engineering mathematics ii · 2018. 1. 30. · engineering mathematics ii prof. dr. yong-su na...
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EEngineering Mathematics IIngineering Mathematics II
Prof. Dr. Yong-Su Na g(32-206, Tel. 880-7204)
Text book: Erwin Kreyszig, Advanced Engineering Mathematics,
9th Edition, Wiley (2006)
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Ch 14 Complex IntegrationCh 14 Complex IntegrationCh. 14 Complex IntegrationCh. 14 Complex Integration
14 1 Line Integral in the Complex Plane14.1 Line Integral in the Complex Plane
14.2 Cauchy’s Integral Theorem
14.3 Cauchy’s Integral Formula
14.4 Derivatives of Analytic Functions
2
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Ch. 14 Complex IntegrationCh. 14 Complex Integration ((복소적분복소적분))
복소적분의 중요성
p gp g ((복소적분복소적분))
복소적분의 중요성
• 실질적 이유 : 실적분 계산법으로 접근이 용이하지 않은 응용분야에서나타나는 일부 적분들을 복소적분에 의해 계산해 낼 수 있기 때문.
• 이론적 이유 : 해석함수의 몇 가지 기본 성질들을 다른 방법들로는증명하기 어렵기 때문.
내용 : 복소적분의 정의, Cauchy의 적분정리, Cauchy의 적분공식
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
(복소)선적분(Line Integral):
복소정적분으로 피적분함수를 주어진 곡선 또는 그것의 일부를 따라 적분한다.
( )dzzfC∫
복소정적분으로 피적분함수를 주어진 곡선 또는 그것의 일부를 따라 적분한다.
• ( ) ( ) ( ) ( )btatiytxtzC ≤≤+= : )nIntegratio of(Path 적분경로
• 방향증가하는가대하여에 : )Sense (Positive tC방향 양의
도함수아닌이연속이고점에서모든곡선매끄러운•
( ) ( ) ( ) 곡선갖는를
도함수아닌이연속이고점에서모든
0 : )Curve(Smooth
tyitxdtdztz +==
곡선 매끄러운
dt
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
1. 선형성(Linearity):
기본적인 속성들은 정의에 의해서 직접적으로 나타난다.
( ) ( )[ ] ( ) ( )dzzfkdzzfkdzzfkzfk ∫∫∫ +=+ 221122111. 선형성(Linearity):
2.
( ) ( )[ ] ( ) ( )ffffCCC∫∫∫ 22112211
( ) ( )dzzfdzzfz
z
z
z∫∫ −=0
0
3. 경로 분할(Partitioning of Path) :
zz0
( ) ( ) ( )dzzfdzzfdzzfCCC∫∫∫ +=
21
C1
C2 Z
z0
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
첫 한첫 번째 계산방법 : 부정적분과 상, 하한의 대체
• 일반적으로 통용되는 개념
단순 닫힌 곡선(Simple Closed Curve) : 자신과 교차하지 않는 닫힌 곡선단순 닫힌 곡선(Simple Closed Curve) : 자신과 교차하지 않는 닫힌 곡선
단순 연결(Simply connected) : 단순 닫힌 곡선이 집합에 속한 점들만 에워쌀 때
Ex 원판(Circular Disk)은 단순연결되어 있지만
• Indefinite Integration of Analytic Functions (해석함수의 부정적분)
Ex. 원판(Circular Disk)은 단순연결되어 있지만, 환형(Ammulus)은 단순연결되어 있지 않다.
g y (해석함수의 부정적분)
( )
( ) ( ) ( ) 존재가해석함수만족하는를내에
해석적내에서영역단순연결
'
:
zFzfzFD
Dzf
=( ) ( ) ( )
( ) ( ) ( ) 성립가대하여경로에모든내의연결하는을와점두내의 0110
1
0
zFzFdzzfDzzD
f
z
z
−=⇒ ∫
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
Ex 1 ( ) iizdzzii 22111 3
13
12 +−=+==
++
∫Ex. 1 ( ) iizdzz33
133 00
+−=+==∫
i⎞⎛
Ex. 4 ( ) πππ iiiiiz
dzi
i
=⎟⎠⎞
⎜⎝⎛−−=−−=∫
− 22LnLn
zizz ArglnLn +=
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
두 번째 계산방법 : 경로에 대한 표현식의 사용
• Integration by the Use of the Path (경로를 사용한 적분)
( ) bC 경로매끄러운구분적으로( )
( )
⎞⎛
≤≤=
Czf
btatzzC
:
; :
함수연속인위에서
경로매끄러운구분적으로
( ) ( )[ ] ( ) ⎟⎠⎞
⎜⎝⎛ ==⇒ ∫∫ dt
dzzdttztzfdzzfb
aC
• 적용하는 과정
A. ( ) ( ) 표시형태로의를경로 btatzC ≤≤
( ) 계산한다를함수dz
B.
C.
( ) . 계산한다를도함수dtdztz =
( ) ( ) . 대입한다를에모든의 tzzzf
D. ( )[ ] ( ) . 적분한다까지에서대해에를 battztzf
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
Ex 5 A Basic Result: Integral of 1/z Around the Unit Circle
.2 1/ ) 0 ,1( 이다적분하면를반시계방향으로따라을원인중심반지름단위원 iz
dzzC
π== ∫
Ex.5 A Basic Result: Integral of 1/z Around the Unit Circle
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
Ex 5 A Basic Result: Integral of 1/z Around the Unit Circle
.2 1/ ) 0 ,1( 이다적분하면를반시계방향으로따라을원인중심반지름단위원 iz
dzzC
π== ∫
Ex.5 A Basic Result: Integral of 1/z Around the Unit Circle
A.
B.
( ) ( )π20 sincos: ≤≤=+= tetittzC it단위원
( ) itietz =B.
C.
( )
( )( ) ( )ite
tztzf −==
1
22
D. idtidtieez
dz
C
itit πππ
22
0
2
0
=== ∫∫ ∫ −
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
Ex.6 Integral of 1/zm with Integer Power m (정수 거듭제곱을 포함하는 적분)
( ) ( )
( ) . 000
적분하여라를방향으로반시계따라
주위를원의인중심이이고반지름이하자이라고때상수일가정수이고이
zf
zzzzfzm m ρ−=
( ) . 적분하여라를방향으로반시계따라 zf
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
( ) ( )
( ) . 000
적분하여라를방향으로반시계따라
주위를원의인중심이이고반지름이하자이라고때상수일가정수이고이
zf
zzzzfzm m ρ−=
Ex.6 Integral of 1/zm with Integer Power m (정수 거듭제곱을 포함하는 적분)
( ) . 적분하여라를방향으로반시계따라 zf
( ) ( ) ( )( )
≤≤+=++= 20 sincos 00
dtid
teztitztzitimtmm
it πρρ
( )( )−
==−⇒
∫ ,
2
0
0
dzzz
dteidzezz
C
m
itimtm ρρ
( )
( ) ( ) ⎤⎡
=
∫∫
∫ ++
22
2
0
11 dtei tmimρ
ππ
π
( ) ( )
( )( )⎨
⎧ −==
⎥⎦
⎤⎢⎣
⎡+++= ∫∫+
12
1sin1cos00
1
mi
dttmidttmi m
π
ρ
( )⎩⎨ −≠
=정수및 10 m
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
Dependence on Path (경로 의존성)Dependence on Path (경로 의존성)
일반적으로 복소선적분은 경로의 양끝점에 의존할 뿐 아니라,
경로의 기하학적 현상에도 의존한다.
Ex.7 Integral of a Nonanalytic Fucntion. Dependence on Path
( ) ( ) ( ) . , * ,21 0 Re 21 적분하라따라를구성된로과따라를까지에서를 CCCCixzzf ba+==
z = 1 + 2i2
y
C*
C2C
2
C1
1 x1 x
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
Dependence on Path (경로 의존성)Dependence on Path (경로 의존성)
일반적으로 복소선적분은 경로의 양끝점에 의존할 뿐 아니라,
경로의 기하학적 현상에도 의존한다.
Ex.7 Integral of a Nonanalytic Fucntion. Dependence on Path
( ) ( ) ( ) . , * ,21 0 Re 21 적분하라따라를구성된로과따라를까지에서를 CCCCixzzf ba+==
z = 1 + 2i2
y( ) ( ) ( )
( ) ( )[ ] ( ) ,21
10 2 : *
1
==+=
≤≤+=
ttxtzfitz
titttzCa
C*
C2
( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ,1 10 :
2121
2121Re
1
1
0
===⇒≤≤=
+=+=+=∴⇒ ∫∫
ttxtzftztttzC
iidtitdzzC*
bC
2
C1
1 x
( ) ( ) ( ) ( )( ) ( )
2211ReReRe
1 , 20 1:
21
2
+=⋅+=+=∴⇒
===⇒≤≤+=
∫∫∫∫∫ idtidttdzzdzzdzz
txtzfitztittzC
1 x
.
20021
있다수다를적분값이따라경로에적분∴
CCC
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
Bounds for Integrals (적분한계값). ML 부등식
( ) ( )부등식 MLMLdzzfC
≤∫
( ) 상수만족하는을곳에서모든위의길이의 : , : MzfCMCL ≤
Ex 8 Estimation of an IntegralEx.8 Estimation of an Integral
다음 적분의 절대값에 대한 상계(Upper Bound)를 구하여라
선분까지의부터 1 0: ,2 iCdzz +∫C∫
1
1
C
1
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
Bounds for Integrals (적분한계값). ML 부등식
( ) ( )부등식 MLMLdzzfC
≤∫
( ) 상수만족하는을곳에서모든위의길이의 : , : MzfCMCL ≤
Ex 8 Estimation of an IntegralEx.8 Estimation of an Integral
다음 적분의 절대값에 대한 상계(Upper Bound)를 구하여라
선분까지의부터 1 0: ,2 iCdzz +∫C∫
1
( ) 2 ,2 2 ≤== zzfL
1
C8284.222 2 =≤⇒ ∫ dzz
C1
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14.1 Line Integral in the Complex Plane 14.1 Line Integral in the Complex Plane e eg a e Co p e a ee eg a e Co p e a e((복소평면에서의복소평면에서의 선적분선적분))
PROBLEM SET 14.1
HW 28 34 (b)HW: 28, 34 (b)
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
단순 닫힌 경로(Simple Closed Path) : 스스로 교차하거나 접촉하지 않는 닫힌 경로
Ex. 원은 단순 닫힌 경로이지만 8자형 곡선은 단순 닫힌 경로가 아니다.Ex. 원은 단순 닫힌 경로이지만 8자형 곡선은 단순 닫힌 경로가 아니다.
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
영역에 대한 정의
• 단순연결영역(Simply Connected Domain):
영역의 모든 단순 닫힌 경로가 오직 영역의 점들만 둘러싸고 있는 영역영역의 모든 단순 닫힌 경로가 오직 영역의 점들만 둘러싸고 있는 영역
Ex. 원, 타원 또는 어떠한 단순 닫힌 곡선의 내부
• 다중연결(Multiply Connected) : 단순연결되지 않은 영역
Ex. Annulus (환형)
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
유계영역(Bounded Domain) : 원점이 중심인 어떤 원의 내부에 완전히 속해 있는 영역
p중연결( p fold Connected)p중연결( p-fold Connected)
• 유계영역의 경계가 공통점이 없는 p 개의 닫힌 연결집합으로 구성되어 있을 때
• 경계의 집합은 곡선, 선분, 또는 점일 수 있다.
• p-1 개의 구멍을 갖게 된다.
Ex. 환형(Annulus)은 이중연결(p = 2)되었다.
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
Cauchy’s Integral Theorem
( )
( )
,
이다대하여에곡선닫힌단순모든있는에
해석적이면에서정의역단순연결가
∫ df
Dzf
C( ) .0 이다대하여에곡선닫힌단순모든있는에 =∫C
dzzfCD CD
Ex.1 특이점들이 없다(완전함수들)특이점들이 없다(완전함수들)
nz zzze ) ( ,cos ,대하여경로에닫힌임의의
이다함수해석적인대해에모든완전함수은
( )∫∫∫ ====C
n
CC
z , , ndzzzdzdze 10 0 ,0cos ,0
대하여경로에닫힌임의의
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
Ex.2 윤곽선 밖에 있는 특이점
: C 단위원
. 2 1
. , cos
1sec 2
32
2 ±=
=±±=
Ciz
Cz
z, π, πz
있다밖에의는점않은해석적이지가
있다놓여밖에의모두점들은이않지만해석적이지는에서
04
,0sec
4
2
2
=+
=∴
+
∫∫CC z
dzdzz
z
Ex.3 비해석적인 함수
( ) ( )( )2
∫∫π
( ) ( )( )
( ) . Cauchy
20 : 20
않는다적용되지정리가의아니므로해석함수가는zzf
tetzCidtieedzz ititit
C
=
≤≤=== ∫∫ − ππ
( ) yf
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
Ex.4 해석성은 필요조건이 아니라 충분조건이다.
( ): 02 단위원Czdz
=∫
( ) . Cauchy 1 0 2 않았다나오지정리로부터의결과는이때문에않기해석적이지이에서z
zfz
zC
==
∫
. Cauchy
충분조건이다필요조건이라기보다는위한성립하기정리가의
조건은해석적이라는가에서 fD∴
Ex. 5 단순연결은 필수적이다.
( ) 2:1),(:,31: 해석적에서포함에단위원환형 idzDzfDCzD =⇒=⎟⎞
⎜⎛ << ∫ π( )
. Cauchy
2 : ), (: ,22
:
필수적이다아주조건은한다는있어야되어단순연결가정의역
없다수적용할정리를의않으므로있지되어단순연결가
해석적에서포함에단위원환형
D
D
iz
Dz
zfDCzDC
∴
⇒⎟⎠
⎜⎝
<< ∫ π
. 필수적이다아주조건은한다는있어야되어단순연결가정의역 D∴
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
경로 독립성경로 독립성
( )
( )
,
독립이다경로에내의적분은의
해석적이면내에서정의역단순연결된가만일
Dzf
Dzf
경로변형의 원리(Principle of Deformation of Path):
( ) . 독립이다경로에내의적분은의 Dzf
변형경로(양끝을 고정한 채 연속적으로 변형한 적분경로)가 항상 f (z)가해석적인 점만 포함하는 한, 선적분의 값은 어떠한 변형이 있더라도같은 값을 유지한다.같은 값을 유지한다
C1
z2
C2
z1
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
부정적분의 존재성
( )영역단순연결 :D
( )( ) ( ) ( ) ( ) 존재에서가부정적분의만족하는를해석적이고에서
해석적에서
' :
DzFzfzfzFDDzf
=⇒
다중연결 영역에 대한 Cauchy의 정리
CCD : 21 영역이중연결갖는를경계곡선안쪽과경계곡선바깥
( ) Dzf
DD
*:
:*
해석적내에서
정의역임의의포함하는경계곡선들을와
( ) ( )dzzfdzzfCC∫∫ =⇒
21
C
1
C2
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14.2 Cauchy’s Integral Theorem14.2 Cauchy’s Integral TheoremCauc y s eg a eo eCauc y s eg a eo e((CauchyCauchy의의 적분정리적분정리))
PROBLEM SET 14.2
HW 16 23HW: 16, 23
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14.3 Cauchy’s Integral 14.3 Cauchy’s Integral FormulaFormula3 Cauc y s eg a3 Cauc y s eg a o u ao u a((CauchyCauchy의의 적분공식적분공식))
Cauchy의 적분공식
( ):
:
해석적에서
영역단순연결
Dzf
D
C
z0
D
( )
( ) ( ) ( ) ( ) ( ).1,2
:
:
00
0
방향이다반시계방향은적분
곡선닫힌단순임의의안의있는둘러싸고를점임의의
해석적에서
∫∫ ==⇒ dzzfzfzifdzzf
DzC
Dzf
π
C
( ) ( ) ( ). 2
,2 0
000
방향이다반시계방향은적분∫∫ −−⇒
CC
dzzzi
zfzifdzzz π
π
Ex.1 Cauchy의 적분공식z
iieiedzzez
zz
C
z4268.4622
2 2 2
20 ===−
==∫ ππ대하여윤곽선에임의의둘러싸는를
Ex.2 Cauchy의 적분공식
dzi
z∫
−2
63
izC∫ −2
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14.3 Cauchy’s Integral 14.3 Cauchy’s Integral FormulaFormula3 Cauc y s eg a3 Cauc y s eg a o u ao u a((CauchyCauchy의의 적분공식적분공식))
Cauchy의 적분공식
( ):
:
해석적에서
영역단순연결
Dzf
D
C
z0
D
( )
( ) ( ) ( ) ( ) ( ).1,2
:
:
00
0
방향이다반시계방향은적분
곡선닫힌단순임의의안의있는둘러싸고를점임의의
해석적에서
∫∫ ==⇒ dzzfzfzifdzzf
DzC
Dzf
π
C
( ) ( ) ( ). 2
,2 0
000
방향이다반시계방향은적분∫∫ −−⇒
CC
dzzzi
zfzifdzzz π
π
Ex.1 Cauchy의 적분공식z
iieiedzzez
zz
C
z4268.4622
2 2 2
20 ===−
==∫ ππ대하여윤곽선에임의의둘러싸는를
Ex.2 Cauchy의 적분공식
⎟⎠⎞
⎜⎝⎛ =−=⎥⎦
⎤⎢⎣⎡ −=
−=
−∫∫ izCizidz
zdz
iz
21 6
83
2121
321
26
03
33
내부에서πππ ⎟⎠
⎜⎝⎥⎦⎢⎣−− =
∫∫iziz izCC 282
212 0
2
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14.3 Cauchy’s Integral 14.3 Cauchy’s Integral FormulaFormula3 Cauc y s eg a3 Cauc y s eg a o u ao u a((CauchyCauchy의의 적분공식적분공식))
다중연결 영역(Multiply Connected Domains)
( ) 2121 해석적안에서영역모양의고리둘러싸인의해에과및과가 CCCCzf
( ) ( ) ( )11
0
0
성립가
점임의의있는영역에그가
dzzfdzzfzf
z
∫∫ +=⇒ ( )
.
22
21 00
0
한다방향으로시계적분은안에서의하고방향으로반시계적분은밖에서의
성립가dzzzi
dzzzi
zfCC∫∫ −
+−
⇒ππ
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14.3 Cauchy’s Integral 14.3 Cauchy’s Integral FormulaFormula3 Cauc y s eg a3 Cauc y s eg a o u ao u a((CauchyCauchy의의 적분공식적분공식))
PROBLEM SET 14.3
HW 14 18HW: 14, 18
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14.4 14.4 Derivatives of Analytic FunctionsDerivatives of Analytic Functionsyy((해석함수해석함수의의 도함수도함수))
해석함수의 도함수
( ) ( )
( ) ( )
해석적이다역시도함수도그갖고도함수를계의모든에서해석적에서영역가
!21
.
ff
DDzf
( ) ( )( )
( ) ( )( )
( ) ( ) ( )( )
( )일반적으로 ,2 ,1 ,!
,2
!2'' ,21'
10
30
020
0
ndzzfnzf
dzzzzf
izfdz
zzzf
izf
n
CC
==
−=
−=⇒
∫
∫∫
+
ππ
( )( )
( )
방향반시계방향적분
경로닫힌단순임의의안의있는속해에전체가내부둘러싸면서를
:
:
,,,2
0
10
0
DDzC
zzif
Cn−∫ +π
방향반시계방향적분 :
Ex. 1 선적분의 계산
( )( ) ( )반시계대하여윤곽선에임의의있는싸고를점 sinh2sin2'cos2cos 2 πππππ
ππ
π=−==
−∫ =iizidz
izzi
Ciz
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14.4 14.4 Derivatives of Analytic FunctionsDerivatives of Analytic Functionsyy((해석함수해석함수의의 도함수도함수))
Cauchy의 부등식. Liouville와 Morera의 정리
• Cauchy의 부등식 : ( ) ( ) nn Mnzf !
0 ≤
• Liouville의 정리
nr
어떤 완전함수가 전 복소평면에서 절대값이 유계이면, 이 함수는 반드시 상수이다.
• Morera의 정리(Cauchy 적분정리의 역)Morera의 정리(Cauchy 적분정리의 역)
( ) ( )
( )
,0 ,
해석적이다에서는
이면대하여곡선에닫힌모든있는에연속이고에서영역단순연결가 dzzfDDzfC
=∫
( ) . 해석적이다에서는 Dzf
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14.4 14.4 Derivatives of Analytic FunctionsDerivatives of Analytic Functionsyy((해석함수해석함수의의 도함수도함수))
PROBLEM SET 14.4
HW 12HW: 12