For learning complex on mathematics [Keywords] Complex, Complex numbers, Complex plane Residue Theorem, Laurent Series, Gamma Function, Beta Function 複素解析 No.2 学習用まとめスライド 「留数定理」から「実積分への応用」まで おまけ「Γ関数とΒ関数」

2. !
"
#
$ sin $
$( − 4$ + 5
-$
!
"
#
$(.
1 + $(0
-$ 123 4 > 6 ≥ 0
!
"
#
sin $
$
-$
!
"
#
$9:;
1 + $
-$

3. ⟺
" = $+&' = ( ) = ( *++' (
$* = &+ ∧ $+ = −&*
" = ( ) D (regular)
⟺" = ( ) D
; ( )

5. ∫"#
"$
% " &" "# "$
D )5 ; ) % " D ) )
*
)
% " &" = ,
(
)

6. ) ( 2

7. 7 6
.
58
Γ Β

9. (
- .
0
9 Residue Theorem
9 Residue 4 )
. 5
-3 5
2 13

10. - Residue Theorem !
"#, "%, "&, … , "( .5 ! !#
"#"%
!%
"&
!&
1 )
!
* " +" = -
./#
(
)
!.
* " +"
* * (
!. ".
6 )
Residue
012 3* "
"/".
=
#
%45
)
!.
* " +"
1 *0 * *
Residue Theorem
)
!
* " +" = %45 -
./#
(
012 3* "
"/".
1

11. Residue Theorem
!
"
# $ %$ = '() *
+,-
.
/01 2# $
$,$+

13. Residue
!"# $% &
&'&( * 2 5
Residue Theorem
)
*
% & +& = -./ 0
('1
2
!"# $% &
&'&(
& = &3 1 7
!"# $% &
&'&3
= 456
&→&3
& − &3 % &
.
2 & = &3 7 9 7=
*3 & & = &3 + 9"/;
:
+& = 3 + /9"/;
+; = / & − &3 +;
!"# $% &
&'&3
=
1
-./
)
*3
% & +&
=
1
-./
<
3
-.
/ & − &3 % & +;
9 → 3
456
&→&3
& − &3 % &
:
!"# $% &
&'&3
= % & 7
1
& − &3
7 =>1
% & =
=>1
& − &3
+ =3 + =1 & − &3 + =- & − &3
- + ⋯
& = &3 6 - 3 :73
!"# $% &
&'&3
= 456
&→&3
& − &3 % & = =>1
- 1
* 3

14. ! = !# $
%&' () !
!*!#
= +,-
!→!#
/
$ − / !
2$3/
2!$3/
! − !#
$
) !
%&' () !
!*!#
= ) !
/
! − !#
1 43/
) ! =
43$
! − !#
$
+
43 $3/
! − !#
$3/
+ ⋯ ⋯ +
43/
! − !#
+ 4# + 4/ ! − !# + 47 ! − !#
7 + ⋯ ⋯
! = !# ) (
! − !#
$) ! = 43$ + 43 $3/ ! − !# + ⋯ ⋯ + 43/ ! − !#
$3/ + 4# ! − !#
$ + ⋯ ⋯
4 8 − 1
2$3/
2!$3/
! − !#
$) ! = $ − / ! 43/ + $! 4# ! − !# + ⋯ ⋯

15. Practice
!". $
%
& ' (' = $
%
('
'* + "
% ( ,(> ") ) % ( ,(> ") )
Residue Theorem
$
%
& ' (' = *01 2
34"
5
,67 8& '
'4'3
,67 8& '
'4'9
= & '
"
' − '9
;<",67 8& '
'4'9
= =>?
'→'9
' − '9 & '
A
B
1
−1
%
A
B
1
−1
%

16. Practice )
Residue Theorem
!
"
# $ %$ = '() *
+,-
.
/01 2# $
$,$+
/01 2# $
$,$3
= # $
-
$ − $3
56-
/01 2# $
$,$3
= 789
$→$3
-
. − - !
%.6-
%$.6-
$ − $3
.
# $
<'. !
"
# $ %$ = !
"
>?@ $
$A
%$
" ( /(> 3)
E
F
G
3
"

17. 3 - ( )
3 -
7D % & D ' &
% & =
)
*+,
-
'
% .
. − &
0. '
1
&
! . ' -% & -
- - 5- & % &
% &
% &
.2&
. = & 1
-
'
% &
. − &
0. = *+, 3 456 7
% .
. − & .8&
= *+, 3 9:;
.→&
% . = *+, 3 % &

18. .
.
D # $ D %
#(')
$ =
'!
+,-
.
%
# /
/ − $ '12
3/
! - # $ 5 4
2 8 1
#4
$ = 567
∆$→:
# $ + ∆$ − # $
∆$
= 567
∆$→:
2
+,-∆$
.
%
# /
/ − $ + ∆$
−
# /
/ − $
3/
= 567
∆$→:
2
+,-∆$
.
%
# / ∆$
/ − $ − ∆$ / − $
3/ =
2
+,-
.
%
# /
/ − $ +
3/

19. .
.
! " 5
⟺ ! " = %&'()
9 "* -
!+
"* =
,
-./
0
1
! 2
2 − "*
- 42
1 1 "* 1 5 1
!+
"* =
,
-./
0
1
! 2
2 − "*
- 42 ≤
,
-.
0
1
! 2
2 − "*
- 42
≤
,
-.
7
*
-.
8
5- 549 =
8
5 5 → ∞
* ! 2 ≤ 81
!+
"* = * 1 9 " ! " = %&'() ∎
"*
1
5

20. ! (
)
! " = $% + $'" + $("(
+ ⋯⋯+ $*"*
= %
2 0* ( ) * ∈ ℕ,$* ≠ %
! " = % 2
1 " =
'
! "
) ) 0
0 1 " = 23456 2
! " 2 " = "'
! " = " − "' 8 "
* − ' 8 " ! " 0
0 ! " = % * ∎

22. 0
- .
1 2-
1 3 .
Exercise!!!!!!

23. ( 8 -
8
8 - ! " 4 "# "$ 5
"# 5 "$ − "# 8 &# 23 "# : &
6
! " =
$
()*
+
&
! ,
, − "
-, " 9 "# 8 ⋯ ⋯ ⋯ ★
4
$
,/"
"# 4 5
$
, − "
=
1
1 − 23 − 2 − 23
=
1
1 − 23
4
1
1 −
2 − 23
1 − 23
=
1
1 − 23
1 +
2 − 23
1 − 23
+
2 − 23
6
1 − 23
6
+ ⋯ ⋯ = 7
89#
:
" − "#
8
, − "#
8;$
5 , 9 & 8 84 , − "# > " − "#
"#
"$
&#
&
"
) . 1 *
! " =
1
2>?
+
@
A 1 7
B93
:
2 − 23
B
1 − 23
B;C
D1
= 7
B93
:
2 − 23
B 4
$
()*
+
&
! ,
, − "#
8;$
-,
= 7
89#
:
! 8
"#
8!
" − "#
8
*
9
! 8
"#
8!
*
" = "# 8

24. ! = !#
!#
!$
%#
%
! = !#
…
!#
!$
%#
%
Residue Theorem
&
%
' ! (! = )*+ ,
-.$
/
012 3' !
!.!- 012 3' !
!.!#
= ' !
$
! − !#
56$

25. 2
2
!"
!#
$"
$#
!
$%
$# , $%
' ! =
#
%)*
+
$%
' ,
, − !
., − +
$#
' /
/ − !
./
5 $%: , − !" > ! − !"
#
, − !
= 2
34"
5
! − !"
3
, − !"
36#
2 $#: ! − !" > / − !"
#
/ − !
= −
1
8 − 89 − : − 89
= − 2
;49
5
: − 89
;
8 − 89
;6<
= − 2
34=5
=#
! − !"
3
/ − !"
36#
2 -
' ! = 2
34"
5
! − !"
3 >
#
%)*
+
$%
' ,
, − !"
36#
.,
+ 2
34=5
=#
! − !"
3 >
#
%)*
+
$#
' /
/ − !"
36#
./
2 2
. $" . , @A / = !"
2. $# , $% - 2 $
-
' ! = 2
34=5
5
! − !"
3 >
#
%)*
+
$
' ,
, − !"
36#
.,
! = !" 6.2

26. ( )
!
"

27. Exercise!!!!!! 2 ] 3
1 . -
[ -
!
!
! − #
# = % & '#
# = %( ) *+, # [# = %]
1 . -
[ -
3 2 E 6
!
# − !
# + !
# = −! &
'#
# − ! )
# = ! ) 01*
!
#
[# = 2]

29. -
.
2
. 2 4 13
. 2 4 -3
Exercise!!!!!!!

30. 7 1- !
"
# $ %$ ( 3
# $ " $& , $( , $) , … (
7 . - 7
0
$ − $,
-&
.-& ) /01 |# $ $3$,
0 (
0
!
"
# $ %$ = (56 7
,
/01 8# $
$3$,

31. !". $
%
&'( )
)*
+)
( % ∶ ) = / )
Practice
1
2
/
%
3
% ) = / 1
) = /
4 ) =
&'( )
)*
=
3
)*
) −
)"
"!
+
)8
8!
− ⋯ ⋯ =
3
)"
−
3
"! )
+
)
8!
− ⋯ ⋯
⟹ ;<= >4 )
)?/
= −
3
"!
3 3
$
%
&'( )
)*
+) = @AB −
3
"!
= −
AB
"
4 ) % )3 , )@ , )" , … )
) − )E
F3 GF3 ( ;<= |4 ) )?)E
)
3
$
%
4 ) +) = @AB I
E
;<= >4 )
)?)E

32. .
!". $
%&
&
'()*
*+ + -+
.*
( ), - ∶ 3, ) > 3, - > 3)
5
*
-(
67
7−7
9
6
'():
:+ + -+
.: = $
%7
7 '()*
*+ + -+
.* + $
67
'():
:+ + -+
.:
7 → ∞
2 6 : = -(
: = -( 3
7'> ?
'():
:+ + -+
:@-(
= ABC
:→-(
: − -(
'():
:+ + -+
=
'%)-
+-(
9
6
'():
:+ + -+
.: = +D(
'%)-
+-(
=
D'%)-
-
7 2 2 : = 7'(E
$
67
'():
:+ + -+ .: = $
3
D
'()7'(E
7+'(+E + -+ (7'(E
.E
= $
3
D
7'%)7 FBG E
7+'(+E + -+ ('( EH)7 IJF E
.E
$
67
'():
:+ + -+ .: ≤ $
LM
NOPQ
RS + TS UR
≤ $
V
W
XN%PY Z[ ]
XSNOS] + TS
U^ ≤ $
3
D
'%)7 FBG E
7
.E
7 → ∞ 7 2- ) > 3, FBG E ≥ 3
$
3
D
'%)7 FBG E
7
.E → 3 ⟹ ABC
7→&
$
67
'():
:+ + -+
.: = 3
∴ $
%&
&
'()*
*+ + -+
.* = ABC
7→&
$
%7
7
'()*
*+ + -+
.* =
D'%)-
-

33. 7
!". $
%
&
'() *
*
+*
- .
$
%
&
'() *
*
+* = -(.
/→&
1→%
$
1
/
'() *
*
+* = -(.
/→&
1→%
$
1
/
23*
− 253*
63*
+*
=
7
63
-(.
/→&
1→%
$
1
/
23*
*
+* + $
51
5/
−239
−9
−7 +9
=
7
63
-(.
/→&
1→%
$
1
/
23*
*
+* + $
5/
51
23*
*
+* ≡
7
63
; $
5&
&
23*
*
+*
<
*%
=/
=1
/−/ 1−1
( (
% = >
=
23?
?
+? = $
5/
51
23*
*
+* + $
=1
23?
?
+? + $
1
/
23*
*
+* + $
=/
23?
?
+?
( (
= 3
/ → ∞
)
-(.
/→&
$
=/
23?
?
+? = %
/ → ∞
1 → %
1 ? = 123A
$
=1
23?
?
+? = $
B
%
23123A
123A 3123A
+A = $
B
%
323123A
+A
1 → %
$
B
%
323123A
+A → $
B
%
3+A ⟹ -(.
1→%
$
=1
23?
?
+? = −3B
∴ $
%
&
'() *
*
+* =
7
63
E −3B =
B
6

34. Exercise!!!!!!! 7 1 4 2
- 7 .
E
! "
#$
$
%&
! + &( )
( "
#$
$
*+, )&
! + &-
%&
) "
.
(/
%0
1 − ) 34* 0 (
- "
.
$
! − 34* (&
&(
%&
- .
E
! "
.
$
&(5
! + &(6
%& 5, 6 ∈ ℤ, 6 > 5 ≥ . ( "
.
$
&<#!
! + &
%& (. < < < !)

36. -
Γ Β
Γ.2 3
Γ.2 4 3 6 8
- 1
Β.2

37. Γin j
Γin
" # = %
&
'
()**#)+ ,* (.( # > &)
7 1! = 1 1 − + 1 − 4 ⋯ ⋯ 4 6 + *
! 7 1 = 1! 1 ∈ ℕ
! 3:
2 y
7 1 = 1 6 7 1 − + 7 & = +
! Γin " : + + = 7 : = :! y ℝ > & d ht
d n d opr ht
① " : + + = %
&
'
()**: ,* = −()**:
&
'
+ %
&
'
()*:*): ,* = :" :
② " + = %
&
'
()* ,* = −()*
&
'
= +
10
en Q
H n d Γin a
/ 6 . 4 63 593 95 8 1> 5< :9 ) -(*(* ) 0

38. . Γ H
Γ
" # = %
&
'
()**#)+ ,* (.( # > &)
" # = 123
4→'
4! 4#
# # + + # + 8 ⋯ ⋯ # + 4
(# ≠ &, −+, −8, … )
" #, 4 = %
&
4
+ −
*
4
4
*#)+
,*
+
" #
= #(>#
?
4@+
'
+ +
#
4
(
)
#
4
ABC > = 123
4→'
D
E@+
4
+
E
− 1FG 4 ≈ &. JKK
8
2 -
H
" # =
L
8 M2N O#
%
P
−Q #)+
()Q
,Q
R
S
P
8 T
3

39. ) 83
! ! " 3 " → ∞ 3 ( .
%! 3 % - 3
83
%! ≃ ()% * %%
+,%
" > . 2 / = "1 3 2
! " + 3 = 4
.
5
+,/
/"
6/ = 4
.
5
+,"1
""
1"
" 61
= ""73
+,"
4
.
5
+," 1,89: 1,3
61
23 " ) 2
; 1 = 1 − 89: 1 − 3 2 2 1 = 3
; 3 = .
8=>
"→5
+,"; 1
= .
2 1 = 3 ( ) 3
; 1 1 = 3 9 1 − 3 = ? 2
; 1 = ? − 89: 3 + ? = ? − ? −
?(
(
+
?@
@
− ⋯ ⋯
=
?(
(
−
?@
@
+ ⋯ ⋯ =
3
(
1 − 3 (
−
3
@
1 − 3 @
+ ⋯ ⋯
23
4
.
5
+,"; 1
61 ≃ 4
3,B
37B
+
,
"
(
1,3 (
61
=
(
"
4
,
"
(B
"
(
B
+,C(
6C ≃
(
"
4
,5
5
+,C(
6C =
()
"
∴ ! " + 3 ≃ ""73
+,"
()
"
= ()" * ""
+,"
"
(
1 − 3 ( = C(
.

40. Β4
Β4
" #, % = ' (
)
*
'
+,-'#./
0 -12'%./
0 30
(56 # > ), 56 % > ))
4 4
" #, % =
9 # 9 %
9 # + %
" #, % = (
)
/
;#./
/ − ; %./
3;
) (
" #, % = (
)
=
>%./
/ + > #?%
3;
+,-' 0 = ;
-12' 0 = / − ;
; =
/
/ + >
" / − @, @ = 9 / − @ 9 @
" / − @, @ = (
)
= >@./
/ + >
3; =
*
-12 *@
.- 8
9 / − @ 9 @ =
*
-12 *@