This document discusses Fourier series and transforms. It begins by introducing periodic functions and their fundamental periods. It then defines Fourier series and derives the formulas for the Fourier coefficients. Several examples of calculating Fourier series are provided. It also covers Fourier series for functions with any period, complex Fourier series, Parseval's identity and its applications, and Dirichlet's theorem. The key topics of Fourier series, Fourier transforms, and their applications in engineering mathematics are covered over multiple sections.

1.
PSUT
Engineering Mathematics
II
Fourier Series and Transforms
Dr. Mohammad Sababheh
4/14/2009

2.
2
2 11.1 Fourier Series
Fourier Series and Transforms
Contents
11.1 Fourier Series ........................................................................................................................................ 3
Periodic Functions ..................................................................................................................................... 3
Fundamental Period .................................................................................................................................. 4
Period of Multiple Functions..................................................................................................................... 5
Fourier Series ............................................................................................................................................ 6
11.2 Functions of Any Period p = 2L .......................................................................................................... 11
11.6a Parseval's Identity ............................................................................................................................. 13
Applications ......................................................................................................................................... 14
11.7 Dirichlet's Theorem ............................................................................................................................. 17
11.4 Complex Fourier Series ....................................................................................................................... 18
11.6b Parseval's Identity ............................................................................................................................. 22
11.9 Fourier Transform ............................................................................................................................... 24
Fourier Transform ................................................................................................................................... 24
Fourier Sine and Cosine Transforms ....................................................................................................... 27
Inverse Fourier Transform ...................................................................................................................... 29
Applications ......................................................................................................................................... 30

3.
3
3 11.1 Fourier Series
11.1 Fourier Series
Periodic Functions
A function is said to be periodic of period if for all x
Example 1
cos
2 cos 2
cos 2 sin sin 2
cos
Hence cos is periodic of period 2
Example 2
sin 4
2
sin 4
2
sin 4 2
sin 4 cos 2 cos 4 sin 2
sin 4
=
Hence sin is periodic of period , observe that 2 is also a period of sin 4

4.
4
4 11.1 Fourier Series
Useful Identities
• sin sin cos cos sin
• cos cos cos sin sin
Notes
• Any function can be considered periodic with period zero, this period is trivial and is not
considered as a period.
• If is a period of , then is a period for any integer .
Proof:
want:
we know that
2
3 2
• If is a period then is not necessarily a period.
Fundamental Period
The most interesting period for a periodic function is the smallest positive period , this period is called
the Fundamental Period.
The fundamental period of
• sin is 2
• sin 3 is

5.
5
5 11.1 Fourier Series
Period of Multiple Functions
If and are periodic of period then so is
Proof
Denote by
want
is periodic of period
If is periodic of period then the graph of repeats itself every units
2
Therefore if we know the curve of a periodic function on , , then we can draw the entire graph.
Exercise
If is periodic of period then
,
0
0,2
0,4
0,6
0,8
1
1,2
‐2 ‐1 0 1 2 3 4 5

6.
6
6 11.1 Fourier Series
Fourier Series
Our purpose is to approximate periodic functions by sine and cosine.
we define Fourier series of the periodic function f(x) by:
cos sin
Fourier coefficients , can be obtained by Euler formulas.
Derivation:
Suppose cos sin 5
*
cos sin 5 2
1
2
*
cos cos cos sin 5 cos
1
cos
*
....
0

7.
7
7 11.1 Fourier Series
In general
Example 3
Find the Fourier series:
1 0
1 0
Solution:
1
2
0
‐1,5
‐1
‐0,5
0
0,5
1
1,5
‐4 ‐3 ‐2 ‐1 0 1 2 3 4
• When the phrase "Fourier series" is
mentioned then we implicitly
understand that is periodic.
• If the period is not given , then we
implicitly understand that its 2
1
2
1
cos
1
sin
Fourier coefficients of f(x), given by the Euler
formulas

8.
8
8 11.1 Fourier Series
cos 1
1
cos 0
1
sin
1
sin sin
1
cos
cos
1
1 cos cos 1
1
2 2 cos
2
1 1
0
4
Now Fourier series
cos sin
4
2 1
sin 2 1

9.
9
9 11.1 Fourier Series
Example 4
Evaluate:
2 3 cos 4 sin
Solution:
Denote function by
* We need to find
* Remember that
1
2
2
* We need to find to be able to find
* However isn't in Fourier form because of " " , so we need to simplify using identity
1
2
1 cos
so 2 3 cos 2 2 cos 2 0
* And now substitute to find ...

10.
10
10 11.1 Fourier Series
Notes
By a trigonometric polynomial we mean a finite part of the Fourier series. For instance:
• 1 sin 3 cos 5
• 2 sin sin 2 sin 3
• 2 sin sin 2 (Trigonometric but not Fourier form)
Notes
• .
• sin sin
0
• sin cos 0
• cos cos
0

11.
11
11 11.2 Functions of Any Period p = 2L
11.2 Functions of Any Period p = 2L
Example 1
Find the Fourier series of
, 1 1
Solution:
In this example, p = 2 (period = 2 )
In this case when p = 2 L
Thus in our example L = 1
1
2
1
3
cos sin
1
2
1
cos
1
sin
, 2
In general

12.
12
12 11.2 Functions of Any Period p = 2L
1
1
sin 0
1
1
cos
1
1
cos
2 cos
2
1 1
4 1
1
3
4 1
cos
cos
2
sin
2
cos
Integration by parts
0

13.
13
13 11.6a Parseval's Identity
11.6a Parseval's Identity
Consider Fourier series and expand it
cos sin cos sin cos 2 sin 2 …
Square it
cos sin 2 cos sin 2 cos sin
2 cos cos sin 2 cos sin
Integrate
cos sin
2 0
2| | | | | |
1
2| | | | | |
1
Parseval's Identity
Standard form
General form

14.
14
14 11.6a Parseval's Identity
Applications
Example 1
From Chapter 11.1 , Example 1
∑ sin 2 1
1 0
1 0
* L.H.S of Parseval's
2 0 0
4
2 1
16 1
2 1
*R.H.S of Parseval's
1
1 2
*Therefore
1
2 1
8
Example 2
From Chapter 11.2 Example 1
1
3
4 1
cos ,
Apply Parseval's
2
1
3
16
2
5
16
2
5
2
9
8
45
1
π
90

15.
15
15 11.6a Parseval's Identity
Example 3
Find
1
Now series is given but not unlike Example 2
Solution:
We need such that
1
1
We attempt with since when integrating by parts , we get in the denominator
Taking
= 0
0
1
sin
1
cos |
1 1
2 1
cos
1
cos
Integration by parts
0

16.
16
16 11.6a Parseval's Identity
Now apply Parseval's
4 1
4
1 2
3
1
6
Exercise
Find
1
1
Example 4
Evaluate
2 sin 5 cos 3 cos 10
Solution:
Let 2 sin 3 cos 3 cos 10
Want
According to Parseval's
2
2 1 1 3
We can't find any
sum using this
method , like ∑

17.
17
17 11.7 Dirichlet's Theorem
11.7 Dirichlet's Theorem
If is a nice function , then
lim lim
2
Suppose that is periodic of period 2 and that is piecewise continuous , that and both exist.
Example 1
Suppose
2 1
sin ;
Plug 0 , 0 = 0
Plug
2
2 1
sin
2
2 1
sin
2 2
,
2 1 1
2 1
sin
2 1
2 2
2 1
2 1 2
1
2 1 4
Plug
0 ,
lim lim
2
2
0

18.
18
18 11.4 Complex Fourier Series
11.4 Complex Fourier Series
cos sin
is called Real Fourier series
Note
cos sin
cos sin
2 sin
2 cos
cos
2
sin
2
1
2
The Complex Fourier Series of is
defined to be

19.
19
19 11.4 Complex Fourier Series
Remark
1
2
1
2
1
cos
1
sin
1
2
, 0
1
2
, 0
, 0
Example 1
Write the complex Fourier transform of
2 sin cos 10
Solution:
2
2 2
1 1 1
2
1
2
1
,
1
2
,
1
,
1
2
Example 2
Find the real Fourier series of
5
Solution:
5 sin cos sin cos 2 2

20.
20
20 11.4 Complex Fourier Series
Example 3
Find the complex Fourier series of
,
Solution:
1
2
1
2
1
2
1
2
1 1 1 1
1
, 0
For 0
1
2
1
2
0
Therefore complex Fourier series is
0
1
,

21.
21
21 11.4 Complex Fourier Series
Note
By a complex trigonometric polynomial , we mean a finite part of
For example
• Trig. 1 5
• Not Trig. 1
• Trig. 11 sin 11
!
Note that a complex Fourier series of a complex trigonometric polynomial is the same function.
Exercise
Show that
0 ,
2 ,

22.
22
22 11.6b Parseval's Identity
11.6b Parseval's Identity
Note
|2 3 | 4 9 13
|3 | |2 3 | 0 3 9
| | 0 1 1
Example 1
1
lets apply Parseval's
| |
1
,
1
,
2
1
2
1 1
2 3
1
6
| |
1
2
| |
| |
Parseval's Identity for complex Fourier series

23.
23
23 11.6b Parseval's Identity
Example 2
Evaluate
1 3 1 cos 4
Solution:
Let
1 3 1 cos 4
Want
| | 2 | |
1 , 1 , 3
1
2
,
1
2
, 1
| | 2 1 1 9
1
4
1
4
2

24.
24
24 11.9 Fourier Transform
11.9 Fourier Transform
Fourier Transform
Example 1
Find the Fourier transform of
1 2 2
0
Then apply Parseval's identity and see what it gives
Solution:
1
√2
1
√2
1
√2
0
0,5
1
1,5
‐4 ‐2 0 2 4
1
√2
| |
Let be defined on ∞, ∞
We define its Fourier transform by
Parseval's Identity

25.
25
25 11.9 Fourier Transform
Note here if we are
asked about 0 , we
take the limit
1
√2
cos 2 sin 2 cos 2 sin 2
1
√2
2 sin 2
Let's apply Parseval's
2 sin 2
4
2 sin 2
2
1) Let's play with
2 sin 2
2
2) Let 2
sin
sin
2
3) Let's find
sin
sin
2

26.
26
26 11.9 Fourier Transform
sin
2
sin cos
2
sin 2
2
Let 2
sin
2
4)
sin
Note
is continuous regardless of
lim ∞ 0
lim lim lim 0
sin
1
t
Integration by parts
2 sin cos

27.
27
27 11.9 Fourier Transform
Fourier Sine and Cosine Transforms
Where did these equations come from?
Recall Fourier transform
1
√2
If is even
1
√2
cos sin
1
√2
cos
√2
sin
2
√2
cos
2
cos
2
cos
2
sin
If is defined on 0, ∞ , we define its
Fourier Cosine Transform by
And Fourier Sine Transform

28.
28
28 11.9 Fourier Transform
Note
Practically
when is even.
when is odd.
Note that when is defined on 0, ∞ , we can consider it even or odd.
Example 2
Find and for
, 0 1
0 ,
Solution:
2
cos
2
cos
2 sin cos
2 sin cos 1
Using limits
0
2
1
1
2
1
2
2

29.
29
29 11.9 Fourier Transform
Inverse Fourier Transform
Useful Rules
2
0
1
√2
2
cos
2
sin
Fourier Inverse Transform
Fourier Inverse Cosine Transform
Fourier Inverse Sine Transform

30.
30
30 11.9 Fourier Transform
Applications
Example 3
Find ;
Solution:
Using the rules
2
0
2
2
2
...

31.
31
31 11.9 Fourier Transform
Example 4
You are given that
2
1
2 2
1
sin
2 sin
1
1
sin
1
2
0 1 0 ? ? ?
The formula of the Fourier Inverse Sine Transform sin is true when is
continuous at . Moreover , recall that is computed for odd function .
If we extend to be odd , we get
Not continuous at 0 when taking , so we use Dirichlet's Theorem.