Engineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/4341.pdf · 2018-01-30 · Ch. 11...

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)

Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, andand TransformsTransforms

11.1 Fourier Series

11.2 Functions of Any Period p=2Ly p

11.3 Even and Odd Functions. Half-Range Expansions

11 4 Complex Fourier Series11.4 Complex Fourier Series

11.5 Forced Oscillations

11.6 Approximation by Trigonometric Polynomials

11.7 Fourier Integral

11.8 Fourier Cosine and Sine Transforms

11.9 Fourier Transform. Discrete and Fast Fourier Transforms11.9 Fourier Transform. Discrete and Fast Fourier Transforms

2

Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, andand TransformsTransforms

11.1 Fourier Series

11.2 Functions of Any Period p=2Ly p

11.3 Even and Odd Functions. Half-Range Expansions

11 4 Complex Fourier Series11.4 Complex Fourier Series

11.5 Forced Oscillations

11.6 Approximation by Trigonometric Polynomials

11.7 Fourier Integral

11.8 Fourier Cosine and Sine Transforms

11.9 Fourier Transform. Discrete and Fast Fourier Transforms11.9 Fourier Transform. Discrete and Fast Fourier Transforms

3

1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))

Integral Transform (적분변환):주어진 함수를 다른 변수에 종속하는 새로운 함수로 만드는 적분 형태의 변환

예) L l T f (Ch 6)예) Laplace Transform (Ch. 6)

• Fourier Cosine Transform (푸리에 코사인 변환)(푸리에 사인 변환)

( ) ( ) ( ) ( )∫∫∞∞

==00

cos2 ,cos : wvdvvfwAwxdwwAxfπ

적분코사인푸리에

( ) ( )

( ) ( ) ( )∫∞

=

2ˆ

. ˆ 2 wfwA cπ

변환사인리에

하자라

( ) ( ) ( )

( ) ( )∫

∫

∞

=

==⇒0

cosˆ2:)(

,cos2 : )(

wxdwwfxf

wxdxxfwff cc π

TransformCosineFourierInverse

TransformCosineFourier

역변환코사인푸리에

변환 코사인 푸리에 F

( ) ( )∫=0

cos : )( wxdwwfxf cπTransformCosineFourier Inverse역변환 코사인 푸리에


• Fourier Sine Transform (푸리에 사인 변환)

( ) ( ) ( ) ( )∫∫∞∞ 2( ) ( ) ( ) ( )

( ) ( )

∫∫

=

==00

ˆ2

sin2 ,sin :

wfwB

wvdvvfwBwxdwwBxfπ

하자라

적분사인푸리에

( ) ( )

( ) ( ) ( )∫∞

==⇒

=

,sin2ˆ : )(

. 2

wxdxxfwff

wfwB

ss

sπ

Transform SineFourier 변환 사인 푸리에 F

하자라

( ) ( )∫

∫

∞

=

0

sinˆ2 : )( wxdwwfxf sπ

π

Transform SineFourier Inverse역변환 사인 푸리에 ∫0π

1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms

Ex 1 Fourier Cosine and Fourier Sine Transforms

((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))

Ex. 1 Fourier Cosine and Fourier Sine Transforms

( ) ( )( ) .

00

구하라변환을사인푸리에및코사인푸리에의함수⎩⎨⎧ <<

=axk

xf ( ) ( )0⎩⎨ > ax


Ex 1 Fourier Cosine and Fourier Sine Transforms


Ex. 1 Fourier Cosine and Fourier Sine Transforms

( ) ( )( ) .

00

구하라변환을사인푸리에및코사인푸리에의함수⎩⎨⎧ <<

=axk

xf ( ) ( )0⎩⎨ > ax

( ) ⎟⎞

⎜⎛== ∫

awkwxdxkwfa sin2cos2ˆ ( )

⎞⎛

⎟⎠

⎜⎝∫ w

kwxdxkwf

a

c

122

cos0 ππ

( ) ⎟⎠⎞

⎜⎝⎛ −

== ∫ wawkwxdxkwf

a

scos12 sin2ˆ

0 ππ


Linearity


Linearity

( ) ( ) ( )

( ) ( ) ( )gbfabgaf

gbfabgaf ccc

FFF

FFF

+=+

+=+

Cosine and Sine Transforms of Derivatives

( ) ( ) ( )gbfabgaf sss FFF +=+

( )

( ) '

,

xf

xxf

∗

∗

연속구분유한구간에서모든가

적분가능절대상에서축연속이고가

( )

( ){ } ( ){ } ( ) ( ){ } ( ){ }2

0 ,

fffff

xfx →∞→∗

FFFF

때일

P !( ){ } ( ){ } ( ) ( ){ } ( ){ }

( ){ } ( ){ } ( ) ( ){ } ( ){ } ( )02''0'2''

' ,02'

22 wfxfwxffxfwxf

xfwxffxfwxf cssc π

+

−=−=⇒

FFFF

FFFF Prove!

( ){ } ( ){ } ( ) ( ){ } ( ){ } ( )0'' ,0''' wfxfwxffxfwxf sscc ππ+−=−−= FFFF


Ex 3 An Application of the Operational Formula (9)Ex. 3 An Application of the Operational Formula (9)

( ) ( ) . 0 구하라변환을코사인푸리에의함수 >= − aexf ax in two ways

( ) ( ) ( )'' '' , 22 == −− xfxfaeae axax 이므로의하여미분에

( ) ( ) ( ) ( ) ( ) 20'2'' 222 +−=−−==⇒ afwffwffa cccc ππFFFF

( ) ( ) 2 22

⎞⎛

=+⇒ afwa c πF

( ) ( )0 2 22 >⎟⎠⎞

⎜⎝⎛

+=∴ − a

waae ax

c πF


PROBLEM SET 11.8

HW 4 18HW: 4, 18

1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))

Complex Form of the Fourier Integral (푸리에 적분의 복소형식)

( ) ( )∫∞1

( ) ( ) ( )[ ]dwwxwBwxwAxf ∫∞

+=0

sincos( ) ( )

( ) ( )∫

∫∞

∞−

= wvdvvfwA

1

cos1π

0 ( ) ( )∫∞−

= wvdvvfwB sin1π

( ) dwdvwvwxvfxf ∫ ∫∞

∞

⎥⎦⎤

⎢⎣⎡ −= )cos()(1

Prove!

xixeix sincos +=

( ) dwdvwvwxvfxf ∫ ∫∞−

∞− ⎥⎦⎢⎣)cos()(

2π

0)sin()(1=⎥⎦

⎤⎢⎣⎡ −∫ ∫

∞∞

dwdvwvwxvf

Prove!

xixe sincos +0)sin()(2 ⎥⎦⎢⎣∫ ∫

∞−∞−

dwdvwvwxvfπ

( ) ( ) ( ) dwdvevfxf vxiw∫ ∫∞ ∞

−−1Prove!( ) ( ) ( ) dwdvevfxf ∫ ∫

∞− ∞−

=π2

Prove!



( ) ( ) dwedvevfxf iwxiwv∫ ∫∞

∞−

∞

∞−

−= ]21[

21

ππ ∞ ∞

( ) ( ) dxexfwf iwx∫∞

−=1ˆ ( ) ( ) dxexfwf ∫

∞−

=π2

( ) ( ) i∫∞

ˆ1( ) ( ) dwewfxf iwx∫∞−

−=21π



( ) ( ) ( ) ddff vxiw∫ ∫∞ ∞

−−1)( I t lF iC l적분푸리에복소•

•

( ) ( ) ( ) dwdvevfxf vxiw∫ ∫∞− ∞−

=π2

: )( IntegralFourier Complex 적분 푸리에복소

( ) ( ) ( ) dxexfwff iwx∫∞

−==1ˆ:)( FTransformFourier변환푸리에

•

( ) ( ) ( ) dxexfwff ∫∞−π2

: )( FTransformFourier 변환 푸리에

( ) ( ) dwewfx f iwx∫∞

−= ˆ1 : )( TransformFourier Inverse역변환 푸리에

Existence of the Fourier Transform (푸리에 변환의 존재)

( ) ( )ff ∫∞−2

)(π

( )

( ) ( ) ( ) ( )

xxf

i∫∞1ˆˆ

존재하며는변환푸리에의

구분연속구간에서유한모든가능이고적분절대상에서축가

( ) ( ) ( ) ( ) dxexfwfwfxf iwx∫∞−

−=⇒π2

1 , 존재하며는변환푸리에의


Ex. 1 Fourier Transform

( ) ( ) . 0 1 1 구하라변환을푸리에함수의인구간에서는이외의그이고에서 ==< xfxfx


( ) ( ) . 0 1 1 구하라변환을푸리에함수의인구간에서는이외의그이고에서 ==< xfxfx

Ex. 1 Fourier Transform

iwx 11111

( ) ( )eeiwiw

edxewf iwiwiwx

iwx

21

21

21ˆ

1

1

1 πππ−

−=

−⋅== −

−

−

−

−∫

ww

iwwi sin2

2sin2

ππ=

−−

=


Linearity. Fourier Transform of Derivatives

• 푸리에 변환의 선형성푸리에 변환의 선형성

( ) ( ) ( )gbfabgaf FFF +=+. 선형연산이다변환은푸리에

• 도함수의 푸리에 변환

( ) xxf ∗ 연속상에서축가함수

( )( )xfx

xxf

0,

'

→∞→∗

∗

때일

절대적분가능상에서축가

( )

( ){ } ( ){ } ( ){ } ( ){ }xfwxfxfiwxf

f

FFFF 2'' ,' −==⇒ Prove!


Ex. 3 Application of the Operational Formular (9)2

구하라변환을푸리에의x− . 구하라변환을푸리에의xxe

( ) )0( 21 4

22

>=−− aee a

waxF( )2a


2

구하라변환을푸리에의x−

Ex. 3 Application of the Operational Formular (9)

. 구하라변환을푸리에의xxe

( ) )0( 21 4

22

>=−− aee a

waxF

( ) ( ) ( ){ } ( )111 ⎫⎧

( )2a

( ) ( ) ( ){ } ( )22

2222

1121'

21'

21 xxxx

iw

eiweexe −−−− −=−=⎭⎬⎫

⎩⎨⎧−= FFFF

4422

2221

21

wweiweiw−−

−=−=


Convolution (합성곱)

( )( ) ( ) ( ) ( ) ( )dfdff ∫∫∞∞

)(C l i합성곱•

• Convolution Theorem (합성곱 정리)

( )( ) ( ) ( ) ( ) ( )dppgpxfdppxgpfxgf ∫∫∞−∞−

−=−=∗ : )( nConvolutio합성곱

Convolution Theorem (합성곱 정리)

( ) ( )( ) ( ) ( )

xxgxf , , 가능절대적분상에서축유한하며연속이고구분가와함수

( ) ( ) ( )gfgf FFF π2 =∗⇒ Prove using x-p=q


PROBLEM SET 11.9

HW 14HW: 14

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