 e jaj2 .
3
pa 6= 0, ,
jz 1=a j 1=jaj2 1, p , .
jz 1=a j = 1=jaj2 1 ! .
a = 0,
jw z j 1.
1 + jaj
dw 1 jaj
4 max dw
dz = 1 jaj , min dz = 1 + jaj .
2z 1 # 2 w = 2iz + 1 # 3 w = iz# 4 w a = ei z a .
2.37. 1 w =
2z
2 + iz
1 aw
1 az
wb
z a . 2.39. w = 1 z . 2.40. w = 2 z 2 + i .
2.38. R2 2 = ei R1 2
R2 bw
R1 az
z+2
iz + 2 2i
2 w0 0
0
0
2.41. 1
a|
2.42.
w = R2 e
w
1
a
aw
i
a
az
z
R2
2
i' z
=e
z1
,
z1 z
1
w b
i z
=e
R2 bw
R2
a R.
'=
j j
arg
z2
a
az
3
w = R2 Rz2
z1
,
z1 z2
1
a = j1 z z j + pjz12 zj1zj j2 1 jz j2 .
p
p
1 2
1
az 1 + 1 a2
1 1 a2
2.43. w =
p 2
, =2
.
a
1 1 a z a
p
p
z+2 3
z2+ 3
2.46. 1 w = z 2 w =
p
3 w =i
p
.
1 + 2 3z
1 2 3z
2.49.
$ $
k=1
z
,
0 ! " #
", !" ! % .
2.50.
w=
R kiz R2
,
z R + ki
w=z .
a
,
az
k
h
|
,
k 6= 0
2
'
jaj sin 2 , )$ $ # . ( ) $i +=2 =2 tg , z2 = ei +=2 =2 ctg , tg 0 =
jaj = sin sin , z1 = e
2
2
2
2
w z1
i z z1 .
=e 0
= tg cos , )$ $ ! $ $
2
w z2
z z2
1
( jaj = sin , )$ $ $$) !
=
2
w z0
1
i +=2 =2 , h = tg . ( jaj sin , )$ =
+ ihz
%0 , z0 = e
z z0
2
2
i +=2 + ,
$ ) . ( ) $ : sin = jaj sin , z1 = e
p
2
2
2
K = cos=2 + pjaj 2 sin 2=2 , )$ $ $ -
cos=2 jj sin =2
w z1
z z1
=K
.
w z2
z z2
2.56. . = 2 arctg 'a=h . = 2a=h + O/'a=h 3 0 !$! a=h . =
3
2h=a + O /'h=a 0 !$! h=a.
2.55.
(
II
217
x0
+ 2 arctg 1 x0xsincos = 2 arctg 11 +
x0 tg 2 # / , x0 0,
0
/ , x0 0.
2.61. 1 w = 20=z # 2 w = 2iz +
1 + 2i .
r
p
2
z h2 R2 i
h
p 2 2 e , = R + Rh 1.
2.63. w =
z+ h R
5
4z 3
4z + 3 .
2.64. h = # w = 2ei
w = ei
4
4z + 3
4z 3
2z ei
z + 24 i
2
2.65. w =
w=
e ,= .
z + 24
3z
3
z z10
z z20
2.66. w =
w=
, % | ( (
z z20
z z10
! , z10 = z1 + u1 z2d z1 , z20 = z1 + u2 z2d z1 , d = jz2 z1 j,
p
1
u1 = r12 + d2 r22 0d2 r1 + r2 2 10d2 r1 r2 2 1
2d
p
1
u2 = r12 + d2 r22 + 0d2 r1 + r2 2 10d2 r1 r2 2 1 #
2d
1 = d + r2 u1 r1 u2 :
pd + r2 u2 t1 u1
2.67. 1 = 2# 2 = 5 + 2 6.
2.57. / =
2.69.
, .
2.71. . 62. " # -
218
.
| , !. "
# 7! % , 8! 9! % (
120
60 .
-
. . /, /0 (
3!29! 0 ( /
/
% ( !.
2.72. 1! w = ei z #
4!
w a
1 + aw
= ei
2!
z a
1 + az .
w 1 = ei z 1 # 3! w i = ei z i #
w+1
z+1
w+i
z+i
2.78. 1! 2! - #
3! %
5 " | 5( " % ! 0
,
0 !#
4! /
5 / " (
5( " a 0 , 0 !
0 !.
2.79. 2! " 5 " /
'1 ,
'2 , '3 OAB / 7 = '1 +
+'2 + '3 !, ` 0" AB , A B | 5 "
'2 '3 AB 0 C . 7ABC .
2.80. 1! w = i z 2
2.81. 1! w =
z2 #
a2
2.82. :( /
q
q
w = z p2 i p2 .
qz
A z + 1 2 .
2! w =
# 3! w =
a
2 z 2
- : u = R cos ' + m cos 2'!, v =
= R sin ' + m sin 2'!. <
w-
w = Rm, ( !: = R 1 + 2m sin !. -
m = 0 - (
/, m = 1=2 | w = R=2.
:(
jz j = r
1 a2 !#
2!
0 - , 0 w = Rmr2 : = Rr 1 + 2mr cos !.
:( arg z =
= | (,
: m u sin 2 v cos 2!2 +
+ R sin u sin v cos ! = 0. = = 0 = 0 / 0 6 u 6 R 1 + m! R m 1! 6 u 6 0.
2.83. :( / ( u = v 2 = 2 cos =3,
jj 3=4 . 63!.
II
2.84. 1! :( /
= R sin ' +
(
sin n'
n
,
% : u = R cos ' +
sinnn'
cos n'
n
219
,v=
0 n 1! , 0
p1 1#
z
2! > / :
= n
u=R
cos ' +
cos n' , v = R sin '
n
p
z = n+1 1!.
,
0 n + 1! (
2.85. 1! jmj 6 1=n. :( / % % !, . . ,
mR
R=n, R n 1!=n.
2! jmj 6 1=n. ? > /
, / ( 0
> / 5 "
!.
2 p3 4 + 1e i=3 z4=3 p .
2.86. 1! w = z 1= # 2! w = p
3 4 2e i=3 z4=3 + 3 3 4
1 + z 2
z2 + 1
2z2 + 3z 2
12 z # 2! w = 22z # 3! w = 2z2 + 3z + 2 .
z + 2iz + 1 # 2! w = 2z + 3iz + 2 .
2.88. 1! w = 2
+22z + i
2z2 3iz + 2
z iz
1
2.89. w =
z+ 1 .
z1= + R1= 2 # 2! w = z1= R1= 2 .
2.90. 1! w = 1=
z p
R1=
z1=+ R1=
p
3=2
2z + p3 i # 2! w = 2z + p3 i 3 #
2.91. 1! w =
2z + p3 2z i3 3 i 2z + p3 i 32=z2 3 ih z p21 i i4
3! w =
2z p3 i # 4! w = i 2z p3r i # 5! w = z p21 + i .
z + 1 2=3
z+1
2.94. w = ei=3
z 1 . 2.93. w = 1 z .
2.87. 1! w =
p
2.94. w = z + i!= i z !. 2.95. w =
p
2.96. w = z + R!= z R!.
2.97. w = ei=8
p z i.
p
z z1 != z2 z !.
2
2.98. w = ei=2arctg 2h=1h
r
z + 1.
z1
p2 2
z2 + h2 . 2.100. w = z z+ h .
r
r
2
z
1
z1= 1 2 + tg 2 .
2
2.101. w =
+ tg . 2.102. w =
2
2
z1= + 1
rh z + 1
i
2
2
=
3
2
=
3
1
z
1
+
z
1
2.103. w =
+ .
2=3 + 1 + z 2=3
3
1
z
pz + 1 2
p
2.104. w = p
z 1 . - ( z %
2.99. w =
p
> 1!, | 2!.
2.105. w = z= 1 z !2 .
220
2.106.
2
z
j j
= R . 64
2
R +4u1=R2 + R 4v1=R2 = 1 jzj = 1 | v = 0,
1 6 u 6 1! "# arg z = $ %&
u2
v2
= 1 " arg z = 0 | " v = 0, u 1, " arg z = | "
cos2
sin2
v = 0, u 6 1! "# arg z = =2 | ( u = 0.
4u 2
4v 2
2.107. 1, 2 )* ( #
+
=1 . 65, 1, 2!
2
R +1=R
R 1=R2
3, 4 ( # - 1 1. . 65, 3, 4!
5, 6 ( # # /( "0 1 1. -1 1, #2$ # /0 (0 !
II
221
7 (! 8 $ (!
9 $ (!
4u2
10 $ # # R +1
+
=R2
11
# #
12 ## # #
h
i
# /( #, 1 21 R + R1 !
13 &# ( / %&
4u2
R
1=R2
4v 2
R +1=R2
R 1=R2
2
4u2
+ R 4v1=R2
R + 1=R2
u2
sin2
+
4v 2
v2
cos2
= 1!
= 1!
=1
= 1.
2
jzj = R R 4u1=R2 +
2
+ R +4v1=R2 = 1 jzj = 1 | u = 0, 1 6 v 6 1, "#
2
2
arg z = %& sinv2 cosu2 = 1 "# ' = 0
' = ( v = 0, " ' = =2 | " u = 0, v 1, "
' = =2 | " u = 0, v 6 1!
2
2
2
jzj = R R +4au2=R2 + R 4av2=R2 =
= 1 jzj = a | v = 0, a 6 u 6 a, "# arg z =
u2
v2
%& a2 cos
= 1 " arg z = 0 |
2
a2 sin2
" v = 0,u a, " arg z = | " v = 0, u 6 a, "# arg z = =2 |
( u = 0 !
2.108. 1
3 0 # $ %&, "# 2$ 0 / 6 7% . #/#" 2.106
# % /& &# # 6 /& jcj 6
/& | #"# /# .
p
p
ei
2.109. 1 w =
z + z2 c2 ! 2 w = a +1 b z + z2 a2 b2!
c
&$ "# $ / & "# &# #
* ( /"%
# /% | # % (.
p %#,
az b z 2
a2 b2
2.110. w =
.
a2 b2
p
2.111. w = A z + z 2 a2 b2 , %/ A | (
b
"! = p 2 a2 p
.
a + k b2 + k2
h
i
2.112. ) ( #
1 12 a + a1 , a
i
0! ( # # "#
1 21 a + a1 - 1 +1,
a 0.
r h
r
1 i
1 + z + 1=z =2
1
1
2.113. w =
.
2.114. w =
a
+
z+z .
5=4 z + 1=z =2
2
a
222
p 2 2 2 2
2.115. w =
z + 1 + a + 1 . 2.116. w = z +1=z + +1= .
z + 1=z
r 2 h z a
i
1
1
1
2.117. w =
2 + 2 + z 2 + z 2 .
2
h
i
i
h2
h2
2.118. 1 +
w + 1 = zi + e
.
41 h
w
e
z
21 h
1 1 a2
2
1
4a
2.119. w + =
z+
2
w 0 = 1 + a
2
2
w
1 + a
z
1 + a
4a
r
1
0
2
, , 2 arccos 6a1 +1 a2a
p
a = 3 8.
1
2z + 1=z a + 1=a + b + 1=b
2.120. w + = 2
w
a + 1=a + b + 1=b
w 0 = a + 1=a + b + 1=b
4
, !,
1=a + b + 1=b
+ 1=a + b + 1=b
2 arccos 4 aa+ +1=a
2 2 arccos 4 a a
:
+ b + 1=b
+ 1=a + b + 1=b
0
2.121. 1 "#! $% C & && $% $ '!
($ 1, $ & ($ 1 ! 2 $ )
* ) $% #%& $) !
$
2 #! $% C & && . 66 !$ & $ &
($ w = 1, (! $ ) & , ($ # ) ) 2 $% $ '! 1,
$ () . 1 , % & # , ( . !
/$$ * ) $% C #%& * ) . & /$$.
2.122. 1 "#! $% C & && !$ & $ &, && $% # ! $ '! ($ 1, (!
($ 1 $ ) $ , ! ! # ) ) , 2 2 + * ) $% #%& * ) # , ( $ ! !
II
223
#! $% C & && . 67
!$ & $ & ($
w = 1, (! $ ) , ($ $ !
2 2 + $ )
# $% C % &
# , ( #! $% C * ) $% #%& * ) # , $% .
2 1 ) $% C #%& * ) # , ( ! $% , &
( ($ 1, 1, (! ($ 1 $ ) $ , ! $% &!
# ) ) , : 2 +
+ , 2 + 2 , . $' & wz z- $
! $% C , % % $
# 2 , 2 + , , & . $' wz , $% C , % $ .
w 1
zei + iei=2 2
2.123.
=
, = ,
0, = + ,
i=
2
w+1
zei + ie
0.
2.124. 1& $) ! (! y = 0, x 6 1=2
y = 0,
x 1=2.
2.125. 3 $) x 1=2 ! $ y = 0, 1=2 6 x 6 1.
2.126. 1& $) ! ) ( y = 0, 1 6 x 1 y = 0,
1 x 6 1.
p
2.127. 4 =n arg z =n ! ( y = 0, n 1=4 6 x
1.
1
2k
p
2.128. 1 1& $) ! (! jwj
n 4 , arg w = n
n
2
=n
k = 0 1 ::: n 1 2 w = 1 +pnz4 z .
p
p
p
1 p
2.129. 1p w = z 2 + z 4 1 = p z 2 + 1 + z 2 1
p 4
p 2 p
2 w = 1 + z 1 z = zp1 2 1 + z2 + 1 z2 .
0
0
0
q
n z n + z n + pz n + z n 2 n + n 2 .
6 * . 7 $' & = 21 zn + z1n #% $ %
1
1
n
$), (! ($ ei=n & ($
+
.
2
n
8 , %) $) p = n + n =2 # )
p
( $ = p + p2 1 . 7 $' & w = pn # $!.
2 w = n=2 + n=2 2=n zn=2 + z n=2 +
p
+ zn=2 + z n=2 2 n=2 + n=2 2 2=n .
2.130. 1 w =n + n
1=n
224
rp
p2 2
pp
2
2
p
pbz2 ++cc2 + paz2++cc2 , w = 1 z2 + c2 + + z2 + c2 + 2 2 ,
p2 2
p
p
p
= 21 a2 + c2
b + c , = 1 a2 + c2 + b2 + c2 .
pp
p2 22
2
2
2.131. w =
2.132. w =
z + c + a + c .r
z + i 2
r z + i 2
1
2.133. w = p
+
1
+
1 .
z i
2 pz i
v
u
z2 1 + z i
2.134. w = u
t p2 + 1
p2 .
z i
z 1
+1 v
r
u
u
z12 + ctg 2 + ctg
u
2
u
2.135.
b 1 w = tr
r 2 2
2
2
b1 + ctg
z1 + ctg 2
2
2
r
q
z12 + ctg 2 + ctg
2
2
v
r
u
u
z12 + ctg 2 + ctg
u
2
b1 w=u
r 2 2 z1 = 11 + zz , b1 = 11 + bb .
tr 2
z + ctg 2 + b + ctg 2
b=1w=
1
2
2
pz2 + 1=z2 + b2 + 1=b2 + pa2 + 1=a2 + b2 + 1=b2
2.136. w =
.
pa2 + 1=a2 z2 + 1=z2
pp
pp
p
1
2.137. w = p p
z4 + 4 + 2 +
z4 + 4
5.
2+ 5
. = z2 !"" ##$$ % $ && ( & ( )0 1 + i+, )0 1 + i+ # $ % & , 2.132, "
( !"" ##$$ %. . / 0 , $ # $ (( , " $#
# $ % #$$ % $
& &( ( &. 1$ $ & % $ % 2 & $ % p
, . p
i
2.138. w = p )z + z 2 1 = + z + z 2 1 = + 2+1=2 =
2
p
p
= pi2 )z + z2 1 =2 + z + z2 1 =2 +.
p
. 3 (4%" 0 = = , = z + z2 1 | 0 , " 0 6 $, ! # & / # $ $ # $ % j j 1, Im 0, 0 /
6 $ $ !"" ##$$ %.
(
$ (( , #, ( $ / #7
$" #$$ % $ & &( #, 1 1+ 2 $# # $ % #
$ !"" ##$$ %.
p
( , . 8 #% 1= 2 # , & w = kw k 0 ##$$ % $ .
p
p
2.139. w = )e i z + z 2 1+ =2 )e i z + z 2 1+ =2 ,
=
.
1
0
h
p
ip
II
225
p
= e i z + cz c , c = a2 + b2 , = arctg ab , p =
= 2 arctg a=b .
2.141. 1 1 # $ % $ $ $#"4 ( &(: #% r12 jwj r22
& & $ #% & r12 6 u 6 r22 /$ #%/ $ 2 -2
( " & & # $ , $ % #% : r1 jwj r2 ,
0 6 arg w 2 $# = , #% | # $ 7/
/ $# $# % $ $ 7( ( #% 2 ( $#,
#, ( # $ #% r12 jwj r22
2 $# a 6 1, $ jz2 1j a # # $ $(. & , 1.39, & 7! $ # $ 7/ jw 1j
a $# a 1, $ jz 2 1j a # # $ %,
$ # $ 7/ jw 1j a # , 7
$ % 2 # $ 7/ , $ , 7! 2&(#
jw 1j a & & % #% -# $ $# % /
/ & & 1- 2&(# $ ! ( ( & & 2- 2&(# ,
! / / & & 1- 2&(# | $ ( ( & & 2-
2&(# .
2.142. 1 1 # $ % $ $ $#"4 ( &(: #$$ w,
& & / #% & ) 1 1+, # $ $ % 2## $
4u2 + 4v2 = 1, & & #% & ) 1 1+, R + 1=R2
R 1=R2
,( ( " & & #$$ # $ ! / / && 2## $ , !( " & & #$$ | / / & &
2## $
2 # $ # $ % ww + 11 R2 $ ww + 11 R2 # $ % , $# R 1 ##$$ % R = 1 $ %
$#, , R 1. & & $ $ "4 $#/ #%
# , $ "4/ , w = 1 $ /-# ,/ ,/ # $
w 1 R2 .
w+1
2.143. 1 2 !$ % $$ & ! # $ #$$ z , && 7! & ) 1 1+, ,( / / & & # $
$# $ ! ( ( , ! / / & & # $ |
$ ( ( .
2.144. 1 2 !$ % $$ & ! # $ , & & 7!
#, (, 4 ( $ $ & , i, 0, i $,$ %, ( ,
###% /$ #%/ $ # #%( # . .
& & # $ $#7 $ ! ( ( $ $ "4 !
& & # $ , .
2.145. !$ % $$ & ! # $ #$$ z , & & 7!
#, ( 1 1+ )1 1. ;#% #, 1 1+ $# & $ $#"4 ( &(: ! / / & & # $ $# $ $ ( ( & & # $ , ! / / & & # $ |
$ ( ( & & % # $ ! / / & & %
# $ | $ ( ( & & # $ . ;#% #, )1 1 $#
$# %: / / & & # $ $ ! ( ( & & # $ , / / & & # $ | $ ! ( ( & &
% # $ / / & & % # $ | $ ! ( (
& & # $ .
2.140. w
15 ..
.
2
2
226
2.146. 1
= const, = const
b=k
2 = e
k = 0 = b
3 = 0 = 2 |
!
4
= e
5 1, 0 = 2 | !%
!
v = 0, 0 u 1
6 ' 1, 0 = 2 | ( !
v = 0, 1
u
7 ' e e ,
= 2 * '
+ + = , e
e .
6 6
1
6
2.147. - 0 arg z + n =n
2.148. 1 !
= C
v 4 u 0, 0 v
u ln r2 , 0 v 2 .
2.149. 5 b = a th 0 =2 , l = a th 0 =2 .
2.150. 1 /
x = C '
0 %1 u2
v2
' 0 1 1
=1
y = C |
cos2 C
sin2 C
u2
v2
0 %1 *
0
+
=1
2
ch C
sh 2 C
2 %
3
5 ln r1
j
0
j
j
j
2 1
3 % !
4
30 14
5
!
!
14 31
2
2
u
v
6 *
+
= 1
2
2
1
h
3
ch
1
14 31 ch h4.
h
ch
sh
h
2.151. 1
=2 u =2, v 0
2
=2 u =2
3
0 u =2, v 0 4
2.152. 1 /
x = C '
0
0
1
2
6 6
y .
u = C, v
0
1
1
u
2
ch
2
%1 '
1
C
+
v2
sh 2 C
2.153. 1
2.154. 1
=
2v
=1
=2
0
cos2 C
2
/
x = C '
+ 1 w = i, 5
: u
a2
y = C '
w = i 5
!
2
2 u 0.
0 %1 *-
u
0
!
14 31
3 1
.
=
y
=
v2
C
sin2 C
|
=1
!
0v
=2, u 0.
!
5 2
2
+ v = 1 + a a = ctg 2C
7
-
jj
7: u + v
b =b
1, b 1 b = cth 2C
2 1
!
v 1
3
! 1
v 1
4 w 1, Re w 0
5 !% .
6 6
jj
2
2
2
6 6
II
2.155. 1
14 31
2
31
.
1
1
1
2.156. w = e 1
iz=h
i z +2=
!
!
!
!
2.157. w =
.
z
227
ch
z
h
1
2.158. w = e2iz= z
.
2
.
2.159. w = e
p . 2.160. w = cos z + 2 =2z .
p
2.161. w = e zi= p21 .
1+i 3
iz + 3i
e i z+i= zi + 2 i
2.162. 1 w =
th
2 w=
i z+i= zi
3
2
4z
2
i z+i= zi
3 w =
.
1 + e i z +i= z i
2e
2.163. 1 w =
r
2.164. w =
2.168.
2iz
cos z
+ e h
2
cos h .
1 + cos z
p
2.166. w =
r
pe
cos 2z + ch 2h2
2.170. w =
r
e2 iz + e2 h1
.
e2 iz + e2 h2
r
cos z cos h1
2.165. w =
.
cos z + cos h2
w=
2
w
2.175.
p
z
1
=
i
ch
2.136.
2
y
ai
+
w
= x =
0,
2
= arcsin
x
6v 1
"
0
sin z
. ,
2
.
= th
a
2.177. w =
2.179. w =
2,
.
-
+
cos 2z + 1
sin=z
1 + sin=z
r
.
cos4=z
cos4=z
2
2
e2 i=z
.
sin
z
-
!
%% ""&'
(
"
&,
% ""
&
"
.
""-
"
r
=
2 +
+
,
u
=
b arcsini sh z=ch a
.
arcsin1=ch a
arcsin1=ch a2 arcsinsin z=ch a2
arcsin1=ch a +
.
arcsinsin z=ch a
arcsinsin z=ch a arcsinsin p=ch a
.
arcsinsin q=ch a arcsinsin z=ch a
p
.
p
.
.
e2 =
z
4
. *%
= a 6 y 1
,
2
$
ch
.
! &
ch a
=
2.178. w =
15
ch
$
w
,
sin z
2.175. w = arcsin
cos=z
cos=z
cos 2z + ch 2h
2.171. w =
cos4=a .
r
cos2=z cos2=b
2.172. w =
.
cos2=z cos2=a
r
ch =h
r
cos4=b .
cos4=a
r
e = e i=z
2.173. w =
1 + cos4=z
cos4=z
2.174. w =
r
2.169. w =
.
+2+i
r
2.167. w =
cos 2z + ch 2h.
cos 2z + ch 2h1
e
i
,
=
2,
!-
228
2.180. w = arcsin e2iz . P e e . = e2iz
0 x =2 , w = arcsin
| =2 u =2, v 0. w =
= arcsin e2iz , , x = 0,
1 y 0! x = =2, 1 y 0 x = =2, 0 y 1!
x = =2, 0 y 1. "#
$ % ## , # ,
' #
.
2.181. 1) * b 2 | '+' # $ : 1 ea ,
0 b! b = 2 | 1 ea %% # % -1 ea.!
b = 2k /k = 2 3 :::) | # $ , % k 1 ea , %%+ % -1 ea. + ,
' ' %% $ # # %% $
, ' ' %% $ | # # %%
$ . .! b = 2k + /k = 2 3 :::, 0 2),
# # $
'
% -1 ea. '+' # $ 1 ea ,
0 !
2) a , % 1 ea, %%+ % -1 e . + %+# + #! a
3) , % 1 e , %%+ % -1 ea ., %# + # +
/:::, 2, 1, 0, 1, 2, :::) + ,
' ' %% $
# # %% # # #. 2 a # ' Ln w,
' # 1 e .
2.182. 1) 3 , # +
', %
+ %% v = 0, 1 6 u 1!
%% 4 , . . ,
' '
%% $ # # %% $ ,
!
2) , % ' %%#
' ' ' 1 u 6 1 1 6 u 1 +
4 %% 1 u 6 1. 5 %% 1 6 u 1
+#.
2.183. 3 , % ', %%+ ## ' % 1 6 v 6 1 + ,
+'
' %% $ +# # %% $ .
6 + +.
2.184. 7# # $ #
# w = 0 w = 1. 6
z - ,
# w- %%#2 2 ' ' ' , $ + # 2k/x + y ) + y =
= 0 /k = 0 1 2 :::).
2.185. r1 = 1, r2 = r3 = 1. 6% $ jz j r1 |
+ '
' w = 1! % $ jzj r2 / $ jzj r3) |
Re w 1=2.
2.186. r1 = 1=2! 2) r1 = 1=/2jaj)! 3) r1 = 1.
p
2.191. 1) r2 = 1=4! 2) r2 = 1=/4jaj)! 3) r2 = 2 3.
2.192. r1 = , r2 = 1. 2.193. r3 = 1=2! 2) r3 = 1=/2jaj)! 3) r3 = 1.
III
229
9 III
3.3. 1) I1 = 2 + i, I2 = 1 + i=2! 2) I1 = i=2, I2 = =2!
3) I1 = iR2 , I2 = R2.
p
3.4. 1) 5/1 i=2)! 2) 2! 3) 2i! 4) 0. 3.5. i. 3.6. 4=3.
Rn+1
3.7. 1)
-/1)n+1 1., n 6= 1! i, n = 1!
n+1
2) 3) 0, n 6= 1! 2i, n = 1.
3.8. 1) 2/1 i)! 2) 2/1 i)! 3) 2/1 + i)! 4) 4! 5) 4i.
3.9. 1) 2i! 2) 2 ! 3) 2Ri! 4) 2Ri.
2i , n 6= 1! 22 , n = 1!
3.10. 1)
n+1
2) /1)n+1 n2+i1 , n 6= 1! 22 , n = 1.
e2 i 1
3.11.
1 + , 6= 1! 2i, = 1.
3.13. 1) j j 6 =2! 2) sin p
0. 3.27. 1) =3! 2) =3! 3) 0.
3.28. * C
0 1
1, I = 2i! % 1 1
0, I = i. 6 ,
$ # #
% + % ' /2i! i! 0! i! 2i).
3.29. 2n 1, n 1! 2, n = 1.
3.30. i=2. 3.31. sin a=a. 3.32. ea /1 + a=2).
3.33. 1) 1! 2) e=2! 3) 1 e=2. 3.35. 1) 2=3! 2) 1 2i=3.
3.40. R = 1. 3.41. 1. 3.42. 0. 3.43. 2.
3.44. e. 3.45. 1. 3.46. 1. 3.47. 1. 3.48. 1=4. 3.49. 1=e.
3.50. 1, jaj 6 1! 1=jaj, jaj 1. 3.51. 1.
3.52. 1) R! 2) R=2! 3) 1! 4) 0! 5) Rk !
6) R, jz0 j 6 1, jzR j , jz0 j 1.
0
3.53. 1) R min /r1 r2 )! 2) R r1 r2 ! 3) R 6 r1 =r2 .
z
1 1+z
3.54. 1)
1 z2 ! 2) ln /1 z)! 3) 2 ln 1 z ! 4) ln/1 + z).
3.55. 7
.
3.56. < / )
, # z = 1.
3.57. < .
3.58 < / )
, # z = 1.
3.59. < / )
, #
z =
p
= e2ki=p /k = 0 1 ::: p 1).
3.60. < / )
, # z = 1 2i 3
z = 1.
3.61. < . 3.62. =#, cn = /1)n .
230
X1
p . z = 1.
n
k2 , k2 + 1, :::, k + 1 2 1
3.64.
1
n
n=1
k
1 ! "
# 1 k k $, k ! 0
2
2
. & 0 k k + 12 k = 2k +2 1 ! 0 ' k ! 1. ( ,
k
k
h 1
i h 1
i
1
1
k k+1 = k2 k + 12 + k2 + 1 k + 12 + 1 + :::
i
h
::: + k + 11 2 1 k + 21 2 3 k + 21 2 2 k + 21 2 1
h
i
2k + 1 k + 11 2 1 k + 21 2 3 k + 21 2 2 k + 21 2 1 0:
* , + ' z = 1. , jz j = 1, z 6= 1,
, -
p , ' ' 1.90
' $ an = 1 n z n , bn = 1=n. / , , ' ! -p1 Sn :
Sn = z z2 z3 + z4 + ::: + 1 n zn =
= z 1 z + z 4 1 z z 9 1 z + ::: zp 1 z
z p+1 + ::: + zn
1
z
1
z
1
z 2p
1
z
p
- p = 3 n 4 1. 5 jSn j j1 zj + 2p + 3, , $p
- z 7 k, jSn j kp k n, #
.
3
5
7
2p+1
2
X1 z , R = 1. 3.68. X1 z , R = 1.
n
n1 n
n
X
X1 z , R = 1.
z
3.69. 1 n
,
R
=
1
.
3.70. +
n
n 1 n
n
n
X
z
:::
n
3.71. a
, - =1, =
n
3.67.
2n
=0
2n+1
2 !
+1 22n
h
1+i
p2 1 + 12 zi +
X
n
1
iX
3.74.
3.73.
=0
6
+ 1!
X
1
n=2
n=1
3in
13n
1
1 2n
2 !
n!
+ 1
n = 1 2::: ! R = jaj.
i
n
1 n1 1 3::: 2n 3 z , R = 1.
:::2n
i
i
2 4
2 + 3 n
cn =
zn =
X
n1 =2
m
1 n
X
3.75. 1 n 1 z n , R = 1.
n=2
=0
n
n zn
1 n a n+1
, R = b .
b
a
2
22n
1
2
0
a
n=0
1
2
2 !
=0
3.72.
=0
1 2n
=0
X1 cnzn, -
n=1
p
1 m m n+ 1 2n2m1 32m , R = 13.
3.76. 2
X1 z
2n+1
n=0 n + 1
2
, R = 1.
III
231
X1 1 n z , R = 1. 3.78. z +X1 1 n ::: n z n , R=1.
n
n n
n
n
X1 1 + z , R = 1. 3.80. X1 z , R = 1.
3.79. ln 2
n
n
n n n
1
X
z
3.81. 1 n
, R = 1.
n
n
n
X1
z
3.82. + 2 1 n
, R = 3.
n
X1 3 z 1 n + z 1 n 4, R = 2.
3.83.
n
X1 n z , R = 2.
3.84. + 1 n
n
p 1
i X
1=3 z 1 n , R = 1.
3.85.
n
n
X1
z
3.86. 1 n
, R = 1.
n
n
X1 n= z 1 n, R = 1. 3.88. 1 + z + z + :::
3.87.
2n+1
3.77.
2
=0
1 3
+1
2n
=1
+ 1! 2
+ 1
+1
1n
1
3
1n
4
2
22n
=0
1
+ 1
1n
2n+2
=0
1+
! 2
2 +1
3
4
1
+ 1
3n+1
=0
1
3
2
=0
1n
+1
=1
sin 1 +
n=0
3.89 1
z4
4
96
2
+
2
4
3
+ :::
3.91. 1 + z + z 2 +
z
2
2
n!
z2
3.93. ln 2 +
2 +1
2
! 2
2n+1
=0
2n+1
2
=0
=0
n
1
2n
z2
8
3.90. 1 + z 2
1
2
z3 58 z4 + 13
z5 + :::
6
30
5
z4
192
z3 + 56 z4 34 z5 + :::
X1
+ ::: = ln 2 +
3.92. z +
cn
n=0 n + 1
z2
2!
3
5
+ 23!z + 95!z + :::
zn+1 ! c0 = 1 ,
2
c0 + nc1 + n n 1 c2 + ::: + n!cn1 + 2n!cn = 1, n 1.
1 z3 1 1 z4
+ 1 + 2 3 + 1 + 2 + 3 4 + :::
3.94. 2
2
n
i
::: + 1 + 12 + 31 + ::: + n 1 1 zn + ::: .
h
z3
z2
1
3.95. 4 2 + 2 2 iz + 1 2 i + 1 + 2 i
+ :::
2
3
1 2
n
i
::: + 1 + 2 + ::: + n 1 1 2 i zn + ::: .
1 z4 1 1 z6
3.96. z 2 1 +
+ 1 + 3 + 5 3 :::
3
2
2n
i
::: + 1 n+1 1 + 13 + 51 + ::: + 2n 1 1 zn + ::: .
h z2
232
h
3.97. 2
1+
1
2
z3
1+
3
n
k=1
3.103. 1 1 +
3
X1
X1
k=1
k=1
1k
+
1
1
i
22k 2k 2k
,
2 !
1k
k=1
22k 22k
1
1k
22k 22k 1 B2k 2k
z ,
2k2k !
4 1 +
X
k=1
1k
1
2k !
1 22k
+
+ :::
5
i
1
1
1
1 z 2n+1
1 + + + + ::: +
+ ::: .
2
3
4
2n 2n + 1
4
zn .
R = .
22k 2k 2k
z ,
2k2k !
1k
1
3
+ 1n+1
1
!
n=1 k=1
3.102. 1 +
2
2
X1 h X n
k k
X1
B z
k
1
X
B
:::
3.98. 1 +
1
1
1 z5
R =
B2k z2k1 , R =
2
R=
2 B2k 2k
z ,
2k !
2
R = .
p5 1
h p n+1 1 p5 n+1 i
p1 1 +2 5
, n 0, R =
.
2
2
5
A
B A , c = 2 C B + 2 A
3.105. c0 = , c1 =
2
3
2
cn + cn 1 + cn 2 + cn 3 = 0 n = 3 4 ::: .
e zt=1 t L z 2n + 1 z L z + n2 L z = 0 n 1 .
3.111.
n+1
n
n 1
1 t
3.104. cn =
z
z
z
4 5 + 4 5 8 9 + ::: + 4 5 8 9:::4n4n + 1 + ::: jzj 1.
nn + 1 1 2 z3 + nn + 1 1 2 nn + 1 3 4 z5 + :::
3.113. z
3!
5!
::: + nn + 1 1 2 nn + 1 3 4 ::: nn + 1 2k 12k z 2k+1 + ::: jz j 1.
2k + 1!
z
3.114.
1 z2 .
m2
m2 m2 22 z4 :::
3.115. 1 z 2 +
2!
4!
2
2
2
2
2
::: + 1 m m m 2 ::: m 2n 2 z 2n + :::
2n!
ab
a
a
+
1
b
b
+
1 2
3.116. w = F a b c z = 1 +
1 c z + 2!cc + 1 z + :::
n 1bb + 1:::b + n 1 zn jzj 1.
::: + aa + 1:::n!ac+c +
1:::c + n 1
3.112. z + z 3 +
5
4n+1
9
III
233
3.117. . !, #! !, ! !# $ ! $ =
= dzd F a b c z: 2
d
d
z 1 z 2 + &c + 1 a + b + 3z ' a + 1b + 1 = 0:
1
dz
dz
( ! $ dzd F a b c z, )!! *+ ! F a b c z,
# + z = 0, , $#$ $ ! +, # +
z = 0, $ !$ 1, # z = 0,
#, - kF a + 1 b + 1 c + 1 z . * ! 3.116, k | $$, dzd F a b c z = kF a + 1 b + 1 c + 1 z. #, z = 0, + ,
k = ab=c.
3.120. 1 4 2 15 3 3.
3.121. 1 1!# $ k + l
2 !# , $ ,, min k l
3 !# $ k l, # k l #-+ +, $#$23+ $
!# , # k = l, )+ +, # k l.
3.122. ( z = 3i | !# 1- $ .
3.123. ( z = 3i | !# 1- $ ) !#$ | !#- 2- $ .
3.124. z = 0 | !#- 2- $ z = k k = 1 2 ::: | !# 1-
$ .
3.125. z = 2 | !# 3- $ z = 2ki k = 0 1 2 ::: | !#
1- $ .
3.126. z = 2k k = 0 1 2 ::: | !# 2- $ .
3.127. z = | !# 3- $ #-4 z = k
k = 0 2 3 ::: | !# 1- $ .
3.128. z = =4 k k = 0 1 2 ::: |!# 1- $ .
3.129. 1!#+ .
3.130. z = k k = 0 1 2 ::: |!# 3- $ .
3.131. z = 0 | !#- 2- $ z = k k = 1 2 ::: | !# 3-
$ .
p
p
1p
3.132. z = 0 | !#- 3- $ z = 3 k z = 3 k1 i 3
2
k = 1 2 ::: | !# 1- $ .
3.133. z = 2k + 1 k = 0 1 2 ::: | !# 3- $ .
2
q
q
p
1
3.134. z = 3 2k + 1 z = 3 2k + 1 1 i 3 k = 0 1 2 ::: |
2
2
2
!# 1- $ .
3.135. z = 4 | !#- 3- $ #$ + * +.
3.136. 5 - *4 ! : * 6 !# 2- $
6 z = 2k =6, !$ | !# 2- $ 6 z = 2k + l
=6 k = 0 1 2 :::.
3.137. #-+ + , )4 - #- ) !#$
.
234
3.138. 1 2 3
3.139. 1 f z = z 2
3.140. ,
!.
4
2
z
f z = 1=z + 1 .
.
X
= 1
1
cn z z0 n
n
=0
c + c0
n
n
ux y = 0
2 + n=1fcn ' x z x0 + iy y0 ( z+ c)n 'x x0 iy y0 ( g:
0
0
* % + x = x0 +
2 , y = y0 + 2i , , %
z0
z0
1
z , - + . u x0 +
2 y0 + 2i = 2 ')c0 + f ( ,
%- % z , - %/ -- .
3.143. z 2 + 2 + Ci. 3.144. zez i=z + Ci. 3.145. 1 + i z 3i + C .
3.146. sin z ch z + C C | % +& 1 +& && .
3.153. 2 2 2 M .
X
3.141. " # . $% % & f z
1
X
X
ak
jzj jaj.
4.3.
X1
1
n
n=2 z
X
IV
1 1 z n jzj 2 1 2n jzj 2.
n+1
2 n=0 2
n=0 z
1k 1 n + k 1 z n jzj jaj 1 1 n + k 1 a
4.1.
4.2.
2
=
k1
n=0
X
a
zk
X
X
k1
n=0
n
z
1 + 1 zn jzj 1 1 + 1 1 n z 1 n 0 jz 1j 1
z
z 1
n=0
jzj 1.
1 1 bn+1
X
n=0
X
1 h 1 + 1 z an i
j
j
j
j
n+1
b a
a b z a
n=0
n=0 b a
1 1 bn1 an1 jzj jaj 1 1 zn +
0 jz aj jb aj
b a
zn
a b
bn+1
4.4.
an
an+1 n
z z a
n
+1
n
a
b +1
X
n=2
jaj jzj jbj.
z n+1
1 + i 1 1 n 2 + in+1 2 in+1
4.5.
z 2
5n+1
1 n 1 1n=0 zn
1 z2n 2n+1 1 jzj 2.
2
n=1
n=0
1 n + 3in z
1
i
+
4.6.
2
4z i
4z i
2n+4
n=0
1 n n
1 z2n+2 jzj 1.
+
X
X
n=1
X
X
X
X
n=0
z
in
n
2
0
0
jz 2j
jz ij
p
5
2
4.7.
h
z 12 a + b
cn1 = 1 n
4.8.
+
h 1=2
X1 i
n=1
c n
+
zn
n=0
235
jzj jbj, ,
1=2
bn1 a +
1=2 1=2
1=2
i
n2 2
n
+
n 2 2 b a + ::: + n a :
1=2
n
n b + n1
X1 X1 cnzn
n=1
c n
zn
IV
1
, ,
n h
cn = p12 i 1n=2
+
X1 =
m=1
n m
cn = 2n cn n = 1 2 ::: c0 = pi2
X
1 2
+
h
1+
1=2
X
m
2
m i
1 1=2 2 m i
m 2 :
m=1
1
1
1
4.9. + z + z 2 +
0 jz j 1.
2
n + 2!z n
n=1
1 n 1
1 + 1 cn zn
4.10.
1
0 jz 1j 1 1
n!z 1n
z
n=0
n=2
n1 1k+1 n 1
jzj 1, , cn = 1 + k + 1! k n = 2 3 ::: .
k=1
1 h 1k1 42k1 sin 1 1k 42k cos 1 i
4.11. cos 1 +
4k2 + 2k!z 24k
k=1 2k 1!z 2
0 jz 2j 1.
1 h 1k1 1=2k 1! + 1=2k + 1!
4.12. z 1 + 2 +
+
z 12k1
k=1
i
21k
+
2
k
2k + 1!z 1 , 0 jz 1j 1.
1 n 1
1
4.13.
cn z + cn zn , , cn = cn = k!n 1+ k! n = 0 1 2 ::: .
n=0
n=1
k=0
X
X
X
X
X
X X
X1 c nz n X1 c nz n
4.14.
n
n
X1
c n c n n
=0
2
2
2
2
=
X
+
=
=0
2
, ,
1
n = 0 1 2 ::: :
2
k + 1!2n + 2k + 1!
k=0
1
1 sin 1 + n2
4.15.
2
X
! 1n 0 jz 1j 1
sin 1 cosz 1 + sin 1 2!z22 cos 1 + 6 sin13!z35 cos 1 + ::: jzj 1.
n=0
n z
236
4.16.
+
z
1
X1
n=1
2n
1n 22nB!2n z2n1 , B2n |
X
X
. -
1 2n 1 h n 22n B2n 2 i 2n1
3.100, 0 jzj 2 3z + 2 z2n+1 +
1 2n! + 2n z
n=1
n=1
jzj 2.
1 bn an
4.17.
n jzj max jaj jbj.
n=1 nz
1 cn
2k1 22k3 + 22k5 ::: + 1k+1 2 ,
,
c
4.18.
2k = i 2
n
3
5
2k 1
n=2 z
1
1
n
c2k+1 = 2c2k k = 1 2 :::, jzj 2 i arctg 21 z2n + cznn ,
n=0
n=1
X
X
c1 = 2i arctg c2k = 22k i arctg
k = 1 2 :::, 1 jzj 2.
1
,
2
1
2
X
k 1
m=0
X
1m
2m + 122m+1
X
!
, c2k+1 = 2c2k
4.19. 1 2 3
z = 1 !" ! ! " #
$ # # 4 5 6 7 8 " *$ +,,
- *. z = 0, / , ! !" ! ! 0" # 9
10 11 , | ,
+ " " 2 +02
!2.
4.20. 1 3 2 , - * $ " " 3 4 ,
" " " * - 5 6 p - p p * " " " 02, !0 " !
1 + 1 = 2
7 8 9 10 , - * " 4 + " "#
11 12, 13 14 , *$ ! " "# - 15
16 17 -
p * " " " , "2, !02
Arcsin 2=2 = =4.
4.23. z = 0, z = 1 | * 0 1- ! , z = 1 | " + !
+ 3- ! .
p
p
4.24. z = 1 i= 2, z = 1 i= 2 | * 0 1- ! z = 1 |
" + ! .
z = 1 | 3- .
z = 2i | 2-
z = 1 | 5- .
4.27. z = i | 1- z = 1 | !! "
.
4.28. z = 1 | !! " .
4.29. z = 1 | !! " .
4.30. z = 2ki k = 1 2 ::: | 1- z = 1 | ,
!! .
4.31. z = 0 | 2- z = 2ki k = 1 2 ::: |
1- z = 1 | , !! .
4.32. z = 2k + 1 i k = 0 1 2 ::: | 1- z = 1 |
, !! .
4.25. z = 1 | 2-
4.26. z = 0 | 1-
IV
p
237
z = 0 | * 3- ! z = 2k i ln 2 + 3 k = 0 1
2 ::: | * 0 1- ! z = 1 | , + ! ! * ".
4.34. z = ki k = 0 1 2 ::: | * 0 1- ! z = 1 | ,
+ ! ! * ".
4.35. z = 0 | . " $ ! z = 1 | " + ! .
4.36. z = 0 | . " $ ! z = 1 | * 1- ! .
4.37. z = 1 | . " $ ! z = 1 | " + ! .
4.38. z = 0 | . " $ ! z = 1 | . " $ !
.
4.39. z = 1 | . " $ ! z = 2ki k = 0 1 2 ::: |
* 0 1- ! z = 1 | , + ! ! * ".
4.40. z = k k = 0 1 2 ::: | * 0 1- ! z = 1 | ,
+ ! ! * ".
4.41. z = 0 | * 2- ! z = 1 | . " $ ! .
4.42. z = 2k + 1 k = 0 1 2 ::: | * 0 1- ! z = 1 |
, + ! !2 * ".
4.43. z = 2k + 1 k = 0 1 2 ::: | * 0 2- ! z = 1 |
, + ! !2 * ".
4.44. z = 0 | * 3- ! z = k k = 1 2 ::: | * 0
1- ! z = 1 | , + ! ! * ".
4.45. z = k k = 1 2 ::: | * 0 1- ! z = 1 | ,
+ ! ! * ".
4.46. z = k k = 0 1 2 ::: | * 0 1- ! z = 1 | ,
+ ! ! * ".
4.47. 5
a 6= m + =2 m = 0 1 2 :::, z = 2k + a z = 2k +
+1 a k = 0 1 2 ::: | 0 * 0 a = m + =2,
m z = 2k + =2 m z = 2k + 1 + =2 !" !* ! * 2- ! z = 1 " " 2 !2 , + ! !
* ".
4.48. 5
a 6= m m = 0 1 2 :::, z = 2k + 1 a k = 0 1
2 ::: | * 0 1- ! a = m, m z = 2k,
m z = 2k + 1 | * 0 2- ! , z = 1 " " 2 !2
, + ! ! * ".
4.49. z = 1 | . " $ ! , z = 1 | " + !
+ 1- ! .
4.50. z = 2 | * 2- ! z = 2 | . " $ !
z = 1 | * 3- ! .
4.51, 4.52. z = 1= k k = 1 2 ::: | * 0 1- ! z = 0 |
, + ! ! * " z = 1 | * 1- ! .
4.53. z = 0 | . " $ ! z = 1 | " + !
+ 1- ! .
4.54. z = 0 | . " $ ! z = 1 | . " $ !
.
4.55. z = 1= k k = 1 2 ::: | . " $0 z = 0 |
, + ! ! . " $02 z = 1 | . "
$ ! .
4.33.
238
2
k = 0 1 2 ::: | z =
= 0 | , z = 1 | .
4.57. z = 1=k k = 1 2 ::: |
z = 0 |
, z = 1 |
.
2
4.58. z =
k = 0 1 2 ::: | z =
2k + 1
= 0 | , z = 1 | .
4.59. ! "
, #" | $ 1-# .
4.60. ! "
$ 1-# ,
"
.
4.61. ! "
, #" | $ 2-# .
4.62. ! "
, #" |
.
4.63. !
" $ 1-# .
2
4.64. z = 1 + 1=k | "
$
1-# #" z = 1 | "
& , $ , #"
.
4.65. '( & & | "
$
1-# #".
4.66. ) "
#".
4.67. 1 !
" $ 1-#
2 # # *(
" ,
# # *(
" $ 1-# .
4.68. ! # # *(
" ,
# # *(
" .
4.69. 1 + z = 1 | $ k = max n m, n 6= m
( n = m, z = 1 | $ k 6 n,
2 $ n m, n m, , n 6 m
n m, z = 1 | m n
3 $ n + m.
4.71. )* : 1 z 2 2 1=z 2 + z 3 1=z n 1.
a
4.72. 1
a 6= 0 az + b a 6= 0
z
4.56. z =
2
4
5
2k + 1
a
a 6= 0 a0 + a1 z + ::: + an zn an 6= 0
z n
a0 + a1 z + ::: + an+m z n+m
a0 6= 0, an+m 6= 0
zn
n
a0 + a1 z + ::: + an z
k 6= l k 6= l
z 1 z 2 ::: z n
3
1
z2
+ C
" " * -
IV
239
n
& am z a0 + za1 z + ::::::+zan z an 6= 0,
1
2
n 1
k 6= l k 6= l.
4.74. 1 z0 | *
2 z0 | $ n, 'z z0 ,
$ nm, 'z m- 0 "
3 z0 | .
4.75. 1 + z0 | $ n, 0 | * "" &,
n, 0 | # ( , . .
f z f z0 = z z0 n 'z, # 'z z0
'z0 6= 0. 1 z0 = 1, 0 &
f z f 1 = zn 'z, # 'z 1 '1 6= 0 2 z0 |
.
4.77. 1 1 2 0 3 0 4 0.
4.78. 1 2
z = 1 $ & 0
1 ez ! 0, x ! 1 ez ! 1, x ! +1
2 z = 0 $ & 0 1
e1=z ! 0, z ! 0, * , y = 0, x 0 e1=z ! 1 z ! 0
y = 0, x 0
3 z = 0 $ & "
1 cos1=z ! 1 x = 0, y ! 0
4 z = 1 $ & i i.
4.79. res 3f z 4z=1 = 1=2 res 3f z 4z=0 = 1 res 3f z 4z=1 = 0.
4.80. res 3f z 4z=i = i=4 res 3f z 4z=i = i=4 res 3f z 4z=1 = 0.
2n!
4.81. res 3f z 4z=1 = 1n+1
res 3f z4z=1 =
n 1! n + 1!
= 1n n 1!2nn! + 1! .
4.82. res 3f z 4z=0 = 1 res 3f z 4z=1 = 1=2 res 3f z 4z=1 = 0.
4.83. res 3f z 4z=0 = 0 res 3f z 4z=1 = 1 res 3f z 4z=1 = 1.
4.84.res 3f z 4z=1 = 2 sin 2 res 3f z 4z=1 = 2 sin 2.
4.85. res 3f z 4z=0 = 1=9 res 3f z 4z=3i = 1=54sin 3 i cos 3
res 3f z4z=3i = 1=54sin 3 + i cos 3 res 3f z4z=1 = 1=27sin 3 3.
4.86. res 3f z 4z=2k+1=2 = 1 k = 0 1 2 :::.
4.87. res 3f z 4z=k = 1k k = 0 1 2 :::.
4.88. res 3f z 4z=k = 0 k = 0 1 2 :::.
4.89. res 3f z 4z=k = 1 k = 0 1 2 :::.
4.90. 1 res 3f z 4z=2 = res 3f z 4z=1 = 0
2 res 3f z4z=2 = res 3f z4z=1 = 143=24.
4.91. res 3f z 4z=0 = res 3f z 4z=1 =
X
1
1
n=0 n! n + 1!
4.92. res 3f z 4z=0 = res 3f z 4z=1 = 0.
4.93. res 3f z 4z=1 = res 3f z 4z=1 = cos 1.
.
240
4.94. res f z z= 3 =
res
f z z=
1
h
=
sin 2
X1
=
n=1
2n
42n
1! 2n!
+
X1
n=0
42n+1
2n! 2n + 1!
i
.
1
k = 1 2 :::.
res f zz=2ki=h = 2ki
4.96. res f z z=0 = 0, n 0,
n 0
1n=2
res f zz=0 = n + 1! , n = 0 n 0 | res f zz=1 =
= res f zz=0 .
1
4.97. res f z z=1=k = 1k+1 2 2 k = 1 2 ::: res f z z=1 =
k
4.95. res f z z=0 = 1=2
X1
1k
1
2 = 6.
k
k=1
4.98. res f z z=k2 2 = 1k 2k2 2 k = 1 2 :::.
4.99. res f z z=0 = 0, n
2k 2k
res f zz=0 = 1k+1 2 22k! 1 B2k , n = 2k k = 0 1 2 :::,
B2k | .
3.100
res f zz=k+1=2 = k + 1=12n n k = 0 1 2 :::.
4.100. 1 1. 4.101. 0 2.
p
4.102. p2e2kai , 1 = 1 Ln 1 = 2ki 0 ! ! ", # $%
1 = 1.
4.103. a b2 =8.
4.104. 1 ! % ! ! " 2 e e ! % ! ! ".
1
4.105. 1 2ki +
4 15! + :::, Ln 1 = 2ki
2 3!
= 22
2 2!1 + 3 14! 5 16! + ::: ! % ! ! ".
res f zz=0 = k, Arctg 0 = k res f zz=1 = 2k +2 1 ,
Arctg 1 = 2k +2 1 .
4.107. res f z z=0 = 0, n 0, res f z z=0 = Ln , n = 1,
1
n
+1
n
+1
res f zz=0 = n + 1
, n 6 2 res f zz=1 = n +1 1 n+1
n+1 , n 0 res f zz=1 = 2ki, n = 1 Ln 1 = 2ki,
res f zz=1 = 0, n 6 2.
4.108. 2c0 c1 .
c '0 a + ::: + ck 'k1 a .
4.109. 1 A'a 2 c1 'a + 2
1!
k 1!
4.110. 1 n 2 n. 4.111. 1 n'a 2 n'a.
4.106.
IV
241
X
A . 4.113. AB . 4.114. k 1n ncn an+1 .
0
' a
R2n
n=1
p
4.115. i= 2. 4.116. 2i. 4.117. i=121.
4.118. i. 4.119. 2i=9. 4.120. 1. 4.121. 0.
4.122. 2n+1 =n + 1!, n 1, 0, n 1.
f 0 + n f ak .
4.123. 32i. 4.124. 0. 4.125.
g 0
a g0 a
4.112.
X
k=1 k
k
0, r 1 1, r 1 # #&+ ##
%& ,$ - ..
p
p
1
e=2 . 4.131. 2 .
1
4.128.
.
4.130.
1
+
2.
4.129.
4
1 + a
1 + e
a2 1
2a
2a + b
4.132. 2 2 3=2 . 4.133.
a b
a a + b3=2 .
2
2
4.134.
1
a2 , jaj 1 a2 1 , jaj 1 0 # ,
jaj = 1, a 6= 1 % a = 1 # /# .
a6 + 1 , jaj 1 a6 + 1 , jaj 1 1 a12 #
4.135.
1
a2
a6 a2 1
2 a6 a2
1
, jaj = 1, a 6= 1 % a = 1 # /# .
2
4.136. , n 0 0, n 0.
n!
4.137. i sign a % a = 0 # # 0.
4.138. 2i sign Im a. 4.140. =27. 4.141. =4a.
1 3 5::: 2n
3
4.142.
, n 1 2 , n = 1.
2 4 6::: 2n
2 2
2
sin = 2n
4.143.
ab a + b . 4.144. 2 . 4.145. n sin =n . 4.146. n sin =n .
1k1
4.148.
2ih
zk . 4.149. 1 3e3 cos 1 3 sin 1 2 3e3 3 cos 1 + sin 1.
eab . 4.152. eab .
4.150. 4 2 cos 2 + sin 2. 4.151.
2e
2b
2
4.154. i, t 0 0, t = 0 i, t 0.
cos 1 1 .
4.155. 2 sin 2 3 sin 3. 4.156.
5
e2
4.127.
p
p
"
sin jtj + ejtjp3=2 sin t + p3 cos t
4.157.
3
2
2
4.158.
4.161.
4
e jtj
4b4
16 ..
sin jtj.
2 + abeab .
2
.
4.159. eab
4.162. b
#
.
1
.
2
4.160.
a.
4.163. .
2b2
2
1
eab .
4.164.
3
.
8
242
p cosp=2
p=2
2 p sin
ap
1 p 0, ,
p , , ! , .
1 1
1 1
4.166. $
cos
.
4.167. $
sin 2p .
p p
2p
p p
4.165. 1
ap
1
$
cos
p = 1 2 .
p 1 p
2p
4.170.
. 4.172. 2 cosp=2 .
sin p
4.168.
1
p
4 cosp=2
sin p
4.174.
,
sin p sin
4.173.
1
p = 1 12 .
6= 0, sinpp , = 0.
4.176. 0. 4.177. =4. 4.178. ctg p. 4.179. ctg p .
p1 p
4.180. 3 p
.
4.181.
2p=2 cos p
1 .
2 sin p
sin p
4
h
i
h
i
p
a p
p
2
1
1
.
4.183.
1
.
sin p
2
sin p
1+a
p
p
ap 1
p
p
34
4.184.
p
.
4.185.
sin
cos
1
.
4.186.
.
sin p 1 + ap+1
sin p
2
2
3
4.187. ' a ( )1 1 , I = p 2
,
a 1
p
4.182.
a2 p1 0 a 1 ) *1 1+ a2 1 a = ei I = p ei3=4 =2
2 sin
a = iy I = p i 2 sign y 1 a 1 I = 0 ) .
1+y
p 1
4.188. ' b ( )0 1 , I =
b )b
sin p
p
p
p
1
1 ,p 1 )b a1 0, b 0 b 1 0 b 1, I =
= b )1 b ctg p ) .
4.189.
. 4.190. 2a ln a. 4.191. 8a )2 + 4 ln2 a .
n sin=n
3
ln a 1 34 .
2a 2
1
a
4.193. . 4.194. ln 2. 4.195. ln2
.
4
2
1+a
1
4.197. 1
+ 1 a = 1 I = 12 2 2aln2 a+ 2 =4
1 a ln a
1 n 2n
1 o
2k + 1
k
4.198.
.
)
1
2n + 1 2a k=0
ln2 a + k + 1=22 2 1 + a2
4.192.
2a
p
3
X
4.199.
1n 1
2n 1 + a2
+ 2a
X )1
2n 1
k=0
k+1
2k + 1
ln a + k + 1=22 2
2
o
.
1 +1 a2 .
IV
1
243
2a a = 1 I = ln 2.
sin a
a
2 th 2 .
ea
1
4.203.
2 ch a=2 . 4.204. ea + 1 . 4.205. 2 cosa=2 .
1+a
4.206. ln 1 + a, 0 a 1 ln
, a 1.
2
2
a
n
n
t
4.208. 1
2 lnn! t , t 1 0, t 1 t = 1, I = 0
n!
n 1 I = 1=2
n = 1.
at
n
e t
4.209.
. 4.210. sin t. 4.211. 1 cos t 2 t sin t.
n!
4.201.
1
4.202.
2
2
4.212.
4.213.
eat
ac
ebt
bc
t
it
pe t 2 ept .
+
b a
ect
cb
.
b
c
n
1 1 1 , t 1 0, t 6 1.
n!
t
4.216. 1
+
a a
4.217.
p
p
Z
u
erf t, erf u = e
x2
dx.
0
4.218. e t erf t. 4.219. sin t, t 0, t .
4.220. 1, 0 t a 0, t = a 1, a t 2a
1=2,
t = 2a 0, t 2a.
4.221. n + 1, na t n + 1a n + 1=2, t = na n = 0 1 2 :::.
x
4.222. 1 erf p . 4.217.
2 t
1
2 2
2 sinnr=a
1
4.223. + 2 1n en t=a
.
a
nr
X
n=1
4.224. Eit,
4.225.
2 sh t .
4.230.
2
Eit =
4.226.
t
2i .
a
Z
t
1
eu
u
du.
0, b 0 arch ab , b 0.
p
z + 1 0 z 0.
4.227.
pz + 1z +1 pz + 1n
1 , a b 0, a b
4.231. 1 p
a b
2 0, a b p 1 , a b.
2
2
2
2
pb
a2
2
4.232. =2J0 ijaj
4.233. !
t b .
" # " $# %& ' ", tlim
!1 f t = 1 ! " & $ ' ) "&, tlim
!1 f t = 0 & '& & ! ,
" & ' ) " ' ! " ' ", f t
t ! 1 % #, % ! ,
16
2
2
244
,
, ,
, |
. |
,
f t ! 1, | , f t ! res ezt ' z !z=0 , |
.
p
at 3
i!
i! 2ai! !2 .
4.235. f t 4 . 4.236. f t p
e
a
2ai! !2
4.239. "
. #
Zx
e
x
x t
t
1e
dt +
Z
x t
t
x
dt.
%x & '
0, & & 4.238 - , -. / &
e
Zx
x
e
x t
t
dt = e
Zx
0
et
hZ "
x lim
"!0
x
t
Z t i
dt + e dt =
x
t
"
x
Zx t
e et
x Z et et dt
= ex "lim
dt
=
e
!0 " t
t
0
1- ' . 2 ,
Z t
Z t
Z t
et et
e et
e
e
dt
=
dt
+
dt
+
dt
t
t
t
t
1
x
0
x
1
4 - , -
1
O 1 +
Zx t
e
1
t
dt:
Zx t
e
x 1 + 1 + ::: + n 1! + n!
dt
=
C
+
e
dt
t
x x2
xn
tn+1
Zx t
e
1
1
C = C n | . 6 - ,
Zx
lim xn ex tne+1 dt = 0
x!1
1
7 & 8
9 .
Z 2 2
z
t
4.242. " . " e
dt, - C &
t
C
. 68 Re z 0 . ;
Zz
ez
0
2
Z1
- 7- z = x + iy
1
t2 dt = ez2 t2 dt Z ez2 t+iy2 dt:
x
0
#
&
21
1
Z
x
1
t + iy
dez
2
t+iy2 =
1
2z
12
1
p
2
Z z2 t+iy2
e
t + iy2
x
2
ez <
dt:
# , - - & . 2
IV
245
' 1 3:::2n 1 Z1ez2 t+iy2 1 3:::2n 1 Z1 ex2 t2
dt
2n
t + iy2n
2n
jt + iyj2n dt:
x
x
1 ex2 t2
Z
dt 2jz1j2n x , -
j
t + iyj2n
x
- , & . /- Re z 0 . Re z = 0,
p
Z1ez2 t+iy2 Z1et2
dt
dt
=
:
t + iy2n
y 2n
2y2n
/ - , -
0
!1 sin
0
h
!27t=7=22 + :::
i
::: + 1 n !22nnt2+n+33==22 + ::: <
q h
qt
i
2
2!t4
t f t = 2 t 1 12!t
+
:::
2
:
35
13579
n
3
=
2
X
1
1 n 3t3nn++3=32=2 < t f t = t5=2
4.245. f t 1 +
0
t43 + t119==2 2 ::: 43tp3=2 .
4.244. f t
!t + 4
!12
3=2
t3=2
4.246. 1. 4.247. 0. 4.248. 4. 4.250. 1< 3. 4.251. 0< 4.
4.252. 2. 4.253. 1. 4.254. n. 4.255. n.
4.257. " . % 1- ' fn z
1- ' e1=z -, z = 0, - K ,
' z 6= 0 8 - ,
' , & " 0
-& N , jfn z e1=z j " - - K
n N . 6 "
min
je1=z j, C | - -z2C
K , - "- .
# . = & - & >-' ., , 1,
. VIII, . 2! .
4.261. 0. 4.262. 2< 1.
4.263. ? - .
4.264. ? .
p I . 69 m = 0<
4.266. ? p 0, +
0, + II m = 2< 0 III m = 1.
246
0, 1= I . 70 m = 0 0, 1= II, m = 2.
p2 + 1=4 I . 71 m = 0
p21=+2 1+=4
II m = 2 ,
6 0,
4.268.
0 1=2 +
4.267.
,
p
p
+ 1=4, 1=2
2 + 1=4 6 0, 0 III, m = 1
2
1=2
+ 1=4 0, 0 IV m = 3.
4.270. , ! " #$ "# a %!
= t cos t=sin t 0 6 t 6 .
!& a + b = 0 ab =
t=sin t
1=2
p
2
2
4.271. , ! "& ' %! !&
b = 0 ab == ttg2 =t
cos t 0 6 t 6 2 .
4.272. (%! , %! )'& b = 0, 0 6
= t sin t 0 6 t 6 .
#* ab =
t2 cos t
2
4.282. z = a +
4.283. z =
1
2
w+
4.284. z = a +
4.286. 1 z =
2
f a + w2!
1!
w
X1 1
n
n=2
w1
1! 2
X1
n=1
n dn
dan
f+f a,2 g + ::: + wn!
da
d
1 nn
+ 1:::2n
22n 1 n!
z = 2 12 w +
a 1 +
2
nn 1 an 1
n!
w2 1 d
2! 22 da
2
1
1
6 = 2
f+f a,n g + :::
wn
X1 1
n nn + 1:::2n 2
22n 1 n!
n=2
+ a 12 , + :::
wn .
2
wn 2 ebz = b
X1
n=0
1 2 n
1 + a 1 , + :::
1 n
n 1 dn
2n dan
an + bn
::: + wn!
n!
w.
4.287. z = a +
X1
V
247
n
sin a.
w n dn 1
n=1
n! dan 1
% '# jz 2 j
r, r 2 , & 1& '# #2 #*, '& z = 2 .
3 '# 1 '# jwj er +2rer , "1&# '# jwj #
4.288. - . . /#'0! w =
z =2
sin z
er +2rer ! #$ #$ ' %' z = 2
)& )% ) z w 1& '# 2!.
/#'0!
2r
r + 1
2r
&
&'&#&
"
r
=
r
,
e
=
r
=
1
19
:::
,
er + er
r 1
2
p 2
2r
1 &'&#& er + er = r 1 = 0 6627::: 6'& )&,
'&7* # 2& & . %& 0 6627::: &! %'
w, '72 dw
= 0, . tg z = z , &# 2 ! # '#
dz
2
2& ) ! z w. )% z = t, ")#& #
2
dw
it + 1
2it
=
0
'
#
ctg
t
=
t
e
=
.
8
, it = r | '
dz
it 1
#! dw
= 0. 8#$ )% w = cost t = er 2+irer ,
dz
p
'# jwj = r 2r r = r2 = 0 6627::: : %' 2!, '& e +e
)&, '#
'# 2&, # 2& , p
, r2 = 0 6627:::
; V
5.1. ( 0 1=2 jz j 1. 5.2. . % '# jz j 1.
5.3. jz j 1. 5.4. <#"' Re z 1.
5.5. =* ! .
5.6. ! "' , '& %' z = 0 1 2 :::
5.7. jz j 1. 5.8. jz j 1.
5.9. ! "' , '& %* '# jz j 6= 1.
5.10. ! "' , '& %' z = 41=4 e2k+1i=n k n = 1 2 :::.
5.11. - e . e. > !
X
X1 a
2!, " jz j 1 2-
X1 a
X1
X1
#, % ! 2! " jzj 1. > ! a
X1 a z & # 2& R 6 1. < jzj
2!, !
& !
1
n=1
an z n
1 zn
1 1=z n
n=1
n
%, )
an 1=z n
n=1
n=1
n
n
X1
n=1
an z n
1 zn
=
n=1
n
n=1
n
248
1
X
1
X1 a z
n=1
an
n,
n=1 1 z
1
n n
zn
,
,
X1
an
1 zn
n=1
an z n
1 zn
X1 a . jzj 1, q
X1 a z X1 a z 1 jzj 6 q 6 2 ,
=
n=1
n=1
n
n
n
n=1
1
=
n n
zn
-
, .
X1 b z , b
P
= ap ,
n=1
p,
n, 11 n.
$ R = min fr 1g, r | an zn.
5.12. 1#
n
n
n
X1 a &z 2# ,
5.13.
R = 1.
5.14.
n
n=0
1
2
n
an =
1n
n!
X1
k=1
, jzj 1, 12 , jzj 1.
z
ln k n
k!
X1
a =
0
k=1
X
n=1
1
k2
=
2
6
'
, jzj 1, 1 1 z 2 , jzj 1.
5.16. z , jz j 1, 1, jz j 1.
z
5.17.
, jzj 1, z 1 1 , jzj 1.
z 1
5.19. 1# 2# jz j 6 r 1 (
jzj R 1' 3# ( ( ( '
.
5.21. * jz j = 1'
.
5.22. * ( Re z , 0.
5.23. *
( Re z 1 + ,
0.
5.24. * (
( '
.
5.25. * -2kn + " 2&k + 1# ".
( ( &k = 0 1 2 :::#.
5.26. / . 5.33. xc = xa = 1. 5.34. xc = 1, xa = 1.
5.35. xc = 1, xa = +1. 5.36. xc = 0, xa = +1.
5.37. xc = 0, xa = 1. 5.38. xc = xa = 1. 5.39. xc = xa = +1.
5.42. xc = 0' 0.
5.43. xc = xa = 2' 0.
5.44. xc = xa = 0' &
# z = &2k + 1#i
&k = 0 1 2 :::#, 0 .
5.45. xc = xa = 0'
0.
5.15.
1
z 2
VI
5.46. xc = xa = 0
5.54. +
6 A 1.
5.55.
a 0.
+
249
! * .
" 0 a 6 Re z 6
" " Re z
5.56. + " a 6 Re z 6 2 a,
a 0.
5.57. + " a 6 Re z 6 1 a,
a 0.
5.58, 5.59. + ! % , - .
5.60. + " " Im z 0 " jzj r, r | "
.
5.61. + 0 a 6 z 6 1
1 6 z 1.
2
3
5.62. /: f 't = et " n t n + e n ' n = 1 2 ::: f 't = 0
t.
5.63. xc = xa = 0. 5.64. xc = xa = 1. 5.65. xc = xa = +1.
5.66. xc = 1 xa = 1. 5.67. xc = 1 xa = +1.
5.68. xc = 1 xa = +1. 5.69. xc = 0 xa = 1.
5.70. 1 * . 5.71. ! .
5.72. 2 z = 0 , *
! .
5.73. 2 * ! .
3 VI
6.9.
1 4 . 2 ,
"
Z=2
sin n x dx
0
h
2 +1
2n!!
2n 1!!
i2
Z=2
sin n x dx
2
0
1
2n + 1
2
h
- -
Z=2
sin n x dx
0
2n!!
2n 1!!
2
i2
1
21n :
5 % 6 2 , %
7 !
" n ! 0.
6.12. 1 ) 2 .
6.14. 1 1 2 ' ! 3 4 .
6.15. ! . 6.16. 1 .
6.17. , p 1=2, " ! , p 1
, p 6 1=2.
6.18. ! , p 1 , p 6 1.
250
6.19.
6.23. jz j
. 6.24. z
6.27. z
. 6.28. z
1
jj 1
. 6.21.
j j 1.
j j 1.
z
j j
1. 6.22. z
j j 1=e.
j j 1.
6.25. z
6.29. z
j j
6.26.
2.
jzj 1.
. 1 s = 1 + 21s + 31s + :::
2 s s, 1 2 s s = 1 + 31s + 51s + 71s + :::! " "# $& " " " n1s , ' n 2. ( & 1 2 s 1 s s s = 1 + 51s + 71s + :::, " "#
" n1s , ' n 2 3. *
1 p1 s 1 p2 s:::1 pms s = 1 + n1s
1
" " "# 1 ' n
+ ,, ' p1 p2 ::: pm .
-&' ' +, Re s 1 + 0
" "# 1
Y
1 ' m !1 , " + , s 1 pms = 1.
6.36.
X
m=1
2 / ' ' ' ' # . 6.13 1
Y
, " 1 pms Re s 1 + , 1 ',
m=1
s # Re s 1.
6.37. . 2 ' & " *# ,
1
Y
0 1 pn 1+ = 1 1+ . 3 &' ' +,
lim
!0
1
Y
n=1
n=1
1 pn 1+ = 0. / ' ' ' 1 pn 1 1 pn 1+ , , -
X1
1
Y
" 1 pn 1 ' , " + , pn 1
n=1
n=1
'4 .
2
2
1
2 ctg a
6.57. 2 . 6.58.
.
6.59. . 6.60. 2 1 + a cth a.
sin
a
8
2
a
sin a
1
a
3
6.61. 2 1 +
.
6.62. .
2a
sh a
32
6.63.
6.64.
6.69.
6.71.
6.74.
1k+1 22k 1 2k
2k!
sin ab
. 6.67.
2 sin a
B2k B2k | 6 ! . 3.100.
= 0. 6.68. = n, = a.
= 1, = 3. 6.70. = 1, = 3.
p
= 3, = 2. 6.72. = 2, = 5. 6.73. = 1, = 1.
p
= 1, = 1. 6.75. = 1, = 2. 6.76. = 12 , = 1.
VI
1
6.77. = m ,
1 2zn
X
=
X1
= 1.
n!
n=0
p
1
4
cos z + cos 2 p4 z + cos3
22
X
4
4
zn
1
=
cos k p8 z,
23 n!
4
n=0
k=1
6.78. = 1.
6.79.
2
ezt dt
n=0
.
ez
0
=
z 2n
22 n!
Z1
= 12 cos pz + cos ipz,
! ,
p4 z + cos p4 z, !
4
=
X1
n=0
z 2n
n!
23
=
p4 1,
. .
. " ,
% z 0. &!
1,
Z1
X1
cos p4 z + cos i p4 z .
1
2
=
.
251
2
ezt dt
.
# $, $
ez
1 !' (, 0
0
Z1
ezt
2
dt
0
Z1
p
2
ezt dt ! 1
) z ! 1:
* # $ , = 1, = 1.
6.81. = max 1 2 .
6.82. 1 = , 6 1 + 2 2 = , = max 1 2 .
6.83. 1 6 , 6 2 2 6 , 6 .
. + z = rei' , ) ,- r #
6.84.
1 z2 ! -$ ! % ) ' = # 1 + r2 .
2n
2
2n
n
* !' (, ) ,- " 0 " n n0 " + " n
n
, # $ ,
"
2 2
2
2 2
1 + n"2 r 1 + r2 1 + +n"2 r :
1
Y
n=1
n
1 + nr2
2 2
=
sin ir
ir
=
er er
2r
. 6.48
6.53, # # , ## ) #$ " % 6.80, , ) % 2 3 f z # 1, ) $ ) 2 3 ez -$ ) 2 3 ez , . . 6 6 .
6.86. . 4 $ M r , | jf z j 5
jzj = r, ) % ) 2 3 f z, M1 r, 1 1 | # # ,6
z
Z
7 % 2 3 f 0 z. 8 # # f z = f 0 t dt + f 0
0
Z M r 6 rM1 rjf 0j , # $ , 1 . & f 0 z =
=
1
2i
f d
,
z 2
! # # 9 5 #% $ 5 $ 3 #
z 0 | ,- , M1 r 6 M r + =0 , . . 1 6 ,
, - , 1 = . : , ) # (7 #(
# #
252
, 1 = .
, . .
6.88. = 1, = 1=e. 6.89. = a, = 1. 6.90. = a, = 0.
6.91. = 0. 6.92. = 0. 6.93. = 1, = 1.
6.94. = 1, = 2. 6.95. = 1 h!'" = cos '.
6.96. = 1 h!'" = cos ', cos ' 0 h!'" = 0, cos ' 0.
6.97. = 1 h!'" = j sin 'j. 6.98. = 1 h!'" = j sin 'j.
6.99. = 1 h!'" = j cos 'j. 6.100. = n h!'" = cos n'.
6.101. = 1=2 h!'" = j sin!'=2"j.
6.102. 1" h !'" = h!'", h!'" 0 h !'" = 0, h!'" 0,
h !'" 6 0, h!'" = 0 2" h !'" = h!'".
6.103. # f !z " = ez P !z ", =2 ' 3=2.
% VII
!z" 0 2" F +!z" 0, F !z" = 1=!z a"n
3" F + !z" = z an , F !z" 0. & ' ' (x
) * + ( ) * .
7.3. 1" F + !z " = '!z "
gk z 1 a , F !z" = gk z 1 a
7.2. 1" F + !z " = !z
1
a"n, F
X gk
2" F + !z" =
X
k
+ g!z", F !z" = '!z" +
1
z
X
ak
X gk
k
+
1
z
ak
+ g!z". gk z 1 a | ( ( . / '!z" k
ak g!z" | ( ( '!z"
,
( .
z
+
ln
z
2
=
z
3
7.4. F + !z " =
, F !z" = z2 +i 1 .
z2 4
z z2 z 2 , , F +!0" = 0, F +!" = ,
!z" = z z2 z 2 .
z
, F !z " = z z i .
z i
X1 an ibnn zn , F !z" = X1 Rn ann ibn
R
z
7.5. F + !z " = ctg z
F + !" = 23 F
7.6. F + !z " =
2
7.7. F + !z " =
1
2
1
2
1
3
2
2
+
2
1
2
n=0
n=1
(:
1
F + !Rei " = 21 !an ibn "ein =
1
n=0
= a40 + 12 '!Rei " + 2i !bn cos n + an sin n"
X
F !Rei " = 21
X1 !an + ibn"ein =
n=0
= a40 12 '!Rei " + 2i
X
n=1
X1 !bn cos n + an sin n":
n=1
VII
253
7.8. 1
z Qk : jz kj =2, I1 z = f z
k, I2 z = 1k f z k. "
, jzj =2, I1 z = I2 z =
= f z. z # $ # Qk , I1z = I2 z = 0.
2 % Qk | jz kj . ' I1 z = f z k + f (z
k 1), I2z = 1k (f z k f z k 1), z 2 Qk1 Qk
I1 z= f z k + f (z k + 1), I2z=1k (f z k f z k + 1),
ec z 2 Qk Qk+1 I1z = f z k, I2z = 1k f z k, z
* Qk Qk1Qk Qk Qk+1 I1z = 0, I2 z = 0, z
$ Qk .
1
7.9. F z =
ln zz + 11 5 F = 21i ln 11 + 12 , F = 21i ln 11 +
2i
1 1 F i = 1 , F 0 = 1 , F 0 = 0.
4
2
z+R
z R
F z = 2i Ln
| z-
z+R
z+R
$ C , $ $ Ln 1 = 0 Ln z R = ln z R +
+ i,C arg z, ,C arg z | - arg z C
F + = 21i ln + RR + 34 F = 21i ln + RR 14
1
7.10.
F = 21i ln
+R
R
F iR = 14
F + iR = 43
+ 14 F 0 = 12
i
F iR = 14 F 0 0 = R
:
F z = 21i ln + RR z-
$
C , $ $ Ln 1 = 0 jzj R
. / 7.10 F + = 21i ln + RR + 41 , F =
i
= 21i ln + RR 34 F 0 = 12 , F 0 0 = R
.
7.12. 1 0, jz j r jz j R, 1=z n , r jz j R
2 1=zn , Im z , 0, Im z
3 0, jIm zj , 1=zn , jIm zj
7.11.
4
1
2iz n
Ln RR + zz 2
Xm
k=1
z 2k 1
2k
1 R2k 1
, m=
hni
2
, Ln RR + zz =
= ln RR + zz + i,C farg z arg g | z-
5
1
2i
:
Z1
1
d
z
=
1
2i
ln 1
z
ln 1
z =
1
2i
ln
z 1
:
z+1
, , !", ! #!" $! !" ! &
! ' , ! $! ln z2 ln z1 =
= lnz2 =z1 , ( , $ . *! $&! !"
&
" .
254
C , Ln 1 = 0.
3
C 2 , |
.
2
n1
F C ! F 0 = 21 i 1 + nR1n !
5 #$% F z &, ' $, $ Ln RR + zz |
z- $ ) $& C ,
& Ln 1 = 0. C
32 , |
. 2
F C ! F 0 = 21i 1 + nR1n
n1
.
. * . 4 5 Ln RR + zz $) jzj R,
,' & ! $, , 1
i, +i. . $ F 1 = 0.
b z
b z
7.13. Ln
=
ln
a z + i/C arg z |
a z
z- $ C , Ln 1 = 0 $
& 0 7.1427.17 .
7.14. b a + z Ln b z = a z .
7.15.
)
1
Xn Ankzk + zn Ln b
1
a
z
!
z
k=1
bnk+1 ank+1
Ank = n k + 1 .
b z
1
7.16.
Ln a z Ln ab zz0 .
z z0
0
1
b z
b z0
7.17.
Ln a z Ln a z
z z0 n
0
2
X1 Xn cnAnkzk + ' z Ln b
+
Xn Ak z z k , Ak =
h
b
1
z0 k1
a
1
z0 k1
i
1
n=1 k=1
k=2
, F z0 = An+1.
0
1
z
,
z
a
k
1
1
1 F + z = ln z + R , F z = ln 1 + Rz ! F + = ln 2R cos '2 +
+ i '2 , F = ln 2 cos '2 i '2 ! 2 F + z = ln R z + i 6 , F z =
= ln 1 Rz ! F + = ln 2R sin '2 + i +2 ' , F = ln 2 sin '2 + i 2 '
) ' = arg .
7.19. 1 F + z = 0, F z = ln 1 1=z ! F + = 0, F = ln 1 1= !
2 F + z = ln z= z 1 ,F z = 0! F + = ln = 1 , F = 0.
hni
m
1 1
1
z 2k1
R+z
7.20. F z =
2
,
Ln
,
)
m
=
2i
nz n
R z
2k 1R2k1
2
7.18.
X
X
k=1
n=1
6 Ln z ln z |
,
, Ln z = ln z + i.
VII
255
a Ln RR + zz , 7.12, 4. ) (
X1
F C . 4 jzj R, F z = cn zn , cn =
n=0
Z
= 21i ln 1 d
.
n+1
p i
C
i
h
1h p 1 p
z
z = 1 2 + pz ln 1 pz
7.21. F + z = 2 + z ln
,
F
p
i
1+ z
i
1+ z
pz.
7.22. F + z 0, F z = Ln z b= z a .
r
za
7.23. F + z 1, F z = 1
.
zb
7.24. F + z = z a 1 b, F z = z a 1 b z a
z b1.
z 0
7.25. F + z = Ln
, F z = Ln zz a0 .
zb
1 ln z 2 2 1 ln 2 2 .
7.27. x2 + x 3 5. 7.28. 1
2 z1
2 1
2
1
z
R+z
7.30. 4 jz j 1 z 2= C F z =
2i ln z 1 Ln R z + F1 z,
F1 z | jzj 1, Ln RR + zz ,
, 7.12. 4. . ( F z 0 R |
0 C .
7.31. D f | , G0 , ( f z , , G.
R2 z
2 ln z 3 ln ez e .
7.37. 1 ln
Rz
z
ez e
2
Z2
1 u
r2 R2
1 Z u d .
7.43. u z =
d
,
u
1
=
2
R2 2Rr cos ' + r2
2
0
0
2
2
7.45. 1 f z = ' z + R =z6, f1 z = ' R =z6 z , v 0 =
= Im f 0 = Im 7' 0 + 18
2 f z = i' z + i R2=z6, f1 z = i' R2 =z6 + i z, v 0 = Im f 0 =
= Im 7i' 0 + i 18.
7.46. 1 f z = z n , f1 z = R2n =z n 0 7.462
7.50 v 0 (.
7.47. f z = z n =R2n , f1 z = 1=z n .
7.48. f z = ln 1 z=R2 ln 1 = 0, f1 z = ln 1 1=z .
q
R
7.49. f z = p 2
f
0 = R1 , f1 z = z z 1 .
R z
7.50. f z ln R, f1 z ln R.
256
7.52. u
1
Z
z = 1
1
y dt
,f
t x2 + y2
u t
z = i1
Z1
1 1
1
u t dt
t x
+
iC
-
, | , ,
1
0.
Z
Z
ut
7.53. f
ut
z=u z+ iv z = 21i u t cth t 2
u1 t = u t + i
u t
7.55.
$
jzj
ei .
z
1
1
, 1
1
dt 21i u1 t th t 2
z
1
=jtj1+
1,
=jtj
, ,
0
,
dt,
.
jz j
=1
7.57. % &, ' jz j 1 ei1 , ei2 .
1
z b
1
7.58. ! z ( a b = arg
, ! z ( 1 b =
arg z b( ! z ( a 1 =
z a
1
arg z a. ) * + , , |
= 1
& , * * z .
1
1
7.59. 1 arg z arg z = 0 arg z arg z = .
z R
2
1 = 1 2 arctg R22Ryjzj2 , ! z . =
7.60. ! z - = arg
z+R
= 1 ! z -. / |
&, ' R,
* , -
1 + ! ! z -
2
!
1
| ! z ..
2
7.61. !
7.63.
ln
jzj ln r
ln r
z =
ln R
7.66. u
8.1. f
z . = 2 arg zz + RR .
z =
jzj
Xc !
n
=1
X1
n=0
z + 1=2
R lnlnjzr jlnlnRR
jzj
r
= .
z.
z an
(+
1 an+1
a
f z = ln 32 +
3=2.
=
p
p
z . = 2 arg pzz + pRR .
) VIII
*
jz aj j aj
1,
8.2.
jzj
7.62. !
f z
1
, &
0 0 1.
X1
n=1
ln
n
2
3
z + 1=2n
(
n
-
,
+
VIII
Z18.15.x dx
x
1
Z
x dx
3
sin
=
xs = es ln x.
s
1
cos
1
4 . 5
6
. 5 &
xs+1
257
et = x
,
,
f z =
4 . $ * * 1
jzj
f z
Z1e
f s = cos 1
Re
t'
f s =
8.19. 0 Re z 1, 1 Re z 1.
8.21. 7 z = 1 | & '
8.21. 3
s 1.
.
zt dt. 8
0
1
1
* * * 3.150
,
& /
. . . 6
+ 1 *
* * 3.150,
1
f1 z =
Xz
n
k=0
Z1
n+1 e t 'n+1 zt dt =
zt:1
0 +z
k 0e t 'k
n
jz j
Xa z
n
0
=
j' n j
+1
n
k=0
n + z n+1
Z1e
t 'n+1
zt dt:
0
n ! 1 jzj r
z2G
fz
C
jzj=
fz
1
cn * f z =
cn z n
6*
,
&
'
. % *
* ' '
. 7
,
,
,
+ * , & &
. 5+
0
2+ ,
,
-
Oz
Z f d
C 1
X z t Z f dz :
&. 7 *, +
cn = 21i
' zt = 21i
Zf
X1 z t f
7
=
1
2i
n=0
zt= d .
e
C
q
nn
n! n+1
,
n+1
nn
n=0
,
n!
X
n=0
,
C
n+1
C,
; Re
z=
C
' zt
=
jz j
jz j +
=
=
1 * +
&, & , *
Re
, , ,
z
Aeqt A |
,
. $
.
8.26.
z-
z=
< Ln
17 ..
1
z z2
z z1
'
|
* *
.
,
1
z z2
Ln
,
z z1
Z1e j'' ztzt j dt
t
0
' 0
z = 1,
258
, F z = a b Ln zz zz + 2bi
G :
F z = a b Ln
2i a b
F z = a b Ln zz zz .
8.28. 1 z = 3 2 z = 2i 3 z = 2i.
! !
w z ".
8.30. # z = 1 $ : z = 1 + t,
1
1
t =
w =
= 12 + 81 t + ::: jtj 1
t
z = a
F
b Ln
z
z
1 z
z
z2
z1
z2
z1
1
+
+
1
2
1
1
2
1
1
2
0
1 2
1
t1=2
= 2t 12 + ::: 0 jtj 1
&' ( $ : z = 2 + t , w = 1 +1 it = 1 it
w2 =
1+ 1
t
z = 2
t + :::, jtj 1.
p
8.31. # p
z = 1 &' $ : z = 1 + t , w = i 2 1
=
t=2 = i 2 1 t=4 + :::, jtj 2 z = 5 &' ( $ : z = 5 + t , w = t=2 1 t =32 + :::, jtj 2, + " $ : z = 5 + t, w = 2i 1 + t =32 + :::, jtj 4 p z = 1 &' (
$ : z = t , w = 1=t t + :::, 0 jtj 4 1=5.
8.32. # z = 1 &' $ : z = 1 + t , w = 1 +
+ t = = 1 + t=2 :::, jtj 1.
8.33. # z = 1
&' $ : z = 1 + t w = !k 1 +
+ t=3 + :::, jtj 1, & ! = 1, p! = e i= . # z = 2 &' (
$ z = 2 + t , w = t= 3 2 1 t =12 + :::, jtj 1
+ "
$ : z = 2 + t
p3
w = !k 2 1 + t=12 + ::: jtj 1:
# z =p1 &' ( $ : z = t , w = 1=t + t =3 + :::, 0
jtj 6 1=2.
8.34. # z = 1 &' $ : z = t ,
1
a+b
1
1
1
1
at = 1 bt = =
1
t + ::: 0 jtj min
:
w=
t
t
2
a b
2
2
2
1 2
2
2
2
4
2
1 2
2
1
2 3
3
2
3
3
6
2
2
1 2
1 2
j j
j j
8.35. z = 0 | - '! , !: z = t , w =
= e =t = 1 + 1=t + 1= 2!t + :::, 0 jtj 1.
8.36. # z = 0 &' ( $ z = t , w = sin t=t = 1=t
1= 3! t + t=5! ::: 0 jtj 1.
8.37. # z = 0 &' ( $ : z = t . w = ctg t = 1=t t=3 + :::,
0 jtj .
8.38. # z = 0 &' ( $ : z = t , w = t 1 t =12 + :::,
jtj p.
8.39. # z = 1 | / ! ! 1-& !
z = 1 . .. 1-& !.
2
1
2
2
4
3
2
4
3
VIII
259
8.40.
z = 0 z = 1 | . . . 1- ' z = 1 |
! ' %& ' ( .
8.41.
z = 0 z = 1 | . . . 1- ' - ( z = 1 + 1=k2 k = 1 2 ::: ! %
.% 1- ' z = 1 | ! ' ( ' ' ( ! ' ' . .
8.42.
z = 0 | . . . 1- ' - ( z = k2 2
k = 1 2 :::
. 1- ' z = 1 ( ' ' ! ' ' . .
8.43.
- ( z = 0, z = 2 | / ! . . . 1- '
z = 1 . . . 2- ' / ! !" z = 1
. . . 5- ' .
8.44.
- ( z = i, z = i | -
. 1- ' z = 0 z = 1 . . .
8.45.
z = 1 | -
!" - . 1- ' z = 0 z = 1 . . .
8.46. w = 1=4, w = 1 | . . . 1- ' .
8.47. w = 2 | . . . 1- ' w = 1 | . . . 2- ' .
8.48. w = 1=4, w = 1 | . . . 1- ' .
8.49. w = 0, w = 1 | . . . 1- ' .
8.50. w = e =2 a = cos | . . . 1- ' .
8.51. w = 1 | . . . n 1- ' . 1 - % %. (
w0 z ' k % . . . 2%+ z w - ' .
8.52. %' ( w0 z %. . . . -,
" %& % . 3 . 2%+ wz, ' " !/ i
17
260
,
.. .
.
wz = w0 + c k =z k + ::: k 1, z =
.. . k 1- o
w = w0 .
8.53. ! " z w # , $ n w% $ # w = 0, w =
z = n, z = . ' w-
( ( 0 u , v = 0
2=n * $ z = n. ! n
+ , (
* 2 , wz | ez , " z w | "
Ln w.
8.54. ! " z w # , $ n w. ' w-
( (
u 0, v = 0
2=n $ " z = a, z = b . 72, n = 3 7 .
8.55. ! " z w $ "
w- ( ( u 1, v = 0,
," 0 z
1 z 1. 1$ #
w = 1
z=
= 1. ! * z = r. arg z = '
+ -
1
11
!1
p
0 1:
z
1
p
1
jj
1
jj
jj
x2
= r+
1 2
jj
1=r 2
+
y2
= r
1 2
1=r 2
= 1%
x2
cos2
'
y2
2
sin
'
= 1.
8.56. wz =
= z + 2z 2 + 3z 3 + ::: | ( +
1
z 2
" " 0 # * . 34, . XIII,
15. 6 0 # $ *
z 1
w- ( (
u 1=4, v = 0. ! " z w $
"
" .
8.57. ! " z w $
2n w- ( ( 1 u , v = 0. 6 2n $
1-
w = 1 2 $ n 1-
x
1
jj
jj 1
1
w = . 6 0 # (
. 73. 7 wz * " 0 ( *,
# * 0 ( T = !z ! = e2i=n , S = 1=z . 8 0 (
0 ( " z w 0,
7
. !" ,
!, $ $0
VIII
261
" $ "
"
* $
w-.
8.58. ! " z w 2n- , $* 2n 1-
w = 0,
, * z = , 2n $
1- , ( "
n
#
w =
$ zk = ei=n ! k ! = e2i=n , k = 0 1 2 ::: n 1 , n | $ -
1
wk0 =
2
n
1
2
n
2
zk0
1
r
$ zk0 = n
1
2
n
1
! k . 6
0 # (
. 74. 6 $ #
.
8.59. ! " z w n- , $* n 1-
w = ,
, * z = , n 1 $ 1-
, # wk = zk 1 1=n
, zk =
= ! k ! = e2i=n1 , k = 0 1 ::: n 2 . 9 " #
w- n 1 $ ( ( , ", ( wk # ( ( , - , w- $ ( (. 6 0 # -
1
1
jj
(
. 75. ;
z 1
+
n = 2 | .
8.60. ! " z w n + 1- , $* n
1-
w = ,
, * z = , n + 1 $ 1- , # wk = z<k 1 + 1=n
, zk = ! k ! = e2i=n+1 , k = 0 1 ::: n . 6 0 # (
1
1
262
. 76. z 1 n =
= 1 | .
8.61. ! z w n- , $% n 1 - w = 1, &'% z = 1, n 1 $( 1-
, ) ( wk = 1 k =2 1 &'( z =
j j
n
k
= cosk=n k = 1 2 ::: n 1 . + ! ) ( w | ( 1 u 6 1=2 1 , v = 0 n 2 ( ( 1=2 1 6 juj 1, v = 0,
(
( ! ' , (&'% 1
u 6 1=2 1 , v = 0, n | $ $ , 1=2 1 6 u 1, v = 0,
n | $ $ . ) wz -
.( 1 ! - .( 1=2 1 . / ( % $ % z = const, w = const. 0 . 77
n
n
n
n
n
z- , &' v 0 v 0 ! n = 5.
VIII
263
8.62. w = 0, w = 1 | . .., w = 1 | & 1- %
% . z w .
8.63. 2 z w ( % . .. 1- w = e 2ti
. .. w = 0 w = 1.
Ln w
,
, w- w=e 2 .
p
8.64. " z #w$
w = 2 & ' . ..
1-
w = 1 . ..
8.65. " z #w$ . .. 1-
w = sin z =z #z |
tg z = z* +$,
. .. w = 1 & w = 0, + . .. #.: -
.. / + .| 0.: 1 , 1941.| . 238$.
p
8.66. z = Arccos w = 1=i ln #w + w2 1$. 7 z #w$ & , . .. w = 1 . .. 1- w = 1,
z = k #k = 0 1 2 :::$*
& w- 1 6 juj 1, v = 0,
ti
k
k
k
k
+ + + k x #k + 1$ # . 78$.
8.67. z = Arcsin w = =2 Arccos w. 7 z #w$
',
Arccos w.
1
i
w
8.68. z = Arctg w = Ln
. 7 z#w$ & ,
2i
i+w
. .. w = i* &
w- u = 0, jv j 6 1, +
k x #k + 1$ # . 79$.
8.69. z = Arcctg w = =2 Arctg w. 7 z #w$
',
Arctg w.
p
8.70. z = Arch w = ln #w + w2 1$* ch z = cos iz . 7 z #w$
', Arccos w.
p
8.71. z = Arsh w = ln #w + w2 + 1$* sh z = i sin iz . 7
z #w$ +
=2
.
264
8.72. z = Arth w = 1=2 Ln 1 + w=1 w th z = i tg iz .
zw Arctg w
=2
.
8.73. z = Arcth w = 1=2 Ln w + 1=w 1 cth z = i ctg iz .
zw , Arth w.
8.74.
zw : w-
# # 1 u 6 1, v = 2k+, k = 0 1 2 :::
% w-
. zw
, wz & ' 2k y 2k + 1
2k v 2k + 2, (' w- , ' 1 6
6 u 1, v = 2k + 1 . . 80 + , &
.
VIII
265
8.75. 1 F |
, ' + , w =
= R & - . - .
+ , zw & % F , '( . F w0 w1 &
' . + , Ln w.
.. . , &
.. ., ( F .
2 - . + , zw & % F ,
( w1 w1 & ' . + , Arcsin w.
.. . w1 .. . . 1 ( &
.. . 1- w1 w+1 w1 .. .
k F , zw & .. . 2k + 1. / + , zw #( ' .
z1 = iz .
8.76. 0 #
wz # ' .. .
1- :
1 z = a, z = b 2 z = a, z = b, z = c, z = 1
3 z = ak z = 1, n .
- . & z- ,
( # #2 1, #
.
8.77. 0 #
wz # ' .. .
2- :
1 z = a, z = 1 2 z = a, z = b, z = 1 3 z = a, z = b, z = c
4 z = ak z = 1, n 3.
- . & z- ,
( # 1, # #
w, !w, !2 w ! = e2 i=3 . & wz & ! # ., %
#., , .
. . 81.
8.78. 4
wz n-, .. . n 1- z = a, z = b, z = c, z = 1.
n z - , ( z = a, z = b, z = c 1.
5 , , . 6#
' # + , !k w ! = e2 i=n k = 0 1 ::: n 1 .
8.79. 4
wz 2 , .. . 5- z = 1, .. . 2- . z = a, z = b
.. . 1- z = c.
2 z- ., ( . a b, b
c c 1. 8 # ' # : w1 + w2 ,
!w1 + w2 , ! 2 w1 + w2 , w1 w2 , !w1 w2 , ! 2 w1 w2 , ! = e2 i=3 , w1 , w2 |
266
r
p
z a
z
. z b2
1, 2, 3 4, 5, 6,
| 1, 3, 2 4, 6, 5
" # $ $ %,
| $ 1, 2, 4, 5
3
c
b
a
c
1
"$ . 82' 2, 3, 5, 6' 3, 1, 6, 4,
|
1, 6, 2' 4, 3, 5 "%. % . 82.
8.80. ( % $ ) " * , % . . . 2-
$ = 0, - . . . 1- $ = 1 . . . 5- $
= . . $ # ) /%$ $
3 , #-
0
% %
$
= 0, 1
.
8.81. ( % $ ) " , . . . 1- $
%
=
" = 0 1 2 ' =
$ ) % 1 ) | $ . . . . $ $
% -$ % , 2 %
. . .
"$ %,
$ % $)% % % - , % $ %
%.
8.82 8 . z
w $ 1 # % $ -
% 3 1 14 " . 83.
wz
z
z 1p
z
y
6x1
z k k
z
F
:::
wz
z
z 1
1
F
8
8.828.87 Fw z w,
. . w- !, Fz | #$ % w z
z - !.
VIII
267
8.83. Fz $ % z -$ - % " 1
x 6 0 y = 0, Fw | w-$ - % 3 i=2 i=24 " . 84.
8.84. Fz Fw $ 1 # % $ -, %1-
p
!
p
2 % $ $ % 30 3 44, 30 3 44,
30 2 3 44, = 2i=3 " . 85.
8.85. z $ % -$ - % 30 3 24,
-$
%
4,
w | $ %
1: 3 3 3 2 +
3 3 2
4 -$ - % 1 ,
2: 3
2 " . 86.
8.86. z $ % -$ - % 3 1 14,
w | % -$ -, %12
3p
4, 3p
4, 30 4
3
4, 30
4,
!
F
F
a=
p
i
p
F
! e
= i1
F
ib ib p
p
51
2
5
w
w
2,
b=
z
w
a a p ib
p
5+1
2
5 + 2.
z
i
a a
#
p
=
= i1
ib
w-$
-
% $ . 87 " . %* .
8.87. . z - , %
. . . "
1- $
%
= k " = i=n
= 1 2 p , w 2 - , % 2 . . . 1-
$ , $
%
= n 1 4
= 2i=n
=0 1 2
1 ,
n
F n
n
z ! ! e k ::: n F n
n
w
=
e
n
n
::: n
268
F
z
z !
n
1
w
z = ,
... 1- = ,
... z
= k . . 88
!
= n,
= n ". # $
"
1
- !
%
, %
n 1 4.
"
. '
= 2 ,
" ) zm
$ : 1m2 =
. + 1 z 2m
z
,
"
1
,
"
1.
-
F
z
w
W w
wz
jz j
n
p=
jzj
n
w
F
jzj
n
wz
m
+
n
n
+
k, |
k0
k , + % , 0 %
k . /
k , !0 k % 1 , ! k+1 | % 20 , ! k+2 |
% 3 . . $
. 0
H
H
H
H
H
H
!
VIII
269
Fw 2n , w- n
% , % %1.
8.88. w = z 2=3 , z = 0 z = 1 | ... 2- .
8.89. w = z n1 =m1 n1 =m1 | !, n=m", z = 0
z = 1 | ... m1 1"-o .
8.90. z = 0 z = 1 | ... 1- 3 z = 1 |
4 4 ) $ .
8.91. w = 1 z=3! + z 2 =5! ::: | $ ) $ 3 z = 1 |
.
8.92. z = 0 z = 1 | ... 1- 3 z = 1 |
4 4
) $ , ! 1-
1
ak 2
zk = 1 + a , ak = 2 + k k = 0 1 :::". 9 k
4.76, ! z = 1 4 !.
8.93. '
n = 0, z = 0 z = 1 | % %
w 13 n 0, z = 0 z = 1 | . . ., zlim
!0 w = zlim
!1 w = 1
z = 1 | 4 4 ) $ 3 n = 1, w = z3 n 1, z = 0 z = 1 | ..., ! w z" <
4 1 4
w- !.
8.94, 8.95. z = 0 z = 1 | ... !
w z", 4 1 4 w- !.
8.96. z = 1 z = 1 | ..., lim w = lim w = 1 z = 0 |
z !1
z!1
1- 4 w z", 4.
8.97. z = 0 z = 1 | . . . lim w = lim w = 1.
z!0
z!1
8.98. =
, Arcsin z . . ... 1- z = 1 ... z = 1, zlim
!1 w = 03
z = 0 | 1- 4 ) $ 4, 4".
8.99. z = i | ... lim w = 13 z = 0 | 2-
z!i
4 ) $ .
8.100. / w z " z w" , ) 4 ) $ ... 0 1". ! z = e , w = ea .
8.101. > ! w z " , 4 ...
4 z = a, z = b ... z = 1. / !
z- ,
z = a, z = b 1. 0
% % ) $ w + 2 in n = 0 1 2 :::". / z = a
z = b < ! ,
.
8.102. > ! w z " , 4 ...
4 z = a, z = b, z = c ... z = 1. / ,
%.
8.103. > ! w z " , 4 ...
4 z = k k = 0 1 2 :::". ?
270
1
z -
, z = k
,
.
,
, !
! " . #
| "
% , .. .
8.104. 1 # '
! "
= 'z z-
,
% ! " z = ' 1 G , wz
! " (
2
'
! "
= 'z,
' Gz * % G , z -
, wz
! " .
3 , ',
. 2. #
wz ! " .
8.105. wz
! " z .
1
8.106. wz
8.107. wz
p = .
.. m n(
8.108. wz
::: n 1.
! "
z = .
2 3
p
! " ! k z m1 =n1 ! = e2i=n ,
k = 0 1 ::: p 1( m1 = m=p, n1 = n=p.
n " ! " ! k ez=n ! = e2i=n , k = 0 1 :::
8.109. wz | n- ! "
z = k k = 0 1 :::. . 1
.. . n 1
% ,
.. .
8.110. w = n Ln z + 2ik k = 0 1 ::: n 1 | n ! " .
8.111. wz
! " z + 2ik k = 0 1 :::.
8.112. 0 % ! " .. . z = 0 1 .
8.113. wz | % ! "
.. . zk = 2ik k = 0 1 :::. .
% , .. . 1 ! " wz
%
w-
% % ,
zk
,
.
8.114. 1 ', 8.113,
.. .
z = k k = 0 1 :::.
8.115. 1 ', 8.113,
.. .
z = k=2 k = 0 1 :::.
8.116. wz
! " z + 2k k = 0 1 :::.
8.117. wz
! " z + k k = 0 1 :::.
8.118. 1 3 r1 = m1 =n1 , r2 = m2 =n2 , r = r1 r2 = m=n |
%
p =. .. m1 n2 , q =. .. m2 n1 . , zr1 r2 p
n-
! " , ! k z r , ! = e2i=np ,
k = 0 1 ::: p 1, zr2 r1 q r! " !1k1 zr , !1 = e2i=nq ,
k = 0 1 ::: q 1. 0 , z , ' %
. # , z 2=3 3=2 = z , z 3=2 2=3 = 3 1z .
2 3 r1 = m1 =n1 , r2 = m2 =n2 , r = r1 + r2 = m=n |
%
p = . .. n1 n2 . , zr1 zr2 p n-
1
1
p
1
VIII
271
! " , ! k z r , ! = e2i=np1 , p1 = . .. m1 n02 +
+ m2 n01 p, n01 = n1 =p, n02 = n2 =p, k = 0 1 ::: p1 1.
3 3 p =. .. n1 n2 N =. . n1 n2 = n1 n2 =p. , wz p N -
! " .
8.119. 3 N =. . m n, p = . . . m n, q = . .. m + n=2
mn=2 1=m + 1=n = = . , wz | N - ! " , N=n .. . n 1- z = 0, N=m .. .
m 1- z = 1 N=q | .. .
1- o z = .
8.120. 0 nm- ! " , .. .
n 1- z = 1, n .. . m 1- o z = 0
.. . mn 1- o, z = .
8.121. ! " , 4 z . 4'
, '
, ! "
* ! " ,
z = 1
.
8.122. 5 ! "
.. . n 1- o
z = 1
n .. . ' z = 0, z = . -
1
1
p
1
1
'
n Ln z
6 1
n sin n = k k = 0 1 2 ::: ' . 7
% w-
% ,
y
0
. 89, n = 2, = w2 |
.
8.123. 5 ! "
.. . z = 1
%
.. . z = 0 z = . 1
%
Ln z
6 1 . 3 z = 0, z =
.. ., %
, z = 1 | % .. . 7
eu sin v = 2k + 1 v = 2k k = 0 1 2 :::
% w-
% ,
' z -
x 0, y = 0 1 x
,y=0
w
% ,
" v = 2k
. 90, = e |
.
8.124. 5 ! " ' ,
Ln Ln z .
8.125. 1 % , .. . n 1 z = 0,
.. . 1- z = 1
2n .. .
z = . '
n
1
1
1
1
6
6 1
1
272
Arcsin z 0 1 . n =
= k=n n cos n = k k = 0 1 ::: w- ,
" y 0 . 91, $ n = 2, = w2 |
$ .
8.126. ' ( ) ) $ ) ( Ln z 0 1=e, e 1
( Ln z , , $ *( . + ( z = 0, z = 1 )
... z = 1=e, z = e | , , ) )
$ ... 1-$ . Im sin w = cos ush v = 2k + 1 u = =2 +
VIII
273
+ k k = 0 1 2 ::: w- , " z- 1 x 0, y = 0 ,
: 0 6 x 6 1=e, y = 0 e 6 x 1, y = 0 | , $/ ,( ( , u = =2 + k . 920 = sin w |
$ .
8.127. 1 ) ) 2/. 3 ( )
) $ ) * ( n z 1, ,(
2/ p
)
1 6 x 1, y = 0. 4
( $2 ) 2/.
8.128. 1 ) ) 2/ 5 ... 1-$
z = 0 z = 1 ... z = 1.
8.129. 1 ) ) 2/ ... z = 1 ), ) ... 1-$ z = 0 z = 1.
m
m
2i
8.130. 3 = , w = Ln z +
n
n
n k k = 0 1 ::: m 1 | m ),( )( 2/0 / , w = Ln z +
+ 2ik k = 0 1 ::: |- ) $ ),( 2/.
8.131. 6 ), ) ), ) 2/
( , ) Ln z.
8.132. 6 ), ) 2/, ,
2 Ln z 2 Ln z.
8.133. w = =2 + 2k, k = 0 1 :::, w = =2 + 2Arccos z 0 w = 3=2 +
+ 2Arccos z.
8.134. w = =2 + k k = 0 1 :::.
8.135. w = i=2 + k k = 0 1 :::.
8.136. w | ) ) 2/ ( , ) Ln Ln z . ) 8.123. 3 = ei = Ln z = ln r + i'
' = Arg z, w = ei = ei i ik . 6 f z = e i 0 6 6 2
, ) ,( ) :
1 2 jwj = e 0
3 jwj = e = ' ! +1 jwj = e = ' ! 10
4 / e = 6 jwj 6 e = .
6 $( $ e k k = 1 2 :::.
8.137. : ( ( wz ) 2/ , jaj 10 jaj 1, . 1 3 2/ n,
. 2 4 | ) ) ) , ( )( 2/.
8.138. 3 jaj 1, ( )( wz
) 2/ 0 jaj 1, . 1 n, . 2 )
$ ),( )( 2/.
8.139. 1 + z = rei' = ei = Ln z . ; $ wz $ ),( )(
2/, ,( zei k
p
jzj 10 = ln r + ' = arg ln r + i', ) 1 ' 1 =2
3=2 k = 0 1 :::. : ... z = 0, | / ) , k = 0
/ 0e = 6 jwj 6 0e = .
Ln
ln
+
+2
+ ln
0
2
3
2
3
2
2
2
ln
2
3
181=2 ..
2
2
.
2
+2
274
2 z 1 = rei'
= ei = ln z 1 .
w z
p ,
i ln +2k jz j 1 = ln2 r + ' + 2n 2 = arg "ln r + i ' + 2n #
z e
$ % =2 ' 3=2, 0 2 k n = 0 1 ::: . () k, n
* . + , 1 = zlim
!1 z ,
- $ $ z ! 1 | / , $
k = n = 0 / jwj = 1 e .
p
fz
fz
8.140. 1 ,
. 1 2
|
a a :::
f z ,
a a :::
1
fz
1
pf z
. ..
1 2
,
.
| , =
| " #
$ ,
, " # $ ,
# .
fz
fz
a a :::
fz
z
a a :::
fz
fz
ki k :::
fz
fz
2
1 2
, Ln
1 2
. .. .
, Ln
# ,
" 2
= 0
1
2
.
3
, ) *
+,
Ln
. . 2 .
, ) * # , "
2k i
+
=0
1
2
.
fz
fz
fz
e
8.141.
,
,
fz
k :::
jz j 1 . ..
z = 0.
jzj
pz
X1 z
n=1
n! ,
8.142. 1 ,
1
= 0
, +
1 -
z
jz j
jz j
2
1-
= jz j
3
, 1 2
2.
z
Ln z + -
Ln , +
. IX
A2 2 = 0- 2 0 A1 A2 1 = 2 = 0A3 3 = 0- 4 0 A2 A4 2 = 4 = 15 0 A2 2 = 2, 0 A4 4 = 06 0 A2 , A3 A4 2 = 3 = 4 = 07 0 A2 2 = 2, 0 A4 A6 4 = 6 = 08 0 A2 2 = 2, 0 A4 4 = 2.
9.5. 6# , # k = 1=nk nk |
1
1
1
1
= 2,
$ + n = 4 n1 =
nk
k=1
= n2 = n3 = n4 = 2 . . n = 3:
9.3. 1
/ 0
3 0
X
IX
n1
n2
1
1
2
2
2
3
n3
P
6
4
1
2
3
4
3
275
3
1 w = h ln z + a, z = e wa =h a | $ % ,
z w | $ ) ) $ % ! = 2hi $$ G $ /)
$ - % T w = w + ! %) - B
$
z
w a
2 w = h ln 11 +
z + a, z = th 2h a | $ %
z w | $ ) ) $ % ! = 2hi $$ G %) - B /, $. 1.
9.7. w = arcsin 2, z = sin w z w | $ ) ) $ % ! = 2 $$ G $ /) $ - )% : T w = w + !,
S w = w %) - B $ 0 u
$ $ )% u = 0, u = , v 6 0.
9.6.
9.8 9 .
Zz
1 w = C z5=6 1 z 1=2 dz, C = B1=6! 1=2 B p q =
0
Z1
= xp1 1 x q1 dx | < 1- z w | ) $0
) ) $ % 2! 2!2i=6 $$ G $ /) $ - )% : T w = w + 2!, S w = we2i=6 %) -
B %
Zz
2 w = C z3=4 1 z 1=2 dz, C = B1=4! 1=2 z w | ) 0
$ ) ) $ % 2! 2!i $$ G $ /)
$ - )% : T w = w + 2!, S w = iw %) - B !
)1,
Zz
3 w = C z2=3 1 z 2=3 dz, C = B1=3! 1=3 z w | ) $ 0
p
) ) $ % 2hi 2hei=6 , h = ! 3=2 $$ G
$ /) $ - )% : T w = w + 2hi, S w = we2i=3 %) - B % .
9 !" # " . 62.
181=2
276
9.9.
" w = 0, w = d = B
, . + 1, # +
1, $ % #
+ = 1, d = =sin ,
" & ''( ), = # + = 2 (
", (+ ,
, ' (, ' ( ' '$ ( (
6
d=
sin
1
1
#
')
2
, (+
=
=
3
2
,
" ' %)
''( . 93 .
" ) ) ) w = 0,
. / (
" '0
, , | ', + -
1
h=
+ 1
. 2 (
+
sin
(
h .
3. 4'' & 5. 6 , , 0 . 148, '. 13.2. 3
= 1 ' ' ) # + = 1 (
' ( h = # = 2 ' '' ,,
)
) '$
) '
h=
sin
, ,
+ 1
h = 4, = 1 , h = , = 1 . . 93 .
2
9.10. 1 9. . 94, 1 # w1 w1 = i 1=2 # w
2 . . 94, 2 # w w .
p
p
2
9.11. 1 w = :arcsin z 1 2z z z 2 ;#
2
w=
2
:arcsin
pz pz z ;#
2
w .
p
h
1+ z
3 w =
ln
p 2pz
1 z
4
p
IX
p
=
t =
z 1
Z
a
1=q
Zz
1
t
z1
t t
= , o
X
q
w= a
1
1
tp
=0
1 .
p
a
= , o w =
t t
q
p
q
0 $ ,
,
1
p
d
B =2 1
Zt
z
2
0
1
1
t
2
dt.
1
'. 1 .
a
2t
1t
9.13. w = , =
+ ln
, t=
1 t2
1+t
9.14. w = 2h= z 2 1 + arcsin 1=z .
9.15. w =
p
w = ia z z 3 =2 1 .
= e2i=q = 0 1 ::: q
1
,
5
p
q
dz
.
z 1
z
z
a
z1
2 w =
dz .
z
p
w = 2harctg 4 z + arth 4 z 2 4 z = #
9.12. 1 w =
t
t
p pz #
h
arth z
2
277
qz 1
z
.
+
ln 1
Xt
q 1
=0
p
tt
ln 1
,
278
9.16. 1
2
w
3
w
2
=
=
2
=
w
Zz
arctg
H
C
0
H
=
w
Z1
2
w
=
C
1
tn
=
w
w
=
C
1
2=n
dt
t
2
t
=
5
4 5
dt
+ 1,
t
a
2
a
=1+
= 1+
h2
H2
h2
H2
C b
b + 1 a + b
=
h3 :
=n dt
2 2=n n sin
n
.
B 1=n + 1=2 1=2
1 = C =
c
,
z2
h2
0
+ 1,
p
=
n
1
1
B 1=10 1=5
a2
C , a, b
p
Zz
5 22=5
=
5 2 5
1 + t
dt
0
t5 4=5 1 + t5
2=5
t2
0
H
C a
a 1 a + b
Zz
h1
2
Zz
2
=
n
B 1=n 1 2=n
z
9.18. 1
+ h arth
a2 z 2
pt dt
,
1 t t a t + b
C
a 1 a + b
9.17. 1
q z
hq z
arctg
+ h arth
,
H a z
q z a z
q
h
z
1 = C =
c
2 2=5 7=10
9=10 4=5
.
9.19. 2=n +
, ! " # $
"#
w 1
9.20.
w
=
CC1
w
=
Zz
z
i
n
B 1=n 1
=
9.21.
f bk
9.22. w =
1,
k=1
,
i
z
2
C0
k=1
tg
=
z
2
k
n
2=n
i n
2
tg
2
2
k
n
dz
n
,
n
2=
sin
dz
=n e i=n
2
C1
,
n
j =
ak
C z
lk k
::: n
= 1 2
=
1 +
2
:
m
= 2
2
i
2
P
5
m
= 2
+ 1,
,
C
=
=n .
2
3
bk
,
.
# .
12
z + 12
z
2
,
C
=
h
4
2 2
f bi f ds
P
ak
bi
ds
2
1
j
.
j, j
j, $
. %
# . &
4
n
"
,
m
Y
2=n
! "
cj ds
z
2
2
sin
Zz mY1
C C0
n
2
n
1+
j
9.24.
=
1
1
2 2=n
cj
ak bi
,
,
, # )
* . &
# .
1,
4
5
IX
9.26. w = C z + 1 1 z 1 2 ,
9.27. w = C
r
z 2 a2
,
1 a2 z 2
279
C = hb + 1
r
b = 1 + 2 .
1 b 1 2 ,
h
.
H
1
2
C = hH , a =
9.28. w = Cz 1 2 z 2 1 , C = hb2 1 1 b2 , b = 1 2.
z + a 1 z b 2
9.29. w = hH
, a b
1 + az p
1 bz
& a 1 b 2 =
h=H , 1 1=a a = 2 1=b b .
1
9.30. 1 w = 1Tn z 21=n , Tn z = 1z + z 2 1 n + z
z2 1 n 2 |
2
& 4)&$
h T zpz2 1 i1=n
n
2 w =
.
n
9.31. % & " Re f bi Re f ds , &* P
/ P . + ak , bi ,
cj , ds &) . , ak cj
, 6 7
, ) ' &. , bi ds
, 6 7
p
p
p
p
p
0
1
1
) .
# " 9.24 .
9.32. % & " Re f bi , Re f ds . + ak , bi , cj , ds &) . 8 9 10 a bi , cj , ds &) .
9.33. % & " Re f bi /
k ,
9.34.
P . ;
&)
.
fb
fd
% &
" Re i Re s
/
. + &) . 8
9 11 , ' $ , &) .
P
w = h1 ln z + 1
9.35. 1
+
h2
C=
2
3
w = h1 ln z2
w=
h1
ln
1 +
z + a1
+
1 + a1 z
h2
2
=
w=f
5
w = h ln Tn z , Tn z
2
z
+
z
" 9.30
6
w = h 1ln z
z
b
1 +
ln + 1
C,
z + C,
e
, f z
i1
4
1
h1
h2
b , b = hh1 + hh2
ln 1
C = h1 ln 1 b2 h2 ln b,
h2
z a2
ln
. % & a1 , a2
1 a2 z
h h
& : a 1 a 2
h1
ln
z
ln
1 + 2
, h1 1=a1 a1
|
|
=
1
b=
r
2
h2
2h1 + h2
h2 1=a2 a2
)/
. 3
& 4)&$ .
# -
280
w = ln z1 + 1z + Az, A = b2 2 1 , b
b+1
2
b
ln
+
= d
b 1 b2 1
8 w = ln z + 1 Az 2 z + const, A = 1=2a a
ln a + 1=2a 1=a + d = 0. , d = 0, a = 1
h
C
h
h
2
9 w = ln z a +
+ Az + const, A =
z a
2a , C = 2a 1 a
a ln 11 + aa + a2 = d
. , h = 0,
h
a = 0, A = C = d=4.
9.36. ! "#
$ " u, z ' $ . 95,
7
1+k
&''$( $ C C 0
$ ' (
. ) * "
2 + i ln $k #
$
, p $ u + *
1= k
$ z "., " , $ .. /0" &. 1 0,
$2 . 162, . 13.25.
Zz
dz
w
9.37. w =
k , k onpe
, z = sn
0
9.38. .
p
K
K
1
0
=
. 96.
z 2 1 k2 z 2
b
, a = K .
a
9.39. .
. 97.
0
IX
19 ..
.
281
282
9.40. z - u
. 98
k1
l
h
v-
9.41!.
k
E
K E
9.41. "
1
1
1
E k 2K
E k2 K
z
v
k
k
k
. 98. #$ l h k
E
K E
$ . I.
9.42. % & . ' 1
1
1
C1 , k b
E k2 K
E k 2K
k
k
k
) ik
1
1
:
E
K E
k
k
k
1! w1! = a
C1 ,k2 b2
ik
1
1
1!K + E . = a/
E
K E
2! w1! = w1=k!
k2 b2
k
k
k
1!K + E = k2 b2 1!K + iK ! + E + iK E ! . 9.41!/
r
3! wb! w1! = ih.
1
2 2! b = k KE . 3 1!,
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
h
+ EK i = a, . . C1 = 2aK . 4 10!.
C1 E K KK
K
. 165!. 89 k 3! :
1rE
1rE
E K !u k K k + K v k K k = 2 a + ih!:
0
0
0
0
0
0
0
0
0
0
0
0
IX
283
3 & 94 . 9:
B e t z A. Konforme Abbildung.| Berlin, 1948.
9.44. " < 1!, 2!, 3! 4! $ . 99,
100. # ) ) u-
9 9 9 1-9 . 3 9 I z- :
w- 9 < $< . I + II . 99, 1!, 2!!, 9 $ $ . I + II . 100, 1!, 2!!, 9 $ )
$ )!. $ $ H , l, h
II
1 2
k
k 2
0
0
H
l
j j
k2
0 0
2 j + 1jj + k2 j k2 + 1 k
r
j j
1 k
2 j + 1jj + k2 j
r
j j
k2
0 0
2 j + 1jj + k2 j k2 + 1 k
r
j j
1 k
2 j + 1jj + k2 j
1
H
r
= 1
|
= k 2
|
k
1
k2
k2 +
1
k
|
h
1
0 k0
k2
0
+ 1 k
1 E k0 2 K
1E
H
H
H H
0
1 E 0
k2 K 0
k0 2
k0 2
1 K0 E0
k0 2
H
H H
k
2
H
|
k0 2
. l" !" # # $". 0 = k0 2 =k2 + :
. . II. = , < 2 <
, $< | < z t: k z2 + k02 t2 = 1. 3 , k
) 2! 4! 0 = k2 + k 2 k
0
w-
w-
dl:4
dl:3
=
k2 +
k2
$
"
w
19
k
=
=
2
l
1 hE
k
02
=
u
$
0
=
u-
w-
=
$
k2
0
1
=
l
=
1 K
$
0
0
$
0
k0 2
0
0
0
E H =
0
=
,
k2
. ()-
0
=
sn u cn u i
dn u
k2 +
k2
=
1
E $u +
0
k
dl:2
dl:1
i
1
sn u dn u i
E ik u
cn u
k
k
1 E k 2 K h 1 E k2 K
k2
k2
1 hk 2 u
02
. &
*) . II,
=
0
1 3
!
. 101 "
w
,
1 E:
k0 2
284
IX
285
9.45. .
0 , e1 , e2 , e3
g2 0. e1 e2 e3 .
Im z 0 ! " 0, !, ! ! , !
# Im ! =! 0$, % 1, e1 , e2 , e3 . &
" % ! # . 102$:
0
0
0
* p
e3 , + e2 e1
# ,
#e1 e3$#e2 e3$$, p *
e1 , + e2 , e3 # #e1 e2 $#e1 e3 $$.
, ! ! #w$ * , !, ! . + / *:
0
!=
1
Z
p
x
e1
dx
x e2 x
! =i
0
e3
Ze3
p
e1
1
. 102
e3
#w$ = e3 + sn2 eK1 w=!
k2 = ee2
k
e1
1
dx
x e2 x e3
e3
e3
x
:
:
2 g3 0, e3 e2 0 e1 ! j! j # k k , , K K $3
g3 0, e3 0 e2 e1 , , ! j! j. 2 g3 = 0, e2 = 0,
e3 = e1 ! = j! j. . ! + . + z- #1 3 4, 5e1 1$,
50 i1$ " #0 2! 2! $, %+ " # . $ # . 103$. 7 ,
e1 , e2 , e3 #e1 + e2 + e3 = 0$ "
z- ,
+,
e1, e2 , e3 1 # . + . 104$.
9.46. . 9 ! . 105. 9 % * ! " II.
0
0
0
0
0
0
286
IX
287
, dw = p z e zdz e z
1
2
Arg dw = + Arg dz
3
21 Arg z ek , k=1
, Arg dw " P BMC
=2,
0, =2, , # . 105 &^
'. (, # , P B
. 106 arg dw = +
+ 2 + =2 1 + 2 +
+ 3 =2 = =2 + 1 + 2
3 =2 = =2 . .
* # # # & # #
! , ! +,
w
X
Z1
!+! = p
0
0
e2
x
e1
dx
x e2 x
e3
!!
Arg 1 =
e3
0
i
=2
Zp
e2
1
e1
dx
x e2 x e3
,
x
:
- g2 = 0, e1 , e2 , e3 ' . #
# # + 60 , g3 0 e2 0, 120 , g3 0 e2 0.
/
# # #
z -# , 0
e1 , e2, e3 . 107, 108.
9.47. 1 w = 3z =e1 42 , # # ! , i! 5
2 w = 0 z=2je2 j3=2 , # # ! = hei=6 , !0 = hei=6 5 h =
p
= a 3=25
288
3 w = 2 z=4e32 , , . 2.
9.48. 1 z = snu k, w = e u=K , ln = 2 K =K Re u K , Im u K
0
0
0
j
j
j
j
0
2 , . 1, ! 0 Re u
K , Im u K , ln = K
K
. 1 # $! % & ' ($ k =
p3 #
p #
= p p 1 , = a b c d |
+ 1
p
4 . 3 ! " # t = z2 + h2
5 z = sn u k, w = iei u= K , ln = 2 KK 3K Re u K , 0 Im u K
p
6 . 5 ! " # t = 1 + z ) k =
= cos
7 " # ! + , #, . 5, - 0
1
K
1
Im u K , . K Im u K ln =
2
2
4
K
0
K
i
u=
K
8 z = k sn u k, w = e
, ln = K jRe uj K , 0 Im u K 0
9 " # ! + , #, . 8, - 0 Im u
1
1
K0
K 0 , . K 0 Im u K 0 ln =
2
2
2K
10 . 8 ! " # t = pz1++1 ) k =
p
= 1= 1 +
z
11 . 7 ! " # t = 11
+ z ) k =
= sin=2
j
0
j
0
2
0
0
2
0
0
2
0
0
X
289
12 # # . 2 # $! t = i sin z, ($ k =
= sh H1=sh H2
13 # # . 7 # $! t = sin z ($ k =
= cos th H
14 # # . 5 # $! pt = p
sin z #
& % & ' ($ k = p p 1 , & =
+ 1
sin 1 sin
= 1 +2sin
sin
h
i
0
u
15 z = C Z u + dnsnu cn
, w = e iu =K , ln = 2 KK , & k #
u
% Z tg = 2K , dn2 = KE .
1 cn u . 3 * K iK 0 , K iK 0 + ei ,
9.49. t =
sn u
ei , & cos = k.
2K arcsin z.
9.50. 4 # ' 9.49 # $! u =
5 ($ k # ' q = e K =K = a2 +1 b2 , & b =
p
= a2 1.
1 dn u , & u = 2K arcsin z, jRe uj K , jIm j K 0 . 5 $
9.51. t =
k sn u
k $ ' , ' 9.50. 6 # z = 1
+ 1 +k k0 .
h
dn u cn u i, t = 1 + cn u
9.52. 1 z = C Z u +
sn u
i sn u
h
1
+
cn
u
u
2 z = C Z u + 2KK 0 , t = sn u
h
i
a
3 z = hk
E u + iE 0 k2 K 0 k0 2 u + cn u dn u . 7 #sn u
+ #$. & , ' ' 9.41.
8 X
10.1. 5# ! # # #! V = i . 9 1 |
! # $$ $ p = 2c. : x + y = C ( !
x y = C.
10.2. 3 z = 0 | # ' , 1 |
$ ! ! 2n ' rn cos n' =
= C | ( ! , rn sin n' = C | z = rei'
V = nz;n 1 .
10.3. 3 + z = 0 z = 1 | + # : 0 Q =, 1 Q =
ln r = ='=Q + C | ( ! ln r = Q'== + C |
. 7 # $ %# % | & $ # # # + = = 0
Q = 0 ' ( + # $ %# | # r = C , & |
' = C . 4 #! V = j + iQj ei' arctg Q=+ =2 z = rei' .
2r
'
0
290
10.4.
a, b |
: a Q , b
Q
-
| a b . 109 ln =
= Q + C | # $ %& ln = Q + C |
iQ
a b
z a = ei . ' % V =
.
z b
2i z az b
10.5. z = 0 | ( % p = 2 r = C cos ' | # $ %& r = C sin ' | . V = e2i' =r2 , V1 = 0.
2 i % 3 4i=25.
10.6. 1 2 0 1 | ( 2R2 2 - . . 1, / - | . 2. 0 | & 3-
2
( x x2R+xy2 = C | # $ %& , y x2Ry
=C |
+ y2
V = 1 R2 e2i' , V1 = 1.
r
2
. 2
. 110
111.
z= R
. 1
z = iR
X
291
10.7. z = 0 | ( % r2 = C cos 2' | # $ %& r2 = C sin 2' | V = 2e3i' =r3 , V1 = 0
. 112 10 .
10.8.
a | 2 % 2 , 1 | 2 % 4 . 3 $ %& | &
jz2 a22j = C , 2 2 | 2& x 2xy=C y = a $
( (4
. a V = 25z =5z 2 a52 ,
V a = i=a ( |
. 113.
10.9.
a, ai | : a 2, ai 2 r4 +
+ C a2 r2 cos 2' + a4 = 0 jC j 2 |
# $ %& ( / / & y = x
4
2 2
4
r + C a r sin 2' a = 0 |
( / / 4az2
( V = 4 4 , V1 = 0, z = 0 | . 114.
z
a
10.10. 1, 0 1 | : 1 Q, 0 Q, 1 Q
+ 1=r2 = C + 2 cos 2' C 0 | # $ %&
p p 2%6r
. C 2 . / r = C r = 1= C r =
tg '
= CC + tg
| ( / / (
'
h 2
i
/% r = 1, V = Q z 2+ 1 , V1 = 0. 1 i |
2 z z 1
. 115.
10.11. i, 0 | : i 2 , 0 4 C r4 2r2 cos 2'
21 = 20 C 1 |p# $ %& C = 0 | 2
y x = 1=2 r =
C sin 2' cos 2' |
C = 0 | r
2
292
2
=
,
2
z z + 1
,
. 116 .
V1 = 0
V
=
p
10.12. 1 i= 2 | 2 , ! z = 0 |
4 , 1 | 4 # r4 +
+ 1=r4 = C 2 cos 4' C 0 | %! & C 4
'( , C 4 | !# p
1
* + ( C % ," r = 4 C r = p
# tg 2' =
4
C
4
r 1
=C 4
| , r +1
h z4 1 i
, V1 = 0.
. ! r = 1 # V = 2
z z 4 + 1
/ 1, i | . 117 .
X
293
10.13. z | 0 Q , z = 1 | 1 Q y = e C ax =Q x : ar cos ' +
Q'=2
+ 2Q ln r = C | r = C a sin
| '
C
!" : yx! 1 ! a , yx!1 !
! C aQ=2 , V = a + Qe2r' , V1 = a z = 2Qa . 118 .
10.14. & z = 0 | 0 ' , 1 | 1 '
'=2
r = C a cos
| x = e ayC = y
'
ar sin ' ln r = C | V = a + ei ' = , V1 = a 2
2r
i
z = 2a . 119 .
10.15. ) * +, + R V1 = a, +2
4
2
+
i
2
+
4
2
2
p ' z = 4a1 'i
16 a R ' . . ' 4aR, jz j = R, . . *
, +, jzj = R ' = 4aR, + ' 4aR, jz j R ,
+ + +,
jzj = R . 0 ., 13, !. III, . 493.
n
X
k + iQk ln z a . 4 1 |
10.16. w z = V eia z +
k
2i
Xn
k
ia
2V e
*
Q1 = Q k
n
k
'1 = X 'k .
k
1
10.17. 1 4 2 3 , w = +
ln z
z 2i
, 6 " .
kr
2 2
2
2
kr
kr
=1
=1
=1
294
10.18. .
.
:
1# $a% p# ! $% pc1 #% 2# $1% p# ! $% pc 1 #%
3# $a% p# ! $1% p=c 1 #, 4# $1% p# ! $1% p=c1 #.
10.19. n- , *+
n .
10.20. , $a | ,
a#: $a% Q .# ! $a % Q .# | , $a% Q .# !
! $a % Q .# | / 0 . , -0
+.
1 : $a% p# ! $a % p #,
+ p, p0 , + a a , +
. 1 jzj = R : $a% p# !
! $a % R2p3=a32#, a 6= 0, $a% p# ! $1% p3=R2#, a = 0. 1 / 0+
: $a% p# ! $a % p0 #% $a% p# ! $a % R2 p3=a32 #, $0% p# ! $1% p3=R2 #.
10.21. 1# 1 + + + jzj = R $ . 10.20#. 1
, , + / , + +
+ . 5 ,
Qk + 1 Q0i = 0, Qk | jz j = R
2
Q0i | jz j = R. 1 jz j = R + %
2# 7 + + + jzj = R. 1 , , +
jzj = R, + + +.
5 1
0
, .k + 2 .i = 0, .k |
jzj = R .0i | jzj = R%
jzj = R + .
10.22. 1# w = V z + c $c | #% 2# w =
ln zz aa + c%
2 i
X
X
X
X
3# w = 2Q ln 8$z a#$z a3#9 + c $ 1 2Q#%
4# w = 2p z 1 a + 2p z 1 a + c%
5# w =
Xn
k=1
h
k + iQk
ln $z ak # + k2+iiQk ln $z
2i
+ 2p z 1 a + V z + c
1
i
a3k#
+ 2p z 1 a +
Xn Qk%
k=1
6# < . = 0, Im p = 0% w =
Q
+ 2 ln z + c.
p
2z
+
X
295
1# w = 2i ln Rz2 aaz + c% 2# w = 2p z 1 a + 2p z 1 a + c,
2
2
a 6= 0 a = R , p = R2 p3 , w = p + p z + c, a = 0 p = p2 .
a
a
2z
2
R
n Q
n
k
2
10.24. 1# w =
ln 8$z ak #$R a3k z#9 + c, Qk = 0%
10.23.
X
X
2
k
k
n Q
m Q
X
X
2# w =
ln 8$z ak #$R a3k z#9 +
ln $z ai # + c,
k
i
Xn Qk + Xm Qi = 0.
=1
=1
2
=1
k
0
2
=1
10.25. 1# w =
2i
ln Rz2
2# w = 2p z 1 a + 2p z
1
a
az
2
i
0
1
2
k=1
+ c%
0
i=1
+ c a = Ra , p = Ra2 p %
2
2
a
2 V ei
2
i
R
3# w = V ei z + z + c% 4# w = V ei z + R Vz e + 2i ln z + c.
az + i
10.26. w =
ln zz + ia
+ c.
2i
iaaz i
Q
Q
ln $z2 1# + c.
10.27. w = ln $z 6 + a6 # + c. 10.28. w =
2
2
Q z2 1
10.29. 1# w = ln 2
+
c% 2# w = Q ln 1 + 44 + c.
2
z +1
2
z
i
n h + iQ
k iQk
k
k
ln
$
z ak # +
ln
$
z a3k # + p 1
10.30. w =
2i
2i
2 z a
k=1
n
p 1
+
V ei z + c. < , = .k = 0,
2 z a
2
n
k=1
X
X
6
X
1 | 2 .k .
k
i
n h
X
iQ
iQ
10.31. w =
ln
$
z ak #+
ln
$
R a3k z # + V e i z
i
i
k
n
X
R V e + c. < , .k = 0, a = 0, p = 2R V ei.
=1
i
2
=1
k +
2
k
k
2
k
2
2
z
k=1
t = f $z# D +
jtj 1. < w = =8f $z#9,
10.32.
=$t# =
Xn h
k=1
k + iQk
2i
ln $t tk # +
tk = f $ak #,
k + iQk
2i
i
ln $1 t3k t# + c
Xn Qk = 0.
k=1
t = f $z# D jtj 1
n
Qk = 0
f $1# = 1, f 0 $1# 0. <
10.33.
X
k=1
296
= ,
i
= Vf e 1 + f V1e t +
w
f z
i
t
t
X
n h
k + iQk
ln
t
2i
k + iQk
k +
t
+ 2i ln 1 k +
10.34. 10.33. = ,
i
i
= Vf e 1 + f V1e t + 2i ln +
0
0
k=1
w
i
t t
c
k
t
= k
f a
:
f z
t
t
0
t
0
c:
= 0 !" # !" # $ 2 1
2 1 i&" '#& " -)'"$" " * $& 1 = 1,
1 =
.
p
p
V
10.35. 1 =
b z 2 c2 cos + ibz a z 2 c2 sin +
ab
p
p
+ const V1 = V e , c = a2 b2 2 wz = a V b az b z2 c2 cos +
p
p
+ ibz a z2 c2 sin + 2i ln z + z2 c2 + const .
p
10.36. 1 wz = V z cos i z 2 c2 sin + const V1 = V ei
p
p
2 wz = V z cos i z2 c2 sin + 2i ln z + z2 c2 + const,
= 2cV sin c | .
10.37. ! " # $% z =
= 21 + 1 $% j 0 j = j1 0 j = R 1, 0 = 1 Re i 0 6
6 2 . ' ( "( V1 = V ei
p2
wz = V R z 0 + z 1 e i +
2
R
p2
p2
+ z Re
+
ln
z
z 1 + c,
0 +
2i
z 1
0+
) = 2RV sin + "" * ! " w0 1 = 0 ! !! , )
! -. / % .
p
10.38. wz = z p=2 + c | #% # / *!%
p
wz = ich 2zp p | #% # / *% .
2 p
p
1
10.39. wz = p z + z 2 c2 =2 ei =2
2
p
z z2 c2 =2 ei =2 + const
| #% !, !! # / *!% tg = ab , = , c =
p
p
p
= a2 + b2 wz = p12 z + z2 c2 =2 + z z2 c2 =2 + const |
#% !, !! # / *% .
10.40. wz "" * !%% " z = ew=v + w=v *%% "
V =f
w
0
0
w z
C
w
V e
w z
V =f
w
az
i
i
0
X
297
.-$1 $ $ *4 )'-) *4
4 4* .
p
10.41. = Arch = ln + 2 1 .-$1 $ $ *4 )'-) *4 4 4* 0 .
10.42. 1 0 " ) * + $ | 1 ' | " $ '#"
+ $ 2 + | $ "$ . 5$ " #
1 )'" ) V = V 1 =
= 2 . 6 $ 7$ ) 1'#4 ' "*. ". 120.
2 0 , '#$ * " " $
$ | '#" V
1
= 2 . 8' )" )' "' - ' $ 7$ ) 1'#4 ' ". 120 )* # '*.
10.43. 0 " ) * + $
| )' " ** * +
v
w
z
z
z
k
Q
k
=
k
x
1
Qi=
i
k
x
=
k
p
"$ " # V 1 = 0. 6 $ ) 4 ". 121.
10.44. 9 ! * ) 1 =
+
+ iQ
z a + iQ
z + a
= 2i ln sin 2! + 2i ln sin 2! + 2Q! +
x
i
V
V
Q=!
w
i
V
z
c:
ctg z2! a 2p ctg z2+! a + + .
10.46. -" # = $. * ) *' &-: )'"- t, ) * ;1 , ;2 ) "$ - ' 4 $ t . <"'
"-= " -: 4 -': ) 4 ;1 , ;2 , "$ " V1 , V2
$": " 14 1 ) '# , ' t 10.45. w =
p
iV z
2
t
f z
S
S
f
S
20 ..
.
c
0
f
0
S
S
298
10.44, 10.45 .
10.47. 1 , M C ! # . $ % & Re u = ! ( % ) # . . 122 & t = f +u = +u + Cu
! # C . - . 122 (
!
j! j. ./ ( 9.45 % e3 0 e2 e1 k2 1=2.
#1 u = 2!
2 f +u = 4M
+ c. 1 = 0 M = 2 & t =
! # u = 2!
0
0
= f +u = +u
1
#0 u=2!
u= 1 1
!
2! #1 u=2!
. 123
,
e3
=! e2 .
M
N
10.51. f +u = +u + +u + Cu + c. 3 4 ) f +u
2
2
% !, & # f +u = 2M 5 +u +u 6 +
+ c. 7 Im u = Im ! | , M & # ! # ,
0
X
% ) # , | . 1
& # 0, !k +k = 1 2 3.
1 = 0, = !k , M = 2
f +u = +u +u !k =
= k 12 u ue =
0
k
= k + i uu j uu
k
+i j k | 1,
2, 3. 9 u !2k +mod ! !
, . . f +u =
= 0. : & .
. 124. ; / # &(
#, / (
/ #! ! +k =
= 1, # /( #! !
+k = 2, !
!
, (! / #( (, (( !
/ # +k = 3. <
& &(
) .
10.52. = 4K
2iK , 2mK + +2n+
+ 1iK m
2+1=k ,
+2m + 1K + niK +m, n | ) . : & .
. 125.
0
0
0
0
0
20
299
300
4K n+2mK + 2iK , , 10.52 2 1ik , 2mK + 2niK
10.53.
0
0
X
301
2m + 1 K + 2n + 1 iK . 126 .
10.54. 2K 4iK , , 10.52
2 1 n+m i, mK + 2niK . ! " . . 127.
+ iQ
10.55. f u =
ln uu + Cu + c. % , = 0, !, ! +
2i
+ ! = ! , & &' &' ( 2+iiQ ,
C "& -)& *, & ln uu + Cu, ln 1 uu + Cu, ln 2 uu +
0
0
0
0
0
3
+ Cu. + f u + 2! = f u , f u =
+ iQ
2i
3
ln ##1
u
1 u
= 2!
= 2!
+ c.
10.56. 1 , 2 , 3 - "( 2 2 & . '. )& */ sn u, cn u, dn u . 128 .
302
10.57.
. 129.
u
10.58. 2!
2 #k v . 130, 1 #4
,
#3
! 1=2" . 130, 2 #1 #2 !
1=2.
10.59. 1 f z = $=2i ln z + c"
2 f z = $=2i ln tz + c, ! tz & D ! ' , ( ! ) "
3 f z = $=2i lnz z1 =z z2 + c, ! z1 , z2 * -
X
303
) & + * & + . . ( & , ! + ), +, (,+
' ), z1
4 f z = =2i ln tz + c, tz
!
.
10.60. f z = #!=i ln z ,
h #0 u =2!
i
#01 u + =2!
1
1
#u =
p
p
$
+ c
4 2 a #1 u =2!
#1 u + =2!
%
f z
R
= i! ln a:
& ! ' (
jRe ipj
)
2a
! - ( #0 u = 0.
. 131. , ' (
10.61. f z = #/uz 0,
h
i
0
0
V
i #1 u a=2!
$ i #1 u a=2! + c
#u =
ke
+ ke
2!
#1 u a=2!
#1 u a=2!
a uz = a + k=z + ::: |
.
!
D
, -)
D !
,
-
B #01 u=2!
10.62. f z = #
ln z , #u = Au +
+ c A =
i
2! #1 u=2!
i!
c 2 !2
,B=
c 2 c 1 . 3 4 ( , c 2 | =
2
4 , ( c 2 c 1 | 4
. 5 A 6= 0,
f z R & ( 4 !
( , ) jc 2 c 1 j=2. , ' (
4
4 ! - ( #0 u = 0 . 132, 1 . 132, 2
( 4 B = 0. 5 A = 0, . . ! ,
R ! !
, 4- ( !
-)- ( ( ( ( , ) !
jc 1 j=2 . 132, 3.
10.63. 1 6& (
= #!=i ln z ,
#u =
, 1 2 = ! 7
# u =2! 2
ln 1
+
2i
#1 u + =2! 2!
u + c = i! ln a
f z =
304
, ln
!
i
u 0 2 2 ,
u = + 1
! !
2
=
! !
#1 u = 2!
#1 u + = 2!
u
!
i!
2 +
0
=! + !0
0
!0 =!
-
i!
0
0.
"
= #2 #, "
" $
0
0
0, ,
+ ,
" ,
% . & #2 =
Zz
R
= 0 ' (
)
" " )
. 133, #
0. & #2 = #1 = # 2 ' (
)
,
$ - "
1
jZj =a2
Zz
=
1
Z
e4
=
R
jZj
0 , "
0 =
0 | - . / (
) 0
-
$
s
,
Z
f z =
4
i
ln 1
s z s0 s z + s0 2
X
a2
305
t
. 134, #
0,
, %
- )
" . & = 0 ) "
-
# 2 # 2 + #2 ,
u
=
2
2
ln
1
a
'
+
2
2
ln
1
'
=
f z = 41u z 2, "
h
5
u = V kei #1 u a= 2! + ke
0
4
2!
#1 u
=
a= 2!
0 #2 #
0.
i #01 u
i
a= 2!
+
#1 u a= 2!
1
2
#1 u a= 2!
+
ln
2 i
#1 u a= 2!
+
u z = a + k=z + ::: ) D " (.
10.64. u = x y , v = x + y , E = ic5, 1, ic5.
1
2q
10.65. u = 2q', v = 2q ln , E = ei' , % % a, 2q
r
10.66. u
%
z b
, v
q
z a
a, 2q .
= 2 arg
b, 2q
r
z
= 2 ln
q
z
a
,
b
E=
z
2q
b a
,
a z b
2
2!
u + C
1,2q.
%
306
= 2q arg z2 a2 , v = 2q ln jz2 a2 j E = 4qz=z2 a2
a2q, a 2q 1 4q . . 113.
10.67. u
10.68. u = jpj=r sin' , v
= jpj=r cos' E = jpje
0 p . 135.R
R
R
10.69. u = r cos ', v = r sin ' E = i1 e i'
r
r
r
0 iR 1 i . . 110, 111.
10.70. u = py + 2q', v = px + 2q ln1=r E = p + 2qei' =r
0 2q 1 2q 1 p . . 118.
n
n
n
10.71. u = py +
2qk 'k , v = px + 2qk ln r1 , E = p + 2qr k ei'k ,
k
k
k
k
k
i'
k
" z ak = re
# ak 2qk 1 p . 136.
i 2
' =r2 ,
2
2
2
2
2
2
X
X
X
=1
=1
=1
X
307
10.72. 1 %
" & # ' , ( 10.18
2 # ( ' ) # ' , ( 10.20 ' * #.
z z
10.73. v = 2qg z a. 10.74. w = 2qi ln
+ c.
z z
0
0
10.75. 1 2 w = 2qi ln
10.76. w
1
+ const, "
= 2qi ln f z
10.77. w = 2qi
10.78. w = 2qi
2d
= B1=2
3=4
+ c.
R2 z0 z
Rz z0
Z1 p
t
1
ln f z
+ c, "
1
ln f z
, "
1 t4
dt
t2
f z
f z =
t = f z
p
z c
,
ab
=z
z
p
2
z 2 R2
R
2
c2
= a b .
2
2
.
+( z =
+ d2 . + 9.17 n = 4 9.52.
z=a k
+ c, " f z = 1 sncnKKz=a
k
k
+( K =K = b=a . + 9.49.
1
z =4a# z z =4a
10.80. w = 2qi ln
+ c, " f z = ## z
f z
z z =4a# z z =4a
! = 2a, ! = 2ib, z = 4a x + iy , z = 4a x + i4b y , z = x +
+ i4b y .
pi
10.81. w =
+ z p ia + c a 6= 0, a = Ra , p = Ra p , w = piz Rpi z +
za
+ c a = 0, c | ) (
. /( ) 10.23, 2.
pi
p i
10.82. w =
+ z a + c a 6= 1, a = Ra , p = Ra p , w = piz
za
Rpi z + c a = 1. /. + 10.25, 2.
p
10.83. w = z cos + i sin z R + const.
p
p
10.84. w =
0
az b z c cos ibz a z c sin 1 + const,
ab
" c = a b .
2K
10.85. w =
cos + i sin cn u, " u = Ka z KK = ab . 2 sn u
+ 10.79.
10.86. 1 2 pi = ei ,
nh 1
i
h 1
i
o
f z = t a =
+
tz cos + i
tz sin + c
tz
tz
10.79. w = 2qi ln
0
0
1
fz
1
0
0
1
0
1
1
1
1
2
0
2
3
0
3
0
0
2
0
2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
nh
i
h
i
o
R
R
2 f z = pitz Rtzpi + c = tz + tz
cos + i tz tz
sin +
i
+ c, " pi = e . 3+ #4 ( #( & # 5#& +6 ( * ( # 7 5' D ( 8 " " , 2
2
2
308
, 11 .
h
i
n
1
10.87. = 2 k ln
+
+ , k
k
k=1
k k = =80, 0 | .
ln 1 j Im 0
10.89. =
: ln j 1 j Im 6 0#
X
w
q i
f z a
f a a
f a a
if
f z a
f
0
v z a
0
p
a a
pf z a
f z a
c
f z a
D
a a
jz
a
z
a
c
z
z
=
x a
2 j
= ln j
=
h
1
x
2 + 2 a = + i .
j j i 6= 0
+
ln
2 j
ln
= 0
%
= ln j 1 j . &
2
2
i = 2 1 2 2R a cos jj + jaj2 a = a i
1
, = 0 2R
, ln
ln j j 8 .
ln jR2 azj 6= 1 j j
ln j j j j
R
2
=
1 =
: ln 1
ln
j j6 "
=
6
1
j
j
6
"
jz aj
jaj2 R2
1
i =
=j j i
6= 1. # =
2R R2
2R jaj cos
+ jaj2
= 1,
, $
= 0.
p2 2
10.91.
1 = ln jz + z2 R j ,
1 = 2pR12 x2 j j .
p2 2
1 = ln 2
10.92.
1 = ln jz + z c j % ,
10.90. 1 %
R
v z a
az
z
R
v z a
Re
z
ln j
a
R =a
R
R
a
a
:
a
R
a
1
R R
a
j je
j j
:
a
R
z
a
z
R
z
v z a
Re
z
R
z
R
v z
a
z
R
R
a
a
a e
a
a
a
v z
x
v z
2
% . & '
1 =
2
2
=
.
1
10.93.
= 21 @g@n
. 10.94.
z
=
10.95.
x
10.96.
=
2
p
pj
11 .
"
". 242, 243.
2
x
x2
1
2
jj
1
R2
c2
j
c
2
R
=
pj
v z
1
2
.
2
a
x
b
2
2
c2
j
R
| % ,
2
c
=
1
= 2R
.
. 10.97. .
R
.
: . , ! ".| $.: %&', 1953. | 1, . 2,
x
X
309
10.98. 1=2R. 10.99. 1=2a b. 10.100. a.
10.103. + ! z # - | '
- / ( ( : # - =
=1
ln j 1 j . 7.58 10.102, ( ' ' ' (' ':
1 = ln , = # -, - = 1 #
Z
z
@
@n
t
z
'
w
2 = ln
'
w
3 =
1
w
dt
z
h
! z
a
z
b
z
a
v
'! z
, =
v
# -, - = #
'! z
a b
1 ln z a1 + '2 ln
'
a
v
=
2
a1
z
+ +
a
X
z
n
:::
'n
ln
n
an 1
z
a
z
# -k , -k =
'k ! z
k=1
ak
i
,
1 ak ,
a
1
= 1#
4 ( ' , / (. 3,
k 0 , n 0
0
. &)
(
k # -k , -0 = n 1.
w
'k
v
'
X
'
k=0
10.104. 1 wz =
i'
w
'
' ! z
id
ln
v
a
ln + const, = ln ln + const, = 2 #
z
d
v z
r
r
r
1
2 = ln ln + const, = ln ln j j + const, |
. . 35 ).
10.106. ( '' ,
10.105,
:
r
p2 2
2
1 = p
p 2 2, = +
1#
id
w z
t z
d
v z
D
t z
t z
D
w z
z
t z
z
+
2 3 = p
'
2+
x
2
R
a
R
a
R
z
t z
2
r
a
z
2
a
p
1,
x2
=
x
+ +
x
R
2
a
a
R
R
1 2, x , x
1 2
x2
R
x
R
=0
x
1
x
( ' ' -
2 #
p
4 = 1 + 2 2 , = + , = 2 2 #
5 = K +u=K , = sn , = 2K =K jRe j K , Im u 6 K .
9.48, 1
h
i
u
6 t = e K u =K , z = 2K h Z u + dnsnu cn
, = e K =K , k u
: K Z = h
, dn = KE . 9.48, 15.
2a
t z
c
z
z
a
c
b
c
0
t
e
a
b
0
z
+
c
u k
e
u
0
2
j
j
0
0
2
10.108. 1 !1 z = 1
1 =
8
=
:
2
ln jz j
,
ln
1
2 ln
1
2 ln
!2 z =
jzj = 1
jzj =
ln jz j
ln
p11 = p22 = p12 = p21 =
1
ln
310
!1 z = 1 !2 z , !2 z = lnlntz
8 t 1
1
1 = 2 = tt ln1
p =p = p = p = 1
: t ln 2 11 22 12 21 ln
tz D 1 jtj , 1
jtj = 1.
n
10.110. v z = Im f z k !k z ,
f z D
j
2
j
0
0
X
k=1
!
pi ,
f z = z a + ::: a 6= 1
piz + ::: a = 1:
10.111. v z
= 2qgz a +
X ! z , ! z = Z @g z ds.
n
k=1
k k
k
1
2
k
@n
= 1, a p , p & & & f z a:
pi
f z = z a + ::: a 6= 1
piz + ::: a = 1:
10.113. 1 $ f a = 0, f b = 1, ! & a 2q , b 2q , & !)
!) *
2 f a = 0, ! & a 2q ,
& !) , *+)
, , , ) D, 4 q,
!) ) *
3 * & . / & ! !, &+ , , , )
D, 4 q + & , &+ ,** ,
!) ) *.
10.114. 0. 10.61 10.63.
n 1
10.115. 1 v z =
k !k z + c, k &* &
k=1
!
n 1
pik k = 2qi i = 1 2 ::: n 1
k=1
. 7.70 c | ) . 4 5 * & D, *+ ! ! k
&& 4 qk k = 1 2 ::: n , 1 2 D, && 4 qk
k = 1 2 ::: n 1 , 4 qn , 1 2 D n | ,) .
2 vz = v0 z 2qgz a , v0 z & &, . 1 ,
a
& & 2qk + 2qk0 , qk0 = 2q @g@n
ds.
10.112. $ f a
X
X
Z
k
XI
10.116. v z
2qgz a + c,
ln
311
= 2q1 ln jrzj + c, q = 0, vz = 2q1 q ln jrzj
2
2
a
= ln
g z a = jz j ln #1 ln z + ln a=2i
ln
# ln z ln a=2i
1
= i , a 1, q 6= 0 & 10.63
!
& 6 gz a = Im 7 i
ln z = 2 2 = 2 |
& = 0 .
10.117. 8 a q a q ,
a | ,
& a. 9 &
u = q=2 ln1=jf z a j + c
f z a D !)
), 5 k & ! 1 .
2
q
z a
10.118. u = ln
+
c. 10.119. u = q ln R az + c.
2
z a
2
Rz a
10.120. u =
q
ln
2
sinz=2a + i sh h=2a
sinz=2a i sh h=2a
+ c.
p
h
q
t + i= k
K z + ib ui, k
10.121. u = ln
p
+
c
,
t
=
sn
2
a
t i= k
KK = 2ab .
0
10.122. 1 g z a D ! "
, ! ! D # a$ 2,
q ! $ n
2 uz = 2 gz a + uk !k z, !k z | # 'k .
X
10.123. u =
q
2
k=1
gz a + u2 r2u1 ln z =r1 + u1 , gz a | ( ln
. ! # 10.116.
j j
r1
! XI
11.3. 1, 2 A =
,a1 + b2 + ia2 b1 -, B = 12 ,a1 b2 + ia2 + b1 -,
1
,a1 + b2 + ib1 a2 -
C = c1 + ic2 $ A1 = 2
1
B1 = 2 ,b2 a1 ia2 + b1 - C1 = 1 ,c2 b1 c1 b2 + ic1 a2 c2 a1 -:
4 / 2 ! ! z- # w =
= e =2 1 1, ! ! #
!
0(( jAj. 1 z- "
0 ! w = ei+=2 1 1.
11.13. 2
" ! (
!
3 4 p 6 R + jaj=R jaj.
1
2
312
11.14. u = x=cos , v = y
x tg , p = 1 + sin =cos .
11.14.
. = ln zz ab z = a
z = b A B ! "#
! $%& % " + 0 . ' % ! %" #
) # p = 1 + 0 =
, %&& # # z , % ! %
% # . * !$ AM . 137 !% " s #
%
AM !% " x M .%% %& A B ) # % #
2 =2
dx
0
= cos
cos2 20
ds
cos2 =2
0
! &!. 0 % # %% ^ %& %%, AB ! !% " !$ ^AB, ,
" %& ) & !% " ! !$ AB. %&
)
p 6 1 + 0 = sec2 0 =2:
1
1
1
11.17. A = 2a d + ic + b3, B = 2a + d + ic b3, F = f + ig .
4
4
2
11.18. b = q2 z 4a % b = a4=q2 z . 0 , & w = ! +
+ q2 zw4 . w = a! + b!4 " ! 2 , % jq22 zj 6=2 1.
11.19. b = a4, $! ! %
q42 1 + jq2 j jq1 j +
+ q2 = 0 % q2 1= q42 = jq1 j2 .
" ! , %
jq1 j + jq2 j 1 % jjq1 j jq2 jj 1.
11.20. 6
% " ! ! %
! ) p , p1 1 ,
q1 z q2 z. z &
! # # ) p, - q1 = pp + 11 e2i , q2 = pp11 + 11 e2i1 . ! 11.7 11.9.
11.28. F z = 2z lnln1=r + 2z ln ln1=R@F
1
e ' cos '
=
2
ln
ln
2
+ 2 lnln R1
@x
r
ln r
1
ei' sin '
@F
=
2
i
ln
ln
2
+ 2i ln ln R1
@y
r
ln r
@F
= 2 ln ln 1r ln 1r + 2 ln ln R1
@z
2i'
@F
= 'z = lne 1=r :
@ z
i