Transcript
!"#$!%& "$(!)*+) (+%*,-#.*#& /!% %*,+)&% 0&#$%,&/ /! #*!.#*&" .&,$)&%!" 1 2&,!23,*#&" /!(&),&2!.,+ /!(&),&2!.,+ /! 2&,!23,*#&" 2&,!23,*#&" #$)"+ /! .*4!%&#*5. 6789 : 8" #&(;,$%+< . = 2 ! ) + " # + 2 ( % ! > + " /!?!) @
AB8 .CDEFGH IGDJKELGH (+
,*-$%&. &+ /01*23 43156*738 9+ /01*23 43156*73 43%7:#&)38 4+ /01*23 2*&6 5:238 )+ /01*23 $1&#$%&2$3 5:238 *+ ;*52*<*%=&4$>% )* :% %01*23 43156*73 *% -321& 2*4=&%#:6&28 -+ ;*52*<*%=&4$>% )* :% %01*23 43156*73 *% -321& )* 5&2 32)*%&)38
?+
@% A&632 %:1B2$43 )* 6& *C52*<$>%.
' ) ) (
4560 )) (cos(4560 °
!3
! !5 $ !5 + # csc # & cot # 3 " " % " 6
2
$$ && %%
* ( , , +
)
sen 390 °
*< $#:&6 &. 17
&+
15
9+
2
4+ ! 4
2
)+
*+ !4i
2i
;*<5:*<=&. )+
D+
E*&
i
=
!1 F <$ A
=
! ## "
i
4
i
2
$ && F 4&64:6* *6 A&632 )* %
i i
3
( )8
det A
;*<5:*<=&. G
H+
E*&
i
=
!1 F <$
6i
ni
!5i
&+ ID
3
7
!3i
5
=
!21 F *%=3%4*< *6 A&632 )*
9+ I(
4+ ?
n
)+ D
*< $#:&6 &. *+ (J ;*<5:*<=&. )+
K+
L&64:6*. M&+
1 i
2
F M9+
1 i
F M4+
i
3
F M)+
i
4
F M*+
i
5
F M-+
i
2510
F M#+
i
-315
;*<5:*<=&. M&+ I(F M9+ I i F M4+ (F M)+ (F M*+ i F M-+ I(F #+ N+
J+
!n " Z
L&64:6*.
4&64:6*.
!
M&+ i F M9+ 4n
i
4 n +1
F M4+
i
4n+ 2
i
F M)+ i ;*<5:*<=&. M&+ (F M9+ i F M4+ I(F M)+ I i 4n +3
!!"! !! !" ! !"
;*<5:*<=&. G !"#$%& ( )* ((
O+
P& -321& 2*4=&%#:6&2 )*6 %01*23 43156*73 39=*%$)3 &6 <$156$-$4&2 6& *C52*<$>% i
1+ 1+
1+
&+
*< $#:&6 &.
i i
1+ i 4
9+
4 + 2i
+ 2i
3
4+
4+
2
)+
i
3
4 3
2 +
3
i
*+
4 3
! 2i ;*<5:*<=&. )+
Q+
L&64:6*.
!
!
!
!!
!
!!! !!! !! !
!!
!!!
;*<5:*<=&. !
!
(G+ L&64:6*.
!
!
!
!! !
!
!
!
!!
!
!
;*<5:*<=&. ! ! ! ! ! ((+ R9=*%#& 6& -321& 2*4=&%#:6&2 )*6 %01*23 43156*73 39=*%$)3 &6 <$156$-$4&2 6& *C52*<$>% 1
i+
1
i!
1
i+
i!
1 i
;*<5:*<=&.
1 5
3 +
i
5
(?+ R9=*%#& 6& -321& 2*4=&%#:6&2 )*6 %01*23 43156*73 39=*%$)3 &6 <$156$-$4&2 6& *C52*<$>% i i
1!
i
1+
1!
i
2
;*<5:*<=&. !1 + 0i (D+ ;*<:*6A& 6&< <$#:$*%=*< *4:&4$3%*< 5&2& x
(
M&+ i x + iy
)
=
,
y ! !
x + 1 + 2 yi
M9+ x 2 ! y 2 + 2 xyi = !ix + y M4+
x
2
+
y
2
=1
! 2 x + yi ;*<5:*<=&. M&+ x
2
=
1
! , y ! F M9+ (0,0 ) , (0, !1) F M4+ x =
3
3
1 =
3
, y
=
0
!"#$%& ? )* ((
(H+ E$
z
*< :% %01*23 43156*73 =&6 S:*.
T%=3%4*):63 ?8 AB6 +JEFMINGOEH
( )
(O+ ,*1:*<=2* S:* 0,0 *< *6 *6*1*%=3 %*:=23 5&2& 6& <:1& )* %01*23< 43156*73<8
!"#$%& D )* ((
(Q+ ,*1:*<=2* S:*
(1,0 )
*< *6 *6*1*%=3 %*:=23 5&2& 6& 1:6=$56$4&4$>% )* %01*23<
43156*73<8 ?G+ ,&)3< 63< %01*23< 43156*73< M&+
F
z + w
M9+
z w
F
M4+
z
=
(3, 2)
3 z ! 4w F
;*<5:*<=&. &+ ?(+ ,&)3 *6 %01*23 43156*73
( x, y ) ( , ) (1,0 ) 8 u v
( x y ) F ,
(
) "
z + z
=
(!1, !4) F 39=*%#&.
(
M)+ !1,0 2 ! 2i F
)
9+
w
F
M*+
(0,!2)
!
z
5 ! 14i F 4+ 12 ! 22i F )+ 1 + 4i F *+ 4 ! 6i
( ) =&6 S:* 1:6=$56$4&=$A3 )* ( x y ) 8
)*=*21$%* *6 %01*23 43156*73
u, v
,
( ) *< 0%$43 W S:* *6 (0,0) %3 =$*%* $%A*2<3 1:6=$56$4&=$A38 u, v
??+ ,*1:*<=2*F )* <*2 53<$96*F S:*.
#
w
X *<=* %01*23 <* 6* )*%31$%& $%A*2<3
=
L3%46:W& S:* *6 %01*23
" C % ?D+ ! z " $
W
! z 1 , z 2 " !F
z + z 1
2
= z + z 1
2
& ('
&+ Y*2)&)*23
9+ Z&6<3 ;*<5:*<=&. &+
?H+ ,*1:*<=2*F )* <*2 53<$96*F S:*.
! z " !F
?K+ ,*1:*<=2*F )* <*2 53<$96*F S:*.
! z " !F z ! z
?N+ L&64:6*.
M&+
3 + 3i 2 ! 4i
F
M9+
z + z =
=
2 x 2 yi
1 ! 3i
!2 ! 2i ;*<5:*<=&. &+ !
! i + 3i $ ?J+ T6 2*<:6=&)3 )* 6& 35*2&4$>% )* %01*23< 43156*73< # & *<. 2 + i i " % &+ 1 ! i 9+ 1 + i 4+ 2 + i )+ 2 ! i *+ 16
3 10
9 +
10
iF
9+
1 2
+i
13
24
?O+
4
(1 ! i)
3
=
3+ i
;*<5:*<=&. 9+
!1 + i
&+ Y*2)&)*23
9+ Z&6<3 ;*<5:*<=&. &+
?Q+ Y*2$-$S:* S:*
z
DG+ Y*2$-$S:* S:*
uv
=
W
8
z
uv
<3% %01*23< 43156*73< 43%7:#&)3<8
!"#$%& H )* ((
D(+ ,&)3 *6 %01*23 43156*73 z
=
1 ! ki 2 !i
4:156$2<* S:*. &+
k
3 =
!
9+
2
k
1
=
!
4+
2
k
F 5&2& S:*
=
)+
0
<*& :% %01*23 2*&6 5:23F )*9*
z
k
=
1
*+
2
k
3 =
2
;*<5:*<=&. )+
D?+ E*&% 63< %01*23< 43156*73<
z
1
= 1+ i
W
z
2
=
1 ! i F
z
1
*6 2*<:6=&)3 )*
*< $#:&6 &.
z
2
&+ !1
9+
4+ !i
i
)+
*+ 1 ! i
1+ i
;*<5:*<=&. 4+ DD+ ,*=*21$%* *6 A&632 )* k 5&2& S:* &+ 9+
z = =
4 + ki 2+i
<*&.
@% %01*23 2*&6 5:23 @% %01*23 $1&#$%&2$3 5:238 ;*<5:*<=&. &+
k
=
2 F 9+ k
=
!8
" i 45 (!3, ! 2) %2 ' F <* 39=$*%*. DH+ X6 <$156$-$4&2 6& <$#:$*%=* *C52*<$>%. $ $# (1,3) (1, ! 3) '& ! 1 3 " &+ $ # + i % & 5 10 '
2
!3 " 9+ # i $ % 10 &
2
!1 3 " 4+ $ # i % & 5 10 '
2
!1 3 " )+ # + i $ % 5 10 &
2
! 3 " *+ $ # i % & 10 '
2
;*<5:*<=&. 4+ DK+ T-*4=0* 6&< <$#:$*%=*< 35*2&4$3%*<.
! 2 $ ! 4 3 $ ! 2 1 $ ! 28 3 $ ' i& M&+ # + i & + # ' i & + # + i & + # ' " 3 % " 3 4 % " 15 4 % " 15 2 % M9+
(
2
3i
!
)(
2+
3i
)(
1+
6i
)
! 5 5$ ! 2 1 $ # , & " 2 3% " 5 2 %
M4+ # , &
(
)
2
M)+ 2 3; 4
;*<5:*<=&. M&+
DN+ ,&)3< 63< %01*23< 43156*73<. T-*4=0*.
2u !
(
v
2
)
!u !
4 15
u
=
! i F M9+
5 + 5 6i F M4+
3 + i; v
=
!
550 23
3 + 3i; w
425
!
=
23
iF
M)+ !4 + 16
3i
2 ! 2 3i
v w
;*<5:*<=&.
(
6+3 3
) ( +
)
6 3 +3 i
!"#$%& K )* ((
(!2 i ) z
DJ+ ;*<:*6A& 6& *4:&4$>%.
+
=
3+ i
;*<5:*<=&. !1 ! i
(2 i ) (1 i )
DO+ ,*=*21$%* z =&6 S:*.
+
+
=
2 + z i
;*<5:*<=&.
3+ i
3
DQ+ E$
&+
z
=
" z % $$ '' *< $#:&6 &. F *%=3%4*< *6 %01*23 43156*73 1! 3 i 1 ! z # &
8i
3
9+
9
8i
3
4+ !
15
8 3
)+ !
9
8i
3
*+ !
9
8i
3
15
;*<5:*<=&. )+ HG+ E*& *6 43%7:%=3 2*-*2*%4$&6
()
p x :
4 x
2
Re
=
! W
+ 2 x + 1 =
63< 52*)$4&)3<.
()
0
q x :
x
! 2 + i = 4 ! 3i
1 + 3i
()
()
,*=*21$%* 6& <:1& )* 63< *6*1*%=3< )* 63< 43%7:%=3< Ap x W Aq x 8 ;*<5:*<=&.
35 2
+ 14i
ABP )EJFEHEOQMINRO SEGDTQFNIM H(+ ,*1:*<=2*F )* <*2 53<$96*F S:*.
! ! ! !F
H?+ ,*1:*<=2*F )* <*2 53<$96*F S:*.
!! ! !F arg
()
HD+ ,*1:*<=2*F )* <*2 53<$96*F S:*.
! ! ! !F
(
z
z
=
arg
z
=
z z
1 2
! arg ( z )
)
=
arg
( ) z
1
+
arg
( ) z
2
HH+ ,*=*21$%* *6 A&632 )* A*2)&) )* 4&)& 52353<$4$>%. M&+ T6 4[24:63 )* 2&)$3 :%$=&2$3 *< *6 43%7:%=3 )* 5:%=3< *% *6 56&%3 43156*73 S:* <&=$<-&4*
z
=
18 2
M9+ !&2& 4:&6S:$*2 43156*73 !" z z z M4+ z = ! + " i *< :% %01*23 $1&#$%&2$3 5:23F <$ W <>63 <$ M)+ z = ! + " i *< :% %01*23 2*&6F <$ W <>63 <$ z ! z
!
=
z
! z
=
ABU .GQMINRO VE !WKEF
(
HK+ X6 <$156$-$4&2 6& *C52*<$>% 1 ! i &+
2 + 256i
9+ !2 + 256i
)
4
+
(
3+i
)
9
4+ !4 ! 512i
<* 39=$*%*. )+
4 ! 512i
*+ !4 + 512i ;*<5:*<=&. 4+
!"#$%& N )* ((
HN+ T6 &2#:1*%=3 )*6 %01*23 43156*73 "#!2 +
9+
+
() ln (3)
+
2
2i
2ln 3
&+
4+
(3 )$% *< $#:&6 &.
()
2ln 3
() ln (3) 2ln 3
)+ *+
;*<5:*<=&. &+ HJ+ ,&)3< 63< %01*23< 43156*73<. i
z 1
=
1 + i;
z 2
2
e
=
3!
;
z 3
=
2 ! 3i
! z $ T%=3%4*< *6 A&632 )* 2 ## && ' z z *< $#:&6 &. " z % &+ 2 + i 9+ 2 ! i 4+ !8 ! i 2
3
3
1
i
HO+ E$ &+
z
1
=
6 2e
!
4
i
W
z
2
9 ! 4
2e
=
*+
1+ i
1 + 2i
;*<5:*<=&. 4+ 10
"1 % F *%=3%4*< *6 A&632 )* $ z ! 2 z ' *< $#:&6 &. #2 & 4+ 32i )+ !16 *+ 4 + 4i 1
9+ !1 + i
1! i
)+
2
;*<5:*<=&. 4+
" 1+ HQ+ ,&)& 6& *C52*<$>% $$ # !1 +
10
3i 3i
% ' F :%& )* 6&< <$#:$*%=*< *C52*<$3%*< *< *S:$A&6*%=* & *66&F ' &
\,T/]\Z^_@TPX. &+ !
3
1 2
+
9+
i
2
3
1 2
+
i
2
4+ !
3
1 4
+
3
1
)+ ! !
i
4
2
*+
i
2
3
1 2
!
i
2
;*<5:*<=&. &+ KG+ T6 2*<:6=&)3 )* 6& 35*2&4$>% )* %01*23< 43156*73<
(1 ! i ) (!1
+
3i
)(
2
!
2i
) *< $#:&6
&. &+
i
4
2e
9+
6
i
2
2e
3
4+
i
3 2e
2
)+
3
i
4 3e
*+
3
i
4 3e
6
;*<5:*<=&. &+ K(+ E$ z 1 = 1 + i W *< $#:&6 &. &+
1
z
2
=
1 ! 2i
9+
e
3
<3% %01*23< 43156*73):63 )*6 %01*23 43156*73
4+
e
)+
3e
*+
e
iz 1 z 2
e +1
e
;*<5:*<=&. 4+ !"#$%& J )* ((
7"
K?+ E$ <* =$*%*% 63< %01*23< 43156*73<
" " 2% 2 %% = $ cos $ ! ' + i sen $ ! '' W 2# # 6 3& # 6 3 &&
z
1
ix
z
2
=
2e 3
F
i
x
!!`
WF &)*1"):63 )*6 %01*23
z z
e
W
9+
5
e
2
4+
5
)+ ObK
5
e
*+ G ;*<5:*<=&. 9+
KH+ E*&% 63< %01*23< 43156*73< i
43156*73
e
&+
e
=
1
W
1 ! 3i
z
2 + i F
=
2
*%=3%4*< *6 &2#:1*%=3 )*6 %01*23
z
1
z
2
*< $#:&6 &.
1
!
z
7
9+
5
e
1
4+ !
5
7
)+ !
5
*+
5
7 5
;*<5:*<=&. 4+ KK+ T6 523):4=3 )* 63< 4:&=23 %01*23< 43156*73< S:* <3% 63< AB2=$4*< )* :% 4:&)2&)3 M4*%=2&)3 *% *6 32$#*%+F 43%34$*%)3 *6 AB2=$4*. &+
i
e
11
1
9+
6
2
3
i +
4+
2
e
i
z
1
=
i
2
F *< $#:&6 &.
3
e
1
)+
2
3
i
!
2
*+
i
e
2
;*<5:*<=&. )+ KN+ E$ <* =$*%*% )3< %01*23< 43156*73<.
z
1
=
!1 ! i W
z 2
=
3 ! i 8
,*=*21$%* *6 1>):63 )*6
2
z
1
%01*23 43156*73
3
z
e
2
8 1
;*<5:*<=&.
! e
4
KJ+ T6 2*<:6=&)3 )* 6& <$#:$*%=* 35*2&4$>% 43% %01*23< 43156*73<. i
e
*< &523C$1&)&1*%=* $#:&6 &. &+ ( 9+ I(
!
2
i
e
!
3
i
e
!
4
i
e
!
9
i
e
i! i
!
8
e
27
4+ G
i
e
!
i
16
)+
e
i!
81
...
i
*+ I *+ I i ;*<5:*<=&. *+
KO+ ,*1:*<=2*F )* <*2 53<$96*F S:* 5&2& =3)3 *%=*23 53<$=$A3 n.
" sen ( $ $# sen (
( ) ) ! i cos ( + i cos
n
) %' ) '&
=
(
(
)
)2
cos* 2 n *
( ( ++ ++ ! -- + isen * 2n * ! -,, ,, ) )2
!"#$%& O )* ((
(
M&+ !1 ! i
KQ+ L&64:6*.
3
)
1
9
F
M9+
(2
2i
+
7
)
;*<5:*<=&. M&+F K(?F M9+ NG+ ,&)3< u = 2 + i -321& 2*4=&%#:6&2. M&+ uv F M9+ u v
2
W
v
2 F
2!i
=
1024 ! 1024i
*156** 6& -321& 536&2 5&2& c&66&2 *6 2*<:6=&)3 *%
;*<5:*<=&. M&+ N(+ ,&)3< u = 2 + i -321& 2*4=&%#:6&2. M&+ uv F M9+ u v
2
W
v
=
3 F
1! i
4F
M9+
i
*156** 6& -321& 536&2 5&2& c&66&2 *6 2*<:6=&)3 *%
(
N?+ R9=*%#& *6 2*<:6=&)3 )* *% -321& 2*4=&%#:6&2 )*.
(
3+i
!1 + i
)
4
3
)
6
;*<5:*<=&. ! 1 i 4
84
ND+ R9=*%#& *6 2*<:6=&)3 *% -321& 536&2 )*.
(1 i ) (!1 ! i ) +
9
75
;*<5:*<=&. NH+ E$156$-$S:* 6& *C52*<$>%.
!
!!
!
22 e
i
7 ! 4
!
!!!
!
;*<5:*<=&. IH I K(?$ NK+ L&64:6*.
!
!
!!
!
;*<5:*<=&. NN+ E$156$-$S:* 6& *C52*<$>%.
!
!"
!" ! !"
!!
! ! ! ! !! !
!!!!! !!!!!! ! !
!
;*<5:*<=&. ! ! ! !" ! NJ+ E$ !!
!
! ! !F
!!
!
!! F
!!
!
!! !
!!
)*=*21$%*
!! !!
! !
! !!
;*<5:*<=&.
NO+ ,*=*21$%* *6 A&632 )* 6& *C52*<$>%.
" $ $ #
4
!
!
!
!!
% 3 ! i "1+ i % ' $ ' ' 3 + i & #1! i &
3
;*<5:*<=&.
3 2
1 +
i
2
!"#$%& Q )* ((
NQ+ E$ V( F V? F VD " &+
z .z
)+
z + z
1
1
=
1
1
z
1
C F
*%=3%4*< *< ZXPER S:*.
9+
z + z 1
2
1
2
!
# !n " N % $
[ ei
"
( ()
*+ ! "
!
4+
= z + z
cos
=
+ isen
cos
( ))
+i
sen
n
JG+ E$ !! ! *< :%& 2&[V 409$4& )* :% %01*23 43156*73 VF *%=3%4*< &+ Y*2)&)*23 9+ Z&6<3 !
]
( )
= cos n
!
+ isen isen
&
(n )('
!! 8
!
;*<5:*<=&. &+ J(+ ,*=*21$%* 6&< 2&[4*< 409$4&< )* (8 ;*<5:*<=&. 1 F !
1 2
3 +
2
i F !
1 2
3
!
i
2
J?+ ,*=*21$%* 6&< 2&[4*< 409$4&< )* !1 8 ;*<5:*<=&. !1 F
1 2
3 +
2
iF
1 2
3
!
i
2
JD+ ,*=*21$%* 6&< 2&[4*< 4:&)2&)&< )* !2 ! 2i W 2*52*% )*9*% 4:156$2 =2*< %01*23< 43156*73< )$<=$%=3< 5&2& *%43%=2&2<* *% :%& 1$<1& 2*4=& *% *6 56&%3 43156*73e OK+ @% AB2=$4* )* :% c*C"#3%3 2*#:6&2 4*%=2&)3 *% *6 32$#*% *<=" :9$4&)3 *%
0, 2) F (0,2
)*=*21$%* *6 2*<=3 )* <:< AB2=$4*<8 AB9 &JKNIMINGOEH
() x
=
cos
()
=
isen isen x
ON+ ,*1:*<=2* S:*.
cosh
OJ+ ,*1:*<=2* S:*.
senh x
OO+ ,*1:*<=2* S:*.
cosh
! !! ! OQ+ E*& !! ! !! ! ! ! E$ !!
!
! !
!
!
!! !
!
2
!
() x
()
( x) ! senh ( x) 2
!! !
! W !! ! !! ! ! !
!
!
=
1
F :% 536$%31$3 )* 43*-$4$*%=*< 2*&6*<8 !F )*=*21$%* 63< A&632*< )* 63< 43*-$4$*%=*< )*6 536$%31$38 ;*<5:*<=&. !! !!! !! !"! !! !!"! !! !" !
!! !
!
!!
!
!
!
!
QG+ T<42$9& :%& *4:&4$>% )* <*#:%)3 #2&)3 S:* =*%#& 532 <36:4$3%*< 1 + 2i W <: 43%7:#&)38 ;*<5:*<=&. ! ! ! !! ! !
!
Q(+ L&64:6*.
ln
(!2)
Q?+ L&64:6*.
ln
(!i )
QD+ L&64:6*.
ln
(!3i )
QH+ L&64:6*.
arcsen arcsen 2
QK+ L&64:6*.
arccos arccos 3
!
()
() !"#$%& (( )* ((
