CHAP. 11 Fourier Series, Integrals, and Transforms · 2018. 1. 30. · B.D. Youn Engineering Mathematics II CHAPTER 11 49 CHAP. 11.8 FOURIER COSINE AND SINE TRANSFORMS An integral

ENGINEERING MATHEMATICS II

010.141

Byeng Dong Youn (윤병동윤병동윤병동윤병동)

CHAP. 11

Fourier Series, Integrals, and

Transforms

2B.D. Youn Engineering Mathematics II CHAPTER 11

ABSTRACT OF CHAP. 11

�Fourier Analysis in Chap. 11 concerns periodic

phenomena—thinking of rotating parts of machines,

alternating electric currents, or the motion of planets.

However, the underlying ideas can also be extended to non-

periodic phenomena.

� Fourier series: Infinite series designed to represent general periodic

functions in terms of simple ones (e.g., sines and cosines).

� Fourier series is more general than Taylor series because many

discontinuous periodic functions of practical interest can be

developed in Fourier series.

� Fourier integrals and Fourier Transforms extend the ideas and

techniques of Fourier series to non-periodic functions and have basic

applications to PDEs.


CHAP. 11.1

FOURIER SERIESInfinite series designed to represent general periodic

functions in terms of simple ones like cosines and sines.


PERIODIC FUNCTIONS

� The smallest period is called a fundamental period.


PERIODIC FUNCTIONS


FOURIER SERIES


EXAMPLE


CONVERGENCE AND SUM OF FOURIER SERIES

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

derivatives.

( )

( )

( )

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

π

π

π π

π π

π π

ππ

π

π

π

π π

π π

π

−

− −

−−

−

=

′

′= −

= −

= −

∫

∫ ∫

∫

∫

1442443


Repeating the process:

Since f" is continuous on [– π, π ]

CONVERGENCE AND SUM OF FOURIER SERIES (cont)

( ) dx nxcosx"fn

1a

2n ∫π

π−π

−=

( )

( )

22n

22n

n

M2

n

M2a

dxMn

1dx nxcosx"f

n

1a

Mx"f

=π

π<

π<

π=

<

∫∫π

π−

π

π−


Similarly for

Hence

which converges.

CONVERGENCE AND SUM OF FOURIER SERIES (cont)

2n n

M2b <

( )

+++++++< L

22220 3

1

3

1

2

1

2

111M2axf


HOMEWORK IN 11.1

� HW1. Problem 9

� HW2. Problem 15


CHAP. 11.2

FUNCTIONS OF ANY PERIOD P=2Lgeneral periodic functions with P=2L.


FUNCTIONS OF ANY PERIODP = 2L


EXAMPLE


FUNCTIONS OF ANY PERIOD (cont)P = 2L

( )

( ) ( ) ( )

( ) ( ) ( )∫∫∫

∫∫∫

∑∑

−−

π

π−

−−

π

π−

∞

=

∞

=

=π

π=

π=

π=

π⋅

π

π=

π=

++=

π=

L

L

L

L

0

L

L

L

L

n

1 n

n

1 n

n0

dxxf L2

1 dx

Lxf

2

1 dvvg

2

1 a

dxL

xncosxf

L

1 dx

L

L

xncosxf

1 dv nvcosvg

1 a

nvsinb nvcosa a vg

L

xv

Proof: Result obtained easily through change of scale.

Same for a0, bn


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

( )

( )

( )

( ) dx P

x2nsinxf

P

2 b

dx P

x2ncosxf

P

2 a

dxxf P

1 a

P

x2ninsb

P

x2ncosa a xf

P a

a

n

P a

a

n

P a

a

0

1 n

nn0

π=

π=

=

π+

π+=

∫

∫

∫

∑

+

+

+

∞

=

a a + P


HOMEWORK IN 11.2

� HW1. Problem 3

� HW2. Problem 5


CHAP. 11.3

EVEN AND ODD FUNCTIONS. HALF-

RANGE EXPANSIONSTake advantage of function properties.


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.

EVEN AND ODD FUNCTIONS


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

=

=

∫

∫∫

−

−

EVEN AND ODD FUNCTIONS


THEOREMS


EXAMPLE


HOMEWORK IN 11.3

� HW1. Problem 11

� HW2. Problem 15


CHAP. 11.5

FORCED OSCILLATIONSConnections with ODEs and PDEs.


Forced Oscillations


HOMEWORK IN 11.5

� HW1. Problem 6


CHAP. 11.6

APPROXIMATION BY

TRIGONOMETRIC POLYNOMIALSUseful in approximation theory.


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=

≈ = + +∑

( )∫π

π−

−= dxFf E2

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 dx

π

ππ

∞

= −

+ + ≤∑ ∫

The square error, call it E*, is

where an = An and bn = Bn.

Parseval’s theorem:


HOMEWORK IN 11.6

� HW1. Problem 2


CHAP. 11.7

FOURIER INTEGRALExtension of the Fourier series method to non-periodic

functions.


FOURIER INTEGRALS

Since many problems involve functions that are nonperiodic and are of

interest on the whole x-axis, we ask what can be done to extend the method

of 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 xω ω ω ω ω∞

= + ∫

( )

( )

( )

0

1

2

1cos

1sin

n

n

a = f x dx

a = f x nx dx

b = f x nx dx

π

π

π

π

π

π

π

π

π

−

−

−

⋅

⋅

∫

∫

∫

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

∞

∑

( ) ( ) ( )

( ) ( ) ( )

1 cos d

1 sin d

A f v v v

B f v v v

ω ωπ

ω ωπ

∞

−∞

∞

−∞

=

=

∫

∫


THEOREM1: 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.

The F.I. equals the average of the left-hand and right-hand limit of f(x)

where f(x) is discontinuous.

( ) ( )∫∫∞→∞−→

+

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

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

0

0

0

0

2A ω f v cos ωv dv

π

f x A ω cos ωx dω Fourier cosine integral

2B ω f v sin ωv dv

π

f x B ω sin ωx dω Fourier sine integral

∞

∞

∞

∞

=

=

=

=

∫

∫

∫

∫



EXAMPLE

Consider

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

( ) 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


HOMEWORK IN 11.7

� HW1. Problem 1

� HW2. Problem 7


CHAP. 11.8

FOURIER COSINE AND SINE

TRANSFORMSAn integral transform is a transformation in the form of

an integral that produces from given functions new

functions depending on a different variable.


FOURIER SINE AND COSINE TRANSFORMS

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

( ) ( ) ( ) ( ) ( ) ( )0

2 2 ˆf x A ω cos ωx dω A ω f v cos ωv dv π π

cf ω∞ ∞

−∞

= = =∫ ∫

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

0 0

0 0

π π 2 2ˆ A ω f v cos ωv dv f v cos ωv dv2 2 π π

2 2ˆ ˆf x cos ωx d cos ωx d π π

ˆ is defined as the Fourier cosine transform of .

is the inverse Fourier cosine transform o

c

c c

c

f

f f

f f

f

ω

ω ω ω ω

ω

∞ ∞

∞ ∞

= = =

= =

∫ ∫

∫ ∫

ˆf .c

f

Then

{ } { }1c c c cˆ ˆ , f f f f−= =F F


FOURIER SINE AND COSINE TRANSFORMS (cont)

Similarly for odd function

( ) ( )

( ) ( )

0

0

2ˆ sine f x sin ωx dx π

2 ˆ sine f x sin ωx d π

s

s

F T f

Inverse F T f

ω

ω ω

∞

∞

→ =

→ =

∫

∫

{ } { }1s s s sˆ ˆ , f f f f−= =F F


NOTATION AND PROPERTIES

{ } { }

{ } { }

{ } { } { }

{ } { } { }

( ){ } { } ( )

( ){ } { }

{ } { } ( )

{ } { } ( )0fω π

2fω f(6)

0f π

2fω f(5)

fω xf(4)

0f π

2fω 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

FFF

FFF

ffff

ffff


TRANSFORMS OF DERIVATIVES


FOURIER TRANSFORM

( ) ( )[ ]

( ) ( )

0

( ) Even function in

1 cos cos sin sin

1 cos

2

G

f x f v v x v x dv d

f v x v dv d

∞ ∞

−∞

∞ ∞

−∞ −∞

=

= +

= −

∫ ∫

∫ ∫1444442444443

ω ω

ω ω ω ω ωπ

ω ω ωπ

( ) ( ) ( )

( ) ( )

( ) ( )

0

cos sin

1 cos

1 sin

f x A x B x dx

A f v v dv

B f v v dv

∞

∞

−∞

∞

−∞

= +

=

=

∫

∫

∫

ω ω ω ω

ω ωπ

ω ωπ

The F.I. is:

Replacing


FOURIER TRANSFORM

( ) ( ) ( )

( ) ( ) ( )

( ) odd function in

1 cos

2

1 sin 0

2

F

f x f v x v dv d

g x f v x v dv d

∞∞

−∞−∞

∞∞

−∞−∞

=

= −

= − =

∫ ∫

∫ ∫1444442444443

ω ω

ω ωπ

ω ωπ

Now,

( ) ( ) ( )

( ) ( )

1

2

1 Complex Fourier Integral

2

i x v

f x F i G d

f v e dv d

∞

−∞

∞ ∞

−

−∞ −∞

= +

=

∫

∫ ∫ω

ω ω ωπ

ωπ


FOURIER TRANSFORM (cont)

( ) ( )

( )

( ) ( )

ˆ Fourier Transform of

1 1

2 2

1 ˆ 2

i v i x

f f

i x

f x f v e dv e d

f x f e d

∞ ∞

−

−∞ −∞

≡

∞

−∞

=

=

∫ ∫

∫

14444244443

ω ω

ω

ω

ωπ π

ω ωπ


NOTATION AND PROPERTIES

{ } { } { }

{ } { }

{ } { }

{ } { } { }g f2 gf

fω f

fiω f

gb fa bgaf

2

FFF

FFF

+π=∗

−=′′

=′

+=+

FF

FF

Documents

CHAP. 11 Fourier Series, Integrals, and Transforms · 2018. 1. 30. · B.D. Youn Engineering Mathematics II CHAPTER 11 49 CHAP. 11.8 FOURIER COSINE AND SINE TRANSFORMS An integral