Литвин І.І., Конопчук О.М., Желізняк Г.О. Вища математика
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31
?CA 3. CA" DA
3.1. =#,! / &!= .
& ( ! ( '! (!, ( -
( (’< (, < ’. 5( (
!, ’
2
+1=0.
2
+1 = 0
'
2
= -1 < ) () !< ') . A !
2
= -1.
5 ( ! ( () !) ') j. 5 -
(9 '( !.
( (< ! (, ) .
&!
) ( a + ib, ( , b – ( !, –
(.
B ) %<!F
, ! b – *!F !
!. !, , !) ') z. !
a
1
+ ib
1
a
2
+ ib
2
) %! ( (,
1
=
2
, b
1
= b
2
, '
( ! < !.
/ «'9» «9» ( ! !.
! z = 0 + i0 < !< 0; ! z = a + i0
)< ( ! ; ! z = 0+ib ) ! -
!) ib. B 0 < <( !,
(!
< (, ! .
! a + ib a – ib ) &$!
!) z
z
. &(, ! z =1 + 2i =1, b = 2, ( '( !
z
=1 – 2,
( ! z =1 – 3i '( !
z
=1 + 3.
# ! ! ') . C -
! ( z = a + ib < =#,!F /F
!.
%, ! &! =#,!%< /%.
(, (, ! '!
) (( ( ( !.
1.
C (' ! z
1
= 2+7i; z
2
=3+5.
?;’;!!:
H ( (( (! 2 – 7, 3 + 5;
z
1
+z
2
= (2 – 7) + (3 + 5) = 2 – 7 + 3 + 5 = 5 – 2.
' ( 9 (! 2-7 3+5 (9) -
)
2
–1.
z
1
z
2
= (2 – 7)(3 + 5) = 6 – 21 + 10 – 35
2
= 6 – 11 + 35 = 41 – 11.
%&%>:
z
1
+z
2
= 5 – 2; z
1
z
2
= 41 – 11.
32
'!, 7 (' ( ! < ( !:
z
z
= ( + b)( – b) =
2
– b + b – (b)
2
=
2
– b
2
(–1) =
2
+ b
2
.
",
zz
2
+
b
2
.
H< <) ) ( ( ( ( (
!.
/ ( !
1
2
z
z
, ( z
1
= a
1
+ib
1
, z
2
= a
-2
+ib
2
( (-
! ('
11
22
iba
ib
! ( ,
' a
1
– ib
1
.
2.
! z
1
= 3 – 4i z
2
= 10 +5i. C ) z
2
–z
1
! z
2
: z
1
.
?;’;!!:
C( ) ( (! 3–4 10+5.
z
2
–z
1
=(10+5)–(3–4)=10+5–3+4=7+9.
W' ! z
2
: z
1
( ! !, -
( :
z
z
2,24,0
25
5510
43
20401530
)43)(43(
)43)(510(
43
510
22
2
1
2
.
%&%>: z
2
–z
1
=7+9 ; z
2
:z
1
=0,4+2,2 .
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0//;;
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z zzzzzzzzzzzzz
.
( < ! !. ,
.
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@! (( 9 ( .
/( ! ( ()< (
( (! ( . / ( , 7
0
=1 ,
1
=,
2
=-1,
3
=- ,
4
= 1 ,
n
= i
4m+k
=(i
4
)
m
i
k
=
k
, ( k = 0, 1, 2, 3.
&(:
24
=
46
=1,
59
=
414+3
=
3
=-;
42
=
410+2
=
2
=-1.
3. !
33
3. C !
2
)32(
512
i
i
z
.
?;’;!!:
59 ( ( , <:
ii
i
z
125
512
9124
512
9124
512
2
.
9 ! !, ( , '-
-5-12, <:
i
iiiii
ii
ii
z
169
169
14425
6016960
125
601442560
)125)(125(
)125)(512(
22
2
.
%&%>
: z =-i.
3.2. -! %!&+% &!: .
! z = a+ib (( -
( ( ! (; b) . A ( ( !
!< ! ' 7.
", ! ( z = a+ib '< ( 7
!) (, b) ' , ! (< ! (,
-
. .
H ( 7 < ) 7),
' – () ), ( – ) ).
&(, ' ! z
1
= 2 + 3i, z
2
= -3 + i, z
3
= -4 – 4i, z
4
= 3i, z
5
= 3 – 2i,
z
6
= 6 (. 1).
Y
Z
4
Z
1
Z
2
Z
6
X
Z
5
Z
3
I.1.
/( ! 7 (< -
( !.
3. !
34
*
! ) ( , 7 ((< (-
! (!) r 'U.
=*!
! (z z 0) ) ! M ((-
( , 7 ((< ( !.
I :
& / <
.
22
bar
&(, ! z = 8 – 6i < ( 10, '
101003664)6(8
22
r
.
@ ! z z 0, ( ( (, !< -
(!. A ! 5 < M
1
= 0 ; M
2
= 2S ; M
3
= -2S, …
M
k
= 2Sk (k = 0, ±1, ±2…). H( ! ! 9 (
(-S; S) ' (0; 2S). + ! '( !.
@ !, 7 ! 7 !. -
, ( ! (9
(. 2) :
b
b
arctg,tg
MM
.
R7 ! 7 9 !, '(
(( . I . 3. E!, 7 (
!
;arctg
b
M
( !
;arctg
b
M
3. !
Y
b z (; b)
r
M
0
. 2
M D
M S D
M SD
M D
35
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b
M
'
,
b
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M
7
a
b
arctg
!
D
.
A !, ( ! -
:
1. 5! < a, b ( !.
2. C
b
arctg
D
.
3. 5, ! 9 !.
4. "'!
M
( ( .
# 9 ' ( !, -
(:
°
¯
°
®
;);(,arctg
;);(,acrtg
"#####$b!!%
b
"#V#$b!!%
b
S
M
' j !) :
°
¯
°
®
.sin
;cos
r
b
r
M
M
3. !
II
III
D
D
DD
M D
M S
–
D
M S D
M D
IV
I
36
4. C !
31 iz
.
?;’;!!:
1. 5! < a=1,
3 b
.
2. C(
.60
1
3
arctg
0
D
3. 5, ! 9 ( ! (. 4).
4. @, 7 ((< (
! V !, '
M
= –
D
= –60
0
.
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3
60
S
M
.
3.3. =!! / &!= .
I . 2. C( ! (9 -
! , b ! r i M !:
= rcosM ; b = rsinM .
A( ! 9 (:
z = a+ib = rcosM+irsinM = r(cosM+isinM).
C ! ( <
=!!F
/F &!= .
", ( , 7' ( '! !-
z = a+ib ( !, ( ( .
5. C !
iz 3
! .
?;’;!!:
C( (
.241313
22
r
C(
63
3
arctg
3
1
arctg
3
1
arctg
S
D
.
5, 7 ((< ( ! !,
()<
,
6
7
6
SS
SM
.
6
7
sin
6
7
cos23
¸
¹
·
¨
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§
SS
iiz
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¹
·
¨
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§
6
7
sin
6
7
cos2
SS
iz
.
3. !
37
, 7' ( ! !-
z = r(cosM+isinM) ( '!, ( ( ! , b
= rcosM; b = rsinM.
6: C ! z = 2(cos330
0
+isin330
0
) '! .
?;’;!!:
C( sin330
0
cos330
0
cos330
0
= cos(360
0
-30
0
)
= cos30
0
=
2
3
;
sin330
0
= sin(360
0
-30
0
) = -sin30
0
= -
2
1
;
(
1
2
1
2;3
2
3
2
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
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§
ba
.
",
iiz 3330sin330cos2
00
.
%&%>:
iz 3
.
%, ! &! =!!%< /%.
5 ! ! ) ( -
, (, ( ( , (' n- . 5( -
, ) (, ( T)
.
& z
1
= r
1
(cos
M
1
+isin
M
1
) z
2
= r
2
(cos
M
2
+isin
M
2
) ,
A(:
))sin()(cos(
2121
2
1
2
1
MMMM
i
r
r
z
z
z
1
z
2
= r
1
r
2
(cos(M
1
+M
2
)+ sin(M
1
+M
2
)).
", !, ( ! ,
( ), ((); ( – ( (, -
().
/ ! ! )(<
( ! . R7 z = r(cos
M
+isin
M
),
)sin(cos))sin(cos(
MMMM
ninrirz
nnn
.
" )
/*F *.
(' n- ! z = r(cosM+isinM)
) :
¸
¹
·
¨
©
§
n
k
i
n
k
rirz
n
n
n
SMSM
MM
2
sin
2
cos)sin(cos
,
3. !
38
(
n
r
! , k = 0, 1,..., n – 1.
7. ! z
1
=12(cos225
0
+isin225
0
)
z
2
=
2
3
(cos75
0
+isin75
0
). C (' z
1
z
2
!
2
1
z
z
. I
'! .
?;’;!!: C)! ( !
! , <:
;399
2
3
2
1
18)300sin300(cos18
))75225sin()75225(cos(
2
3
12
00
0000
21
iii
izz
¸
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¹
·
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2
1
2
3
8)150sin150(cos8
))75225sin()75225(cos(
2
3
:12
00
0000
2
1
iii
i
z
z
¸
¸
¹
·
¨
¨
©
§
%&%>:
i
z
z
izz 434;399
2
1
21
.
8. "'! z = (2(cos24
0
+isin24
0
))
5
. 5(( '!
.
?;’;!!: C(:
.31616
2
3
2
1
32
)120sin120(cos32))245sin()245(cos(2
00005
ii
iiz
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¹
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31616 iz
.
9. "'!
10
3 i
.
?;’;!!: C9 !
i3
! :
.330
6
11
6
2;
6
3
1
;213
0
2
2
SS
SM
S
D
arctgr
3. !
39
A(:
)sin3302(cos3303
00
i
,
10
3 i
=(2(cos330
0
+isin330
0
))
10
=
=2
10
(cos10330
0
+isin10330
0
)=2
10
(cos(60
0
+9360
0
)+isin(60
0
+9360
0
))=
=2
10
(cos60
0
+isin60
0
)=1024
3512512
2
3
2
1
ii
¸
¸
¹
·
¨
¨
©
§
.
%&%>:
10
3 i
= 512+512
3
.
10. "'!
4
81
. 5(( '! -
! .
?;’;!!: C9 ! –81 ! :
–81 = 81(cosS + isinS) .
A(:
,
4
2
sin
4
2
cos8181
44
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¹
·
¨
©
§
k
i
k
SSSS
( k=0, 1, 2, 3 .
/ k = 0:
.
2
23
2
23
2
2
2
2
3
4
sin
4
cos3
2
0
4
sin
2
0
4
cos3
0
ii
iiz
¸
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¹
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¨
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/ k = 1:
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23
2
23
2
2
2
2
3
24
sin
24
cos3
1
iiiz
¸
¸
¹
·
¨
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/ k = 2 :
2
23
2
23
2
2
2
2
3
4
sin
4
cos3
2
iiiz
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S
S
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3. !
40
/ k = 3 :
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23
2
23
2
2
2
2
3
2
3
4
sin
2
3
4
cos3
3
iiiz
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¹
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¨
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.
3.4. ;! / &!= .
I()! ) & =
( , (
.D (9
M
= cosM+isinM.
C ( (<, 7 ! zz0
(, z = r(cosM+isinM) = r
e
iM
, ) &;!F /F ;&*.
&( ! ) (
! . 5( , ) ( T<
.
&
21
2211
,
MM
ii
erzerz
, (:
.1...,,1,0,
;
;
;
2
)(
2
1
2
1
2
1
)(
212121
21
2
1
2121
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errez
e
r
r
er
er
z
z
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n
k
i
n
ni
n
inn
n
in
i
i
i
iii
SM
M
MM
MM
M
M
MMMM
11. /( !
6
5
4
S
i
e
'! .
?;’;!!: C( (!
6
5
;4
S
M
r
,
,2
2
1
4
6
5
sin4
;32
2
3
4
6
5
cos4
¸
¸
¹
·
¨
¨
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S
S
b
a
!
ie
i
2324
6
5
S
.
%&%>:
i232
.
3. !