For learning complex on mathematics [Keywords] Complex, Complex numbers, Complex plane Cauchy, Riemann, Cauchy&Riemann equation 複素解析 No.1 学習用まとめスライド 「複素数の学び直し」から「複素積分の触り」まで

2. 3
0 17 5 -9 3
Augustin Louis Cauchy
- aei r
W e e p
-e ei d 3
e
3
9 1 2
9 k o
9 -
3 3 . .
3 . 3 8 .
3 -
9

3. 6 8
. - 6 062 4 81 6 8
Georg Friedrich Bernhard Riemann
We e
e e 8
7 e d a
5 p
7 i 6 8 k
p 1
7 d p
8 9 8
7 8
3 4

4. 1 2
3
4
4 .

6. 1 -
0 .
1 Complex number
1 4 Conjugate
1 23 )( 23 Complex plane
-. Polar form

7. 1 Complex number
! = # + % & ' = () ! + *+ ! & '
Real part 7 Imaginary part
(#, %) *
/ ' :
* 1
' '0
= −/
.
!/ ±!0 = #/ ±#0 + %/ ±%0 '
!/!0 = #/#0 −%/%0 + #/%0 +#0%/ '
-
!/ = !0
⟺ 4
() !/ = () !0
*+ !/ = *+ !0

8. ! "#$ % &' %
% = #$ % +&' % *"

9. ) ) Conjugate
)
! 9 "!
"! = $% "! + '( "! ) * = $% ! − '( ! ) *
! "!
2
̅! = ̿! = !
. ( $% ! =
!. ̅!
/
, '( ! =
!1 ̅!
/*
. 1 - 2 3 4
2 4 = 5 ⟺ 2 74 = 5

10. ! "#$ % &' %
% = #$ % +&' % *"
̅% = #$ % −&' % *"

11. ! "#$ % &' %
% = #$ % +&' % *"
̅% = ̿% = %

12. .- Complex plane
12 Imaginary axis
32 Real axis
! = # + %& * #, %
()*
# + %& 2 : #, % 1
( = ! = #) +%) = ! ̅! = ̅!
+ = ,(- ! = ./012
%
#
1
1

13. 1 Polar form
!, # $ = ! + #' 1 (
4 ( ) ( 1 (
1
$ = ( *+,)+',-.) = (/')
(= (*-, ) = (∠))
- .
$3 = (3 *+,(3))+',-.(3)) = (3/'(3))
∵ 5657 = 8687 cos(<6 + <7) + =sin(<6 + <7) = 8687@A BCDBE
)
3

15. 2
.0 -
42 1 3
1
- .5
Exercise!!

16. -
! = # $ () $,! ∈ ℝ
.
-(
( = # ) () ),( ∈ ℂ
) = $ + !, ( = - + .,
1) ( 6 -,. / - $,! ,. $,! 1
2 ( = # ) / ) = $,! 1 ( = -,.
/
!
$
.
-
) (
:
$,! -,.
#

17. 2
! =
#$ +#&'+#('( +⋯+#*'*
+$ ++&'++('( +⋯++,',
-./' = 012'
+132'
(
/45' = 012'
−132'
(2
785' = 0/45'
-./'
. -
-./9' = 01' +13'
(
/459' = 01'
−13'
(
7859' = 0/459'
-./9'
! = 1' = 1:;2< = 1: -./<+/45<
⟺ >
! = 1:
#?@ ! = < + (A,

18. ) (
! = #$%&
2
& = '!
3
& = (')*
= '#$% (
')*
= '#$% (+)*
= '#$% (+) *+,-.
.
! = #$%(+) *+,-.
10
1
&
2 ≠ 0 *
( = 5 2=
! = #$% &
6
7
7 = #$% 5 = 8 2 6 2
,-
& 2 1
1 .13 & ,-- 1 . *
!
( 2

19. 2 Exercise!! 4 [ ..
3 ) ( 1
! = # $ = $%&
$ ≠ ( E # -
3 ) ( 1
! = # $ = $) E # -
3 ) ( ]1
! = $
&
* * ] -

20. 5 ( 24
E
)
! "
#$%
&'!
()* #+ , "
#$%
&'!
# ()* #+
) )
! --
, ()* -. / 012 -. 3 *425 -.

22. - . 03
2 4 5 4
- 13
5
.
Exercise!!!

23. !"#
$→$&
' $ = )
⟺$ $& 3 .
- ' $
)
+ = ' $ D
⟺+ = ' $ D $&
$ → $& ' $ 2
!"#
$→$&
' $ = ' $&
'′ $& = !"#
$→$&
' $ −' $&
$−$&
+ = ' $ D (regular)
⟺+ = ' $ D
3
+ = ' $ $ = $& 3 ⟺ '′ $& 1 . -

24. ! = #+%& = ' ( = ' )+*&
.
'′ ( = ,-.
∆(→1
' (2∆( 3' (
∆(
3. ∆( = ∆)+&∆* → 1 C -
∆(=1
( )
( )
( ) ∆* = 1 1 ∆) → 1 2
'′ ( = ,-.
∆)→1
# )+∆),* −# ),*
∆)
+&
% )+∆),* −% ),*
∆)
= #) +&%)
( ) ∆) = 1 1 ∆* → 1 2
'′ ( = ,-.
∆)→1
# ),*+∆* −# ),*
&∆*
+&
% ),*+∆* −% ),*
&∆*
= −&#* +%*
#) = %* ∧ #* = −%)
( ) ( ) 1 -

25. (
C
!" = $% ∧ !% = −$"
.
(=) * R :5
∆( = ) * + ∆* − ) *
= !"∆" + !%∆% + - $"∆" + $%∆%
= !" + -$" ∆" + !% + -$% ∆%
= !" + -$" ∆" + −$" + -!" ∆%
= !" + -$" ∆" + !" + -$" -∆%
= !" + -$" ∆" + -∆%
= !" + -$" ∆*
5 : 4()′ * =!" +-$" 5 (=) *
( 2 4*:)
( = ) * - 4* ⟺ !" = $% ∧ !% = −$"
( = ) * R
C ",% %,"4 :5
1
!"" = $%"
!%% = −$"%
1
!"% = $%%
!%" = −$""
⟹ 3
△! = !"" +!%% = 5
△$ = $"" +$%% = 5
3
!,$ )
harmonicfunction 4*

27. 3
! = # $ )$ = $% ( ⟺ $ = $% # $ 3
. 3
$=$% < # $ ) > (
# $ = '()$ $ = *+
,
-
. *
. 3 ) / 3
$=$% < 2 ) 7
# $ = /0/
,
$
=
,
01)
,
$
$ =
,
*.
* ≠ %
- 4(3 ∈ . 3
# $ )$ = $% () # $% 7 $ = $% 2 7
# $ =
01)$
$
$ = % 2415
$→%
# $ = , ( # $ = 7
⁄01)$
$ 9:; $ ≠ %
, 9:; $ = %
<
=−.−-. . -.
# $ = '()$
<
=
# $ = /0/
,
$
=
,
01)
,
$
−. .
$ = *+
,
-
.
$ =
,
*.

28. Pole
!"#
$→$&
' $ ∼
)*
$− $&
*
* : * 5
<> <
e 5
Essential singularity
' $ $=$& 2*
$ → $& < 8
() ' $ = -
.
$ $ = &
$ = /-01
,-
.
$ = 3-04
: 2 :<5 :
-
.
$ = 567
.
/-01
= 567
.
/
89:1−:";1
⟹ =
3 = -
89: 1
/ ⟺ 89:1 = /!;3
4 = −
:";1
/
⟺ :";1 = −/4
*
89:?
1+:";?
1 = .
/?
!;3 ?
+/?
4?
= .
∴ /?
=
.
!;3 ? +4?
BC;1 =
:"; 1
89: 1
∴ 1 = tanG.
−
4
!;3
?HI
/I
?
1I
-
.
$ > $ = & &
: :
5/ 1
2 5

29. 1 Exercise!!! E 3 5..
2 - R
R] 5 C R]
! "# $ %&'
!
#
( )*+ # , #
!
$
2 C R]
# = . E EC R] [
2
R]
! /)*+ #
$ 0*1 +234!
# ( ##

31. 4 0
- .
43 5.
- 21
3 5
Exercise!!!!

32. -
-
!
"
# $ %$ = '()
∆$ →,
-→.
/
012
-
# $0 ∆$0
$-
$,
$032
$0
"
∆$0 = $0 −$032
∆$ ≥ ∆$0
678 9:: 0
1 4
!
"
# $ %$=!
"
; <,> +@A <,> %<+@%>
= !
"
; <,> %<−!
"
A <,> %> +@ !
"
; <,> %>+!
"
A <,> %<
B
"
# $ %$
"
.
!
"
# $ %$=!
C
D
# $ E
%$ E
%E
%E "∶ $=$ E =< E +@> E C≤E≤D

33. !
"
# $ %$ = !
"'
# $ %$+!
")
# $ %$
"
")
"'
-
!
"
*'#' $ +*)#) $ %$=*'!
"
#' $ %$+*)!
"
#) $ %$
.
!
+"
# $ %$ = −!
"
# $ %$ "
−"
24
!
"
# $ %$ ≤ !
"
# $ %$
" # $ ≤ .
!
"
# $ %$ ≤ ./ (/ ∶ " )

34. - .
4 2
!
"
# $ %$=!
"
' (,* +,- (,* %(+,%*
= !
"
' (,* %(−!
"
- (,* %* +, !
"
' (,* %*+!
"
- (,* %(
# $ =$/
= (+,* /
=(/
−*/
+, /(*
*
("0
"/
"1
3"0,"/,"1
2
"0 ∶ 4,4 → 0,4 → 0,0
"/ ∶ 4,4 → 4,0 → 0,0
"1 ∶ 4,4 → 0,0
3
# $
3"0
!
"0
# $ %$ = !
4
0
(/%(+!
4
0
0−*/ +/,* ,%* =
0
1
+, 0−
0
1
+, = −
/
1
+
/
1
,
3"/
!
"/
# $ %$ = !
4
0
−*/ ,%* +!
4
0
(/ −0+/,( %( = −
0
1
,+
0
1
−0+, = −
/
1
+
/
1
,
3"1 )
!
"1
# $ %$ = !
4
0
6$/
$7(8,(
%
%(
(+,( %( = !
4
0
/,(/ 0+, %( = /, 0+,
0
1
= −
/
1
+
/
1
,
3 1 (

35. ! " = ̅"% = &−() % =&% −)% −( %&)
)
&
*+
*%
*,
3*+,*%,*,
.
*+ ∶ 0,0 → +,0 → +,+
*% ∶ 0,0 → 0,+ → +,+
*, ∶ 0,0 → +,+
3
! "
3*+
2
*+
! " 3" = 2
0
+
&%
3&+2
0
+
+−)%
−%() (3) =
+
,
+( +−
+
,
−( =
5
,
+
%
,
(
3*%
2
*%
! " 3" = 2
0
+
−)%
(3) +2
0
+
&%
−+−%(& 3& = −
+
,
(+
+
,
−+−( = −
%
,
−
5
,
(
3*, )
2
*,
! " 3" = 2
0
+
67"%
"8&9(&
3
3&
&+(& 3& = 2
0
+
−%(&%
++( 3& = −%( ++(
+
,
=
%
,
−
%
,
(
12 2 4 3 . - (

36. D # $ D %
&
%
# $ '$ = )
) 4-1.
&
%
# $ '$ = &
%
/'0−2'3 +5&
%
/'3+2'0
. ( 4 - )
&
%
#'0+7'3 = 8
9:
70 −#3 '0'3
6
&
%
# $ '$ = 8
9:
−20 −/3 '0'3+58
9:
/0 −23 '0'3 [ ]
% => # $
D CR
/0 = 23 ∧ /3 = −20
3 [ ] 0 ∎
%9:

37. D 7 # $ D %
&
%
# $ '$=)
3
%*
%+
%*
%+
%
%*
%+
%,
-
%*
# $ '$=-
%+
# $ '$
&
%*
# $ '$=&
%+
# $ '$
&
%
# $ '$=.
/0*
,
&
%/
# $ '$
# $ =$/ / )
1
2
3 = *
%
[ ]
&
%
# $ '$=)
[3] $=678 )≤8≤+:
&
%
# $ '$=-
)
+:
;$/
$0678
'
'8
678 '8
=-
)
+:
67/8 <7678'8=7-
)
+:
67 /=* 8'8=)

38. Exercise!!!! 5 -. 4
! "
#
$% & '& (# ! + * + + ,* )
, "
#
$% &,
'& (# 3 8 )
+ "
#
&.!
'& (# ,* − ,* , )