010.141FOURIER SERIES, INTEGRALS, AND TRANSFORMSMODULE 3ADVANCED ENGINEERING MATHEMATICS

010.141

FOURIER SERIES, INTEGRALS, AND TRANSFORMS

MODULE 3

ADVANCED ENGINEERING MATHEMATICS

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 nx

n = 1

¥

å

Euler Formulas

Fourier coefficients

a , a , b are the Fourier coefficients.

( ) ( )

( )

0

1 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 =

( )

- -

ìí î

p

p 0

Example: Rectangular wave

FOURIER SERIES (cont)

( ) [ ]

0 0

0

n

1 1

a = k dx+ dx 0

2 2

1 1

a = 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

b = 1 sin x dx

1 1

= sin x dx + sin x dx

s nx cos nx

= n n

= 1 1 1 1 = 4k odd n

f x n

k n k n

k co

k for n

n

p

p

p

p

p

p

p

p p

p

p p

-

-

-

-

æ é ù é ù ö

ç ê ú + -ê ú ÷

ç ë û ë û ÷

è ø

+ + +

ò

ò ò

EXAMPLE 1: RECTANGULAR WAVE

( )

( )

( ) ( )

0 n n

n = 1 n = 1

2n+1 n = 0

n = 0

x =a + a cos cos

= sin 2 1

4 sin 2 1 2 1

4 1 1

sinx+ sin 3 sin 5

2 1+1 5

f nx b nx

b n x

k n x

n

k x x

p

p

¥ ¥

¥

¥

+ +

= +

+

æ ö

= ç + + ÷

è × ø

å å

å å

L Fourier series:

ORTHOGONALITY OF SINE, COSINE FUNCTIONS

Two functions f_m and f_n of the same form are orthogonal if ò f_m f_n dx = 0 for all m ¹ n and ò f_m f_n 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

CONVERGENCE AND SUM OF FOURIER SERIES

CONVERGENCE AND SUM OF FOURIER SERIES

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

( ) ( ) ( )

n

0

a 1 cos nx dx

f sin nx

1 1 sin nx

dx dx

f sin nx

1 1

'sin nx dx

1 'sin nx dx f x

x f

n n

x f

n n

n f

p

p

p p

p p

p p

p p

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)

( ) ^{x cos nx dx}

"

n f a 1

n 2

ò

p

p

p

-

=

( )

( )

n 2

M 2 M

2

dx M n

dx 1 nx cos x

"

f n

a 1

M x

"

f

p

p

<

p

=

<

ò ò

p

p - p

p -

Similarly for

Hence

which converges. (see Page 670)

CONVERGENCE AND SUM OF FOURIER SERIES (cont)

n 2

n M b < 2

( )

ø ç ö

è

æ + + + + + +

+

< L

2 2

2 0 2

3 1 3

1 2

1 2

1 1 1 M 2 a

x f

FUNCTIONS OF ANY PERIOD

P = 2L

FUNCTIONS OF ANY PERIOD (cont) P = 2L

( )

( ) ( ) ( )

( ) ò ( ) ò ( )

ò

ò ò

ò

å å

- -

p

p -

- -

p

p -

¥

=

¥

=

p =

= p

= p

= p

× p p

= p

= p

+ +

=

= p

L

L L

L 0

L

L L

L n

1 n

n 1

n

n 0

dx x L f

2 dx 1 x L

2 f dv 1 v 2 g

a 1

L dx x cos n

x L f

dx 1 L L

x cos n

x 1 f

dv nv cos v 1 g

a

nv sin b nv

cos a

a v g

L v x

Proof: Result obtained easily through change of scale.

Same for a₀, b_n

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

( )

( )

( )

( )

P x sin 2n

x P f

b 2

P dx x cos 2n

x P f

a 2

dx x P f

a 1

P x in 2n

s P b

x cos 2n

a a

x f

P a

a n

P a

a n

P a

a 0

1 n

n n

0

= p

= p

=

÷ø ç ö

è

æ p

p + +

=

ò ò ò

å

+ + +

¥

=

a a + P

EXAMPLE OF APPLICATION

Find the Fourier series of the function

Solution:

( )

ïî ïí ì

<

<

<

<

-

=

=

<

<

=

7 x 1 7 x 4 1

1 a

6 P 4 x 1 1 x

f

1 4 7

x f(x)

( ) ( )

( ) _ú

û ù êë

é p

p - p -

- p ú= ú û ù ê

ê ë

é p

p - p =

=

ú ¹ û ù êë

é p

p - p -

= p ú ú û ù ê

ê ë

é p

p -

=

ú= ú û ù ê

ê ë é

- +

=

=

ò ò

ò

ò ò

ò ò

ò

3 s 7n 3 co

s n co 3

cos 4n n 2

dx 1 3

x sin n 3 dx

x sin n 3

dx 1 3

x sin n x 3 f

b 1

0 n 3 sin 7n 3

sin n 3

sin 4n n 2

dx 1 3

x cos n 3 dx

x cos n 3

a 1

0 dx 1 dx

6 1 dx 1 x 6 f

a 1

7

4 4

1 7

1 n

7

4 4

1 n

7

4 4

1 7

1 0

EXAMPLE OF APPLICATION (cont)

3 cos n n

n cos 2 1

b

3 in n n s

n cos 2 1

a

0 a

3 cos n 3

s 7n co

3 cos n n

3 cos cos 4n

3 in n 3 s

sin 7n

3 in n s n 3 cos

sin 4n

n n 0

÷ p ø ç ö

è æ

p p

= -

÷ p ø ç ö

è æ

p p - -

=

=

= p p p p

p =

= p p p p

p =

x

-3 1 4 7

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

EVEN AND ODD FUNCTIONS

( ) ( )

( )

h

dx x g 2 dx x g

L

L

L

0 L

L

=

=

ò

ò ò

- -

EVEN AND ODD FUNCTIONS

EVEN AND ODD FUNCTIONS

SUM

SUM

SUM

SUM

Forced Oscillations

Forced Oscillations

Forced Oscillations

Forced Oscillations

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 BY

TRIGONOMETRIC POLYNOMIALS

( )

n 1

( ) x cos sin

N

n n

f x F A A nx B nx

=

» = +

å

( )

ò

p

p -

-

= f F dx

E ²

PARSEVAL'S THEOREM

( ) ^{( )}

2 2 2 2

0

n 1

2a a_n b_n 1 f x dx

p

p

¥

= -

+

å

ò

The square error, call it E*, is

Where a_n, b_n are the Fourier coefficients of f.

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 INTEGRALS

FOURIER COSINE AND SINE INTEGRALS

FOURIER COSINE AND SINE INTEGRALS

FOURIER COSINE AND SINE INTEGRALS

( ) ( ) ( )

f x =

ò

( ) ( )

( )

0

1 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 +₀ a cos nx + b sin nx_{n n}

n = 1

¥

å

( ) ( ) ( )

( ) ( ) ( )

1 cos d

1 sin d

A f v v v

B f v v v

w w

p

w w

p

¥

- ¥

¥

- ¥

=

=

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).

( ) ò ( )

ò

¥ -

®

+

b

b 0 0

a a

dx x

f lim

dx x

f lim

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

If f(x) is even

If f(x) is odd

( ) ( )

( ) ( ) ( )

( ) ( )

( )

( )

(

)

f

dv ωv sin v

π f ω 2

B

integral cosine

Fourier

dω ωx

cos ω

A x

f

dv ωv cos v

π f ω 2

A

0 0

ò ò ò

ò

¥

¥

¥ -

¥

¥

¥ -

=

=

=

=

FOURIER COSINE AND SINE INTEGRALS

EXAMPLE

Consider

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

( )

f = ^-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

( ) ( ) ( ) ( ) ( )

( )

π dv 2

ωv cos v π f

ω 2 A dω

ωx cos ω A x

f

0

c w f

=

=

=

ò ò

¥

¥ -

¥

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

ˆ

. of transform cosine

Fourier the

as defined is

ˆ

d ωx ˆ cos

π 2

d ωx ˆ cos

π x 2

f

dv ωv cos v π f

dv 2 ωv cos v π f

2 2 π

ω 2 A

π ˆ

0 0

0 0

c

c c

c

f f

f f

f

w

w w

w w

w

ò ò

ò ò

¥

¥

¥

¥

=

=

=

=

=

Then

FOURIER SINE AND COSINE TRANSFORMS (cont)

Similarly for odd function

( ) ( )

( )

( )

π x 2

f T

sine F

Inverse

dx ωx sin x

π f 2

fˆ T

sine F

0 c 0

c

ò ò

¥

¥

w w

=

®

= w

®

NOTATION AND PROPERTIES

{ } { }

{ } { }

{ } { } { }

{ } { } { }

( )

{ } { } ( )

( )

{ } { }

{ } { } ( )

2

0 π f

f 2 ω

f (5)

f ω x

f (4)

0 π f

f 2 ω

x f (3)

g b

f a

bg af

(2)

g b

f a

bg af

(1)

ˆ ˆ ,

ˆ ˆ ,

c 2 c

c s

s c

s s

s

c c

c

s 1 s c

1 c

s s

c c

- ¢ -

¢¢ =

-

¢ =

-

¢ =

+

= +

+

= +

=

=

=

=

- -

F F

F F

F F

F F

F F

F F

F

F F

F

f f

f f

f f

f f

FOURIER TRANSFORM

( ) [ ( ) ( ) ]

( ) ( )

4 4 4

4 3

4 4 4

4 2

1

w

¥

¥ -

¥

¥

¥ -

¥

ò ò

ò ò

w - w p w

=

w w

+ w w

p w

=

in Even 0

0

v d x v

cos v

f 1 d

v d x sin v sin v f x cos v cos v

f 1 d

x f

( ) [ ( ) ( ) ]

( ) ( )

( ) ò ( )

ò ò

¥

¥ -

¥

¥ -

¥

= w

= w

+

=

v d ωv sin v

π f B 1

v d ωv cos v π f

A 1

dx ωx sin ω

B ωx cos ω

A x

f

0

The F.I. is:

Replacing

FOURIER TRANSFORM

( ) ( ) ( )

( )

( )

(

)

2 x 1

f

v d v x cos v

f 2 d

x 1 f

in odd

4 4 4

4 3

4 4 4

4 2

1

w

¥

¥ -

¥

¥ -

¥

¥ -

¥

¥ -

ò ò

ò ò

- w p w

=

- w p w

=

Now,

( ) ( ) ( )

( )

f 2 d

1

dω G

i ω 2 F

x 1 f

v x

ò

ò ò

¥

-

¥

¥

¥ -

p w

=

w p +

=

FOURIER TRANSFORM (cont)

( ) ( )

( )

( ) ( )

= p

w p

= p

ò

ò ò

¥

¥ -

º w

¥

¥ -

-

¥

¥ -

d e

2 fˆ 1 x f

dv e

v 2 f

d 1 2 e

x 1 f

x iω

f of Transform Fourier

fˆ

v iω x

iω

4 4

4 3

4 4

4 2

1

NOTATION AND PROPERTIES

{ } { } { }

{ } { }

{ } { }

{ ^{f g} } ² { } ^f { } ^g

f ω

f

f iω

f

g b

f a

bg af

2

F F

F

F F

F

+ p

=

*

-

¢¢ =

¢ =

+

= +

F F

F

F