Fourier Series Student.pdf

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CHAPTER 4

FOURIER SERIES

CONTENTS PAGE

4.1 Periodic Function 1

4.2 Even and Odd Function 3

4.3

Fourier Series for Periodic Function 7

4.4

Fourier Series for Half Range Expansions 18

4.5

Approximate Sum of the Infinite Series 24

Exercise 28

References 30



CHAPTER 4: FOURIER SERIES SSE 1793

1

4.1 Periodic Function

A function ( ) f t is periodic with period of T if

, 0 F t T f t T

Example 4.1.1

Show that ( ) cos2 f x x is a periodic function and find its

period.

Solution:Since ( ) cos2 f x x , then

cos2 f x T x T .

What is the value of T , such that cos2 cos2 x T x ?

Note that :

cos2 ( ) cos2 cos2 sin 2 sin 2cos2 1

cos 2 if sin 2 0

x T x T x T T

xT

The smallest value for T is 1 , known as fundamental period.Therefore, ( ) cos2 f x x is a periodic function with

period of 1.

This is true for 1,2,3,...T




2

Example 4.1.2

Determine whether sin5 f x x is a periodic function. If

yes, find the period.

Solution:

sin5 f x T x T

sin5 cos5 cos5 sin5 x T x T

sin5 x if cos5 1sin5 0

T T

cos5 cos 2 1

5 2

2

5

T

T

T

Therefore, sin5 f x x is a periodic function with the

period of2

5

.




3

4.2 Even and Odd Function

f x is an even function if f x f x , for all . x (I)

f x is an odd function if f x f x , for all . x (II)

Geometrically,

The graph is symmetrical about the y-axis.

The graph is symmetrical about the origin.

NOTE: If none of (I) and (II) are satisfied, then f x is called

neither even nor odd.

Example 4.2.1

Determine whether the following functions are even, odd,

or neither.

(i) 2 f x x

(ii) sinh x x

(iii) 1

3 sin g x x x

(iv) t f t e

Even

Odd




4

Solutions:

(i) 2 2 f x x x f x .

2 f x x is an .

(ii) sinh x x

sin(0 )

sin( ) (by trigonometric identity)

x

x

h x

Or, refer to the graph.

( ) sinh x x is an .

(iii) 13 sin( ) g x x x

13

13

sin

( sin )( )

x x

x x g x

13( ) sin g x x x is an .

.2 2 2

1

-1

h(x)

xπ -π 2π

Symmetrical about

.

From (ii),

sin( ) sin x x




5

(iv) ( ) t f t e

Argument 1: Test for even.

Note that ( ) ( ) f t f t only when 0t . (recall definition:for a function to be an even function, it must satisfies

( ) ( ) f t f t for all t ).

Argument 2: Test for odd.t e is never equal to

t

e , ( ) ( ). f t f t

Argument 3: From the graph, t f t e is neither

symmetrical about the y-axis nor about the origin.

Conclusion: t f t e is neither even nor odd.




6

4.2.1 Properties of Even and Odd Functions

Even Even Even Even ×/÷ Even Even

Odd Odd Odd Odd ×/÷ Odd Even

Even Odd Neither Odd ×/÷ Even Odd

Integral properties

i) For even function: 0

2l l

l f x dx f x dx

Example:2( ) f x x

2 22 2

2 02 x dx x dx

ii) For odd function: 0l

l f x dx

Example: ( ) sin f x x 0

2 2

02 2

sin sin sin

0.

x dx x dx x dx

NOTE: It is important to understand the integral properties

of even and odd function before you can proceed to the

next section.

-2 2




7

Example:

If ( ) f x is even, then ( ) sin 0

l

l f x nx dx

.

0( ) cos 2 ( ) cos

l l

l f x nx dx f x nx dx

If ( ) f x is odd, then

0( ) sin 2 ( ) sin

l l

l f x nx dx f x nx dx

( ) cosl

l f x nx dx

Even X Odd ---> dd

Even X Even --->

Odd X Odd --->

Odd X Even --->




8

4.3 Fourier Series for Periodic Function ( ) f x , consisting of

Sin and Cos.

Fourier series for ( ) f x given by

0 cos sin2 1

a n n f x a x b xn n

l l n

where 0 , andn na a b can be expressed as following:

0

1 l

l a f x dxl

1

cosl

l

n xa f x dxn

l l

1

sinl

l

n xb f x dxn

l l

where

2

T l and T is period for periodic function.

If ( ) f x is an even function, then 0nb .

Hence, its Fourier series is given by :

0 cos2 1

a n f x a xn

l n

where 0

0

2 l a f x dx

l

0

2cos

l na f x x dxn

l l

Even X OddOdd.

Recall properties.

Even

properties

( I )

( II )




9

If x f is an odd function, then 0 0, 0na a .

Hence, its Fourier series is given by:

1

sinn

n

n x f x b

l

and 0

2sin

l

n

n xb f x dx

l l

Summary:

( ) f x Formula

Neither I

Even II

Odd III

Recall

properties

Odd X Odd Even

( III )




10

Example 4.3.1

Find the Fourier series for

f x x , 1 1 x

2 f x f x .

Solution:

i) Determine T and l .

T = 2, l = 1.

ii) Draw the graph of ( ) f x .

iii) Check whether ( ) f x is an odd or even function (or

neither). Identify the corresponding formulae.

From the graph, ( ) f x is an odd function. Hence, the Fourierseries formulae are given by (III):

1 3 5-1-2-3 x

f ( x)




11

iv) Calculate the Fourier series of f(x).

1

sinn

n

f x b n x

.

0

2sin

l

n

n xb f x dx

l l

1

0

1

0

1

2 2

0

2 2

2 sin

2 sin

cos sin2

cos sin2 (0 0)

cos2

2( 1)n

x n x dx

x n x dx

n x n x x

n n

n n

n n

n

n

n

Therefore,

1

2 ( 1)sin

n

n

f x n xn

.

Note: It is not wrong to use the full range of [-1,1] when computingnb . The

result will be the same; it’s just that you may encounter a longer calculation.

sin 0

cos ( 1)

for 0,1,2,...

n

n

n

n

Remarks:




12

Example 4.3.2

Find the Fourier series for

0

0

x x f x

x x

2 f x f x

Solution:


2 , .T l




From the graph, ( ) f x is an even function. Hence, the

Fourier series formulae are given by (II):

π 2 π 3 π - π -2 π

x

f (x)




13


0 cos2 1

a n f x a xn

l n

00

0

22

0

2

2

2 1( 0)

2

l

a x dxl

x dx

x

0

2( )cos

l n xa f x dxn

l l

0

2cos

n x x dx

0

2cos x nx dx

2

0

2 sin cos x nx nx

n n

2 2

2 sin cos 1n n

n n n

2

2(cos 1), 1,2,3,....n n

n

2

4, odd

0, even

nn

n




14

Therefore,

0

1

( ) cos2

n

n

a n x f x a

l

1

cos2

n

n

a nx

2

4, odd

where

0, evenn

na n

n

Example 4.3.3

Show that the Fourier series of

( ) sin ,2 2

( ) ( )

f x x x

f x f x

is given by2

1

2 4 1( ) cos2 .4 1n

f x nxn

Solution:


, .2T l




15


2 3 0 2 3

1



From the graph, ( ) f x is an even function. Hence, theFourier series formulae are given by (II):


20

0

20

2sin

2

4 4cos

a x dx

x

2

0

2

0

4sin cos(2 )

2

(sin( 2 ) sin( 2 ))

na x nx dx

x nx x nx dx




16

2

0

2

2 cos( 2 ) cos( 2 )

1 2 1 2

2 1 1 4

1 2 1 2 (1 4 )

x nx x nx

n n

n n n

Hence,2

1

2 4 1( ) cos2 .

4 1n

f x nxn

Example 4.3.4

Find the Fourier series of

, 0

0, 0

x x f x

x

2 f x f x

Solution:

i) Determine T and l .2 , .T l


-3π

. 2π π

f(x)

x-π 3π -2π . 4π ..




17



( ) f x is neither even nor odd function. Hence, the Fourier

series formulae are given by (I):


0 cos sin2 1

a n n f x a x b xn n

l l n

0

1 1l

l a f x dx f x dxl

0

0

10 x dx dx

021

2

x

21

2 2

1

cosl

nl

na f x x dx

l l

01cos x nx dx

0

2

1 1cos

x sin nx nxn n

2

2 2

1 1 10 sin cosn n

nn n




18

2 2

2

2

1 1 1cos

11 cos

1 1 1 , 1, 2, 3,...n

nn n

nn

nn

Remarks: sin sin 0, 1,2,3,..n n n

cos cos 1 , 1, 2,3,..n

n n n

01

sinnb x nx dx

0

2

2

1

1 1cos sin

1 10 sin 0 cos sin

1cos

1cos

111

n

n

xnx nx

n n

n nn n

nn

nn

n n

0

1

1

21

cos sin2

12cos 2 1 sin

4 2 1

n n

n

m

m

a n x n x

f x a bl l

m x mxmm




19

4.4 Fourier Series for Half Range Expansions.

Sometimes, if we only needs a Fourier series for a function defined

on the interval [0 ,

l ], it may be preferable to use a sine or cosineFourier series instead of a regular Fourier series.

This can be accomplished by extending the definition of the

function in question to the interval [−l , 0] so that the extended

function is either even (if we wants a cosine series) or odd (if we

wants a sine series).

Such Fourier series are called half-range expansions.

Fourier Cosine series form

0

1

cos2

n

n

a n x f x a

l

where

00

2 l

a f x dx

l

0

2( )cos , 0,1,0...

l

n

n xa f x dx n

l l

Fourier Sine series form

1

sinn

n

n x f x b

l

where 0

2sin , 1, 2,3...

l

n

n xb f x dx n

l l




20

Example 4.4.1 (Final Exam Sem II 2004/05)

Consider the function

( ) ; 0 . f x x x

a) Sketch the appropriate graphs in the interval

-3 3 x for each of the following cases:

i)

The periodic extension of ( ) f x .

ii) The odd 2 -periodic extension of ( ) f x .

iii) The even 2 -periodic extension of ( ) f x .

b) Find the Fourier cosine series of ( ) f x .

Solution:

a) The original graph of ( ) f x is given by

0 x

( ) f x

Therefore, the solution to a) is just an extension to the

original graph.

i) The periodic, range -3 3 x .

0

2 3 2 3 x

( ) f x




21

ii) The odd 2 -periodic, range-3 3 x .

1 - Transform the original graph into an odd graph.(Remember, the odd graph is symmetry about the origin.)

0

x

( ) f x

0

x

( ) f x

2 – Extend the odd 2 -periodic graph to the

range-3 3 x . (Repeat the shape).

0

2 3 2 3 x

( ) f x

iii) The even 2 -periodic, range-3 3 x .

1 - Transform the original graph into an even graph.(Remember, the even graph is symmetry about y-axis.)

2 period.




22

0

x

( ) f x

0

x

( ) f x

2 – Extend the even 2 -periodic graph to the

range-3 3 x . (Repeat the shape).

0

2 3 2 3 x

( ) f x

b) Fourier cosine series.

0

1

cos2

n

n

a f x a nx

00

2 22

0

2( )

2 2

2 2

.

a x dx

x x

0

2

0

2 ( )cos

2 sin cos( ) bypartsor tabular

na x nx dx

nx nx x

n n




23

2 2

2

2

2

2 cos sin 0 cos00

4, odd2 1 ( 1)

0 , even

4; 1,2,3,..

(2 1)

n

n

n n n

nn

nn

mm

Therefore2

1

4 cos(2 1)( ) .

2 (2 1)m

m x f x

m

Example 4.4.2

Find the half range Fourier series for

1 f x x 0 1 x .

Solution:

i) Fourier Cosine.

To get the Fourier cosine series, we need to extend ( ) f x to

be an even function as shown below:Remember!

Even function

Symmetry about y-axis.




24

Hence, the Fourier cosine series of ( ) f x is

0

1

cos , 12

n

n

a n x f x a l

l

where

1

00

12

0

2 1

2 12

a x dx

x x

and

1

02 1 cosna x n x dx

1

2 20

1

2 2 2 2 2 2

21

1 1

2 1 sin cos

1 1 22 cos 1 1

1 4cos 2 1

2 2 1

n

m

x n x n xn n

nn n n

f x m xm




25

ii) Fourier Sine.

To get the Fourier sine series, we need to extend ( ) f x to

be an odd function as shown below:

Hence, the Fourier sine series of ( ) f x is given by

1

sinn

n

n x f x b

l

where

0

2sin , 1

l

n

n xb f x dx l

l l

1

02 1 sin x n x dx

1

2 2 0

2 2

1 22 cos sin

2 2 20 sin cos 0 0

2

xn x n x

n n

nn nn

n

1

2( ) sin

n

f x n xn

.

1 2 3 -3 -2 -1 x

f(x) Remember!

Odd function

Symmetry about the

origin.




26

Example 4.4.3

Given that 1 f x , 0 x .

Find the odd and even extension of ( ) f x .

Solution:

i) Odd extension Fourier sine series.

To get the Fourier sine series, we need to extend ( ) f x to

be an odd function as shown below:

0 0na a

0

21 sinn

n xb dx

f(x)

π -π

1

.

f(x)

x2π π 3π -3π -π -2π .

1




27

00

2 2sin cos

2 2cos 1 1 1

4 , odd

0, even

n

nx dx nxn

nn n

nn

n

4

1

2 21 1 sin

n

f xn

ii) Even extension Fourier cosine series.To get the Fourier cosine series, we need to extend ( ) f x to

be an even function as shown below:

00

21a dx

2

0

2cosna nx dx

0

2 1sin

0

nx

n

0 12

a f x

x

f(x)

π 3π -3π -π




28

4.5 Approximate Sum of the Infinite Series

We can find the approximate sum of the infinite series byusing Fourier series for ( ) f x .

NOTE:

If; 0

( ); 0

A l x f x

B x l

, then (0)

2

A B f

.

Example 4.5.1

If the Fourier series for

0,

1, f x

1 0

0 1

x

x

, ( 2) ( ) f x f x

is given by

1

1 2 1( ) sin(2 1) ,

2 (2 1)n

f x n xn

show that the approximate sum of this infinite series is

1 1 11 ......

3 5 7 4

0 x is not in the interval.




29

Solution:

i) Expand ( ) f x .

1

1 2 1( ) sin(2 1)

2 (2 1)n

f x n xn

1 2 1 1sin sin 3 sin 5 ...

2 3 5 x x x

ii) Choose an appropriate value of x .

Let1

2 x , then we obtain

1 1 2 1 3 1 51 sin sin sin ...

2 2 2 3 2 5 2 f

or

1 2 1 11 ...

2 3 5

1 11 ...

3 5 4




30

Example 5.5.2

i) Show that

22

21

( 1)4 cos ,

3

n

n

x nx xn

ii) By choosing an appropriate value of x , show that

1 2 2

2 21 1

( 1) 1, and

12 6

n

n nn n

Solution:

i) Since2( ) f x x is an even function, its Fourier series is

govern by:0

1

( ) cos2

n

n

a n x f x a

l

32

00

2 2

3o

xa x dx

32

3

22

3

and

2

0

2cosn

n xa x dx

2

2 3

0

2

2 2 2sin cos sin

2 2cos

x xnx nx nx

n n n

nn

2 2

4 4cos 1

nn

n n




31

Therefore,

22

21

4( ) 1 cos

3

n

n

f x x nxn

.

ii) Choose an appropriate value of x

.

Let x .

2

2

21

41 cos

3

n

n

nn

2 22

21

( 1)4

3

n

n n

2 2

21

1 2

3.4 6n n

.

Let 0. x

2

21

40 ( 1)

3

n

n n

2

21

11

12

n

n n

or 2

1

21

11

12

n

n n

.




32

Example 5.5.3 (continuation of Eg. 4.4.1)

If the Fourier cosine series of function

; 0( )

; 0

x x f x

x x

is given by2

1

4 cos(2 1)( )

2 (2 1)m

m x f x

m

, find the sum of

2 2 2

1 1 11 ...

3 5 7

.

Solution:

1 – Expand ( ). f x

2 2 2

4 cos3 cos5 cos7( ) cos ...

2 3 5 7

x x x f x x

2 – Choose an appropriate value of x .

Let 0 x .

2 2 2

4 1 1 1(0) 1 ...

2 3 5 7 f

Recall function ( ) f x . 0 x is not in the interval. Hence,

how do we find (0)? f

( ) ( )(0)

2

x x f




33

Therefore,

2 2 2

4 1 1 11 ...

2 3 5 7

2

2 2 2

1 1 1 21 ...

3 5 7 4 8

Exercise:

1. The graph of ( ) f x is shown as below. Find its Fourier series.

(0,2) ( ,2)

( ,0)

( ) f x

x

Ans. 1

21

3 1 ( 1) ( 1)2 cos sin

2 ( )

n n

n

nx nxn n

2. Write the Fourier series for ( ) sinh f x x , x .

Ans. 1

21

2sinh( 1) sin .

1

n

n

nnx

n




34

3. Show that if ( ) 0 f x when 3 0 x , ( ) 1 f x when

0 3, x and1

(0)2

f , then

1

1 2 1 (2 1)( ) sin .2 2 1 3n

n x f x n

4. Recall Example 4.3.3. Using an appropriate value of , x find the

sum of the following series

2

1 1 1 ( 1)... ...

3 15 35 4 1

n

n

Ans. 1

2 4

.

5. Consider the following graph of ( ). f x

x2 2

( ) f x

0

a) Write the function of ( ). f x

b) Determine the Fourier series of ( ). f x

Ans. 1

sin(2 1)2 .

2 2 1n

n x

n

6. Write the Fourier Sine series for ( ) (0 ) f x x x .

Ans. 1

sin2

n

nx

n




7. Calculate the Fourier sine series of the function

( ) f x x x on (0, ) . Use its Fourier representation to the

find the value of the infinite series3 3 3 3

1 1 1 11 ...

3 5 7 9

Ans. 3

1

8 sin(2 1)( )

(2 1)n

n x f x

n

, the sum of series is

3

32

if 2

x

8. Let h be a given number in the interval (0, ) . Find the Fourier

cosine series of the function

1 if 0

( ) 0 if

x h

f x h x

Ans.

1

2 1 sin( ) cos

2 n

h nh f x nx

nh

END

REFERENCES

1. Normah Maan et. al., (2008), Differential Equations Module, Jabatan

Matematik, UTM.

2.

Nagel et. al., (2004), Fundamentals of Differential Equations, 5th

ed.,

Addison Wesley Longman.

3. Ruel V.Churchill and James W.B., (1978), Fourier Series and Bondary

Value Problems, McGraw Hill.

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Fourier Series Student.pdf