010.141 fourier series, integrals, and transforms … · 2018. 1. 30. · fourier series,...

010.141

FOURIER SERIES, INTEGRALS, AND TRANSFORMS

MODULE 3

ADVANCED ENGINEERING MATHEMATICS

2B.D. Youn2010 Engineering Mathematics II Module 2

Fourier Series, Integrals, and Transform

Ø Fourier series: Infinite series designed to represent general periodic functions in terms of simple ones (e.g., sines and cosines).

Ø Fourier series is more general than Taylor series because many discontinuous periodic functions of practical interest can be developed in Fourier series.

Ø Fourier integrals and Fourier Transforms extend the ideas and techniques of Fourier series to non-periodic functions and have basic applications to PDEs.


PERIODIC FUNCTIONS

Ø A periodic function is a function such that f(x + p) = f(x) where p is a period.

Ø The smallest period is called a fundamental period.


FOURIER SERIES

Assume that f(x) is periodic with period 2p and is integrable over a period.

f(x) can be represented by a trigonometric series:

i.e., the series converges and has sum f(x)

( ) ( )f x = a + a cos nx + b sin nx0 n nn = 1

¥

å


Euler Formulas

Fourier coefficients

a0, an, bn are the Fourier coefficients.

The corresponding series is the Fourier series

( )

( )

( )

01

2

1 cos

1 sin

n

n

a = f x dx

a = f x nx dx

b = f x nx dx

p

p

p

p

p

p

p

p

p

-

-

-

×

×

ò

ò

ò

TRIGONOMETRIC SERIES WITH A PERIOD OF 2p


( )f x = k, < x < 0

k, < x < - -ì

íî

pp0

Example: Rectangular wave

FOURIER SERIES (cont)

( ) [ ]

0

00

n

1 1a = k dx+ dx 02 2

1 1a = cos x dx= 0 =0

k

f x n

p

p

p

p

p p

p p

-

-

- =ò ò

ò

Fourier Coefficients( )

( )

( )

n

0

0

0

0

1b = sin x dx

1 1= sin x dx + sin x dx

s nx cos nx= n n

4k= 1 1 1 1 = odd n

f x n

k n k n

k co

k for nn

p

p

p

p

p

p

p

p p

p

p p

-

-

-

-

æ öé ù é ù+ -ç ÷ê ú ê úç ÷ë û ë ûè ø

+ + +

ò

ò ò


EXAMPLE 1: RECTANGULAR WAVE

( )

( )

( ) ( )

0 n nn = 1 n = 1

2n+1n = 0

n = 0

x =a + a cos cos

= sin 2 1

4 sin 2 12 1

4 1 1sinx+ sin 3 sin 52 1+1 5

f nx b nx

b n x

k n xn

k x x

p

p

¥ ¥

¥

¥

+

+

= ++

æ ö= + +ç ÷×è ø

å å

å

å

L

Fourier series:


ORTHOGONALITY OF SINE, COSINE FUNCTIONS

Two functions fm and fn of the same form are orthogonal if ò fm fn dx = 0 for all m ¹ n and ò fm fn dx = a for all m = n.

( )

( )

2

2

cos mx cos nx dx 0 for all m n

cos nx dx 0 for all m n

sin mx sin nx dx 0 for all m n

sin nx dx 0 for all m n

cos mx sin nx dx

p

p

p

p

p

p

p

p

p

p

-

-

-

-

-

= ¹ ±

¹ =

= ¹

¹ =

ò

ò

ò

ò

ò 0 for all m n= =


CONVERGENCE AND SUM OF FOURIER SERIES


CONVERGENCE AND SUM OF FOURIER SERIES

Proof: For continuous f(x) with continuous first and second orderderivatives.

( )

( )

( )

n

0

1a cos nx dx

f sin nx1 1 sin nxdx dx

f sin nx1 1 'sin nx dx

1 'sin nx dx

f x

xf

n n

xf

n n

fn

p

p

p p

p p

p p

pp

p

p

p

p p

p p

p

-

- -

--

-

=

¢æ ö

¢= -ç ÷è ø

= -

= -

ò

ò ò

ò

ò

1442443


Repeating the process:

Since f" is continuous on [– π, π ]

CONVERGENCE AND SUM OF FOURIER SERIES (cont)

( ) dx nxcosx"fn1a 2n ò

p

p-p-=

( )

( )

22n

22n

nM2

nM2a

dxMn1dx nxcosx"f

n1a

Mx"f

=pp

<

p<

p=

<

òòp

p-

p

p-


Similarly for

Hence

which converges. (see Page 670)

CONVERGENCE AND SUM OF FOURIER SERIES (cont)

2n nM2b <

( ) ÷øö

çèæ +++++++< L22220 3

131

21

2111M2axf


FUNCTIONS OF ANY PERIODP = 2L


FUNCTIONS OF ANY PERIOD (cont)P = 2L

( )

( ) ( ) ( )

( ) ( ) ( )òòò

òòò

åå

--

p

p-

--

p

p-

¥

=

¥

=

=p

p=

p=

p=

p×

pp

=p

=

++=

p=

L

L

L

L0

L

L

L

Ln

1 n n

1 n n0

dxxf L21 dx

Lxf

21 dvvg

21 a

dxL

xncosxf L1 dx

L

Lxncosxf 1 dv nvcosvg 1 a

nvsinb nvcosa a vg

Lxv

Proof: Result obtained easily through change of scale.

Same for a0, bn


ANY INTERVAL (a, a + P)P = PERIOD = 2L

( )

( )

( )

( ) dx P

x2nsinxf P2 b

dx P

x2ncosxf P2 a

dxxf P1 a

Px2ninsb

Px2ncosa a xf

P a

an

P a

an

P a

a0

1 n nn0

p=

p=

=

÷øö

çèæ p+

p+=

ò

ò

ò

å

+

+

+

¥

=

a a + P


EXAMPLE OF APPLICATION

Find the Fourier series of the function

Solution:

( )ïî

ïí

ì


EXAMPLE OF APPLICATION (cont)

3ncos

nncos12 b

3nins

nncos12 a

0 a

3ncos

37nsco

3ncos ncos

34ncos

3nins

37nsin

3nins ncos

34nsin

n

n

0

p÷øö

çèæ

pp-

=

p÷øö

çèæ

pp-

-=

=

p=

ppp=

p

p=

ppp=

p

x-3 71 4

f(x)


EVEN AND ODD FUNCTIONS

Examples: x4, cos x

f(x) is an even function of x, if f(-x) = f(x). For example, f(x) = x sin(x), then

f(- x) = - x sin (- x) = f(x)

and so we can conclude that x sin (x) is an even function.


Properties

1. If g(x) is an even function

2. If h(x) is an odd function

3. The product of an even and odd function is odd


( ) ( )

( ) 0dx xh

dx xg2dx xg

L

L

L

0

L

L

=

=

ò

òò

-

-


SUM


Forced Oscillations


Consider a function f(x), periodic of period 2π. Consider an approximation of f(x),

The total square error of F

is minimum when F's coefficients are the Fourier coefficients.

APPROXIMATION BYTRIGONOMETRIC POLYNOMIALS

( ) 0n 1

( ) x cos sinN

n nf x F A A nx B nx=

» = + +å

( )òp

p-

-= dxFf E 2

Read Page 503 to see the derivation procedure of Eq. (6)


PARSEVAL'S THEOREM

( ) ( )2 2 2 20n 1

12 n na a b f x dxp

pp

¥

= -

+ + £å ò

The square error, call it E*, is

Where an, bn are the Fourier coefficients of f.


FOURIER INTEGRALS

Since many problems involve functions that are nonperiodic and are ofinterest on the whole x-axis, we ask what can be done to extend the methodof Fourier series to such functions. This idea will lead to “Fourier integrals.”


FOURIER INTEGRALS


FOURIER COSINE AND SINE INTEGRALS



( ) ( ) ( )0

f x cos sin dA x B xw w w w w¥é ù= +ë ûò

( )

( )

( )

01

2

1 cos

1 sin

n

n

a = f x dx

a = f x nx dx

b = f x nx dx

p

p

p

p

p

p

p

p

p

-

-

-

×

×

ò

ò

ò

( ) ( )f x = a + a cos nx + b sin nx0 n nn = 1

¥

å

( ) ( ) ( )

( ) ( ) ( )

1 cos d

1 sin d

A f v v v

B f v v v

w wp

w wp

¥

-¥

¥

-¥

=

=

ò

ò


EXISTENCE

If f(x) is piecewise continuous in every finite interval and has a right hand and left hand derivative at every point and if

exists, then f(x) can be represented by the Fourier integral.

Where f(x) is discontinuous, the F.I. equals the average of the left-hand and right-hand limit of f(x).

( ) ( )òò ¥®¥-® +b

0 b

0

a a

dxxf limdxxf lim


For an even or odd function the F.I. becomes much simpler.

If f(x) is even

If f(x) is odd

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) integral sineFourier dω ωx sinωB xf

dv ωvsin vfπ2 ωB

integral cosineFourier dω ωx cosωA xf

dv ωv cos vfπ2 ωA

0

0

ò

ò

ò

ò

¥

¥

¥-

¥

¥

¥-

=

=

=

=



EXAMPLE

Consider

Evaluate the Fourier cosine integral A(w) and sine integral B(w).

( ) 0 k 0, x e xf kx >>= -

Integration by parts gives

For Fourier cosine integral,

Fourier cosine integral is


EXAMPLE

Integration by parts gives

For Fourier sine integral,

Fourier sine integral is

Laplace integrals


FOURIER SINE AND COSINE TRANSFORMS

For an even function, the Fourier integral is the Fourier cosine integral

( ) ( ) ( ) ( ) ( ) ( ) ˆπ2 dv ωv cos vf

π2 ωAdω ωx cosωA xf

0

wcf=== òò¥

¥-

¥

( ) ( ) ( ) ( )

( ) ( ) ( )

( ).ˆ of transformcosineFourier inverse theis

. of transformcosineFourier theas defined is ˆ

dωx cos ˆπ2 dωx cos ˆ

π2 xf

dv ωv cos vfπ2 dv ωv cos vf

π2

2π ωA

2π ˆ

00

00

c

c

cc

c

ff

ff

ff

f

w

wwww

w

òò

òò¥¥

¥¥

==

===

Then


FOURIER SINE AND COSINE TRANSFORMS (cont)

Similarly for odd function

( ) ( )

( ) ( ) dωx sin f̂π2 xf T sine F Inverse

dxωx sin xfπ2 f̂ T sine F

0c

0c

ò

ò

¥

¥

ww=®

=w®


NOTATION AND PROPERTIES

{ } { }{ } { }

{ } { } { }{ } { } { }

( ){ } { } ( )

( ){ } { }

{ } { } ( )

{ } { } ( )0fω π2fω f(6)

0f π2fω f(5)

fω xf(4)

0f π2fω xf(3)

gb fa bgaf(2)gb fa bgaf(1)

ˆ , ˆ

ˆ ,ˆ

s2

s

c2

c

cs

sc

sss

ccc

s1

sc1

c

sscc

¢+-=¢¢

¢--=¢¢

-=¢

-=¢

+=++=+

==

==--

FF

FF

FF

FF

FF

FF

FFFFFF

ffff

ffff


FOURIER TRANSFORM

( ) ( ) ( )[ ]

( ) ( )4444 34444 21

w

¥

¥-

¥

¥

¥-

¥

òò

òò

w-wwp

=

ww+wwwp

=

in Even

0

0

vdx vcos vfd1

vdxsinvsin vfxcosvcos vfd1 xf

( ) ( ) ( )[ ]

( ) ( )

( ) ( )ò

ò

ò

¥

¥-

¥

¥-

¥

=w

=w

+=

vd ωvsin vfπ1 B

vd ωv cos vfπ1 A

dx ωxsin ωBωx cosωAxf0

The F.I. is:

Replacing


FOURIER TRANSFORM

( ) ( ) ( )

( ) ( ) ( ) lvd v xsinvfd21 xf

vd v xcosvfd21 xf

in odd4444 34444 21

w

¥

¥-

¥

¥-

¥

¥-

¥

¥-

òò

òò

-wwp

=

-wwp

=

Now,

( ) ( ) ( )

( ) ( ) IntegralFourier Complex dve vfd21

dω GiωF21 xf

vxiωòò

ò

¥

¥-

-¥

¥-

¥

¥-

wp

=

w+p

=


FOURIER TRANSFORM (cont)

( ) ( )

( )

( ) ( ) wwp

=

pw

p=

ò

òò

¥

¥-

ºw

¥

¥-

-¥

¥-

de f̂21 xf

dve vf21de

21 xf

xiω

f of TransformFourier f̂

viωxiω

444 3444 21


NOTATION AND PROPERTIES

{ } { } { }{ } { }

{ } { }{ } { } { }g f2 gf

fω f

fiω f

gb fa bgaf

2

FFF

FFF

+p=*

-=¢¢

=¢

+=+

FF

FF

010.141 fourier series, integrals, and transforms … · 2018. 1. 30. · fourier series,...

Documents