UNIVERSAL COLLEGE
OF ENGINEERING AND
TECHNOLOGY
Utsav Shah
(120406111055)
FOURIER SERIES
Guided by
kinjal patel
Fourier Series
f  
be a periodic function with period
2
The function can be represented by
a trigonometric series as:


n 1
n 1
f    a0   an cos n   bn sin n
We want to determine the coefficients,
an
and
bn
.
Let us first remember some useful
integrations.
Determine
1
a0 
2
a0

a0

f d

is the average (dc) value of the
function,
f   .
Determine
am 
1


 

an
f   cos m d
m  1, 2, 
Determine
bm 
1


 

bn
f   sin md
m  1, 2 ,
The coefficients are:
1
a0 
2


f d


am 
f   cos m d

 
bm 
1
1



 

f   sin m d
m  1, 2, 
m  1, 2, 
We can write n in place of m:
1
a0 
2


f d


an 
f   cos n d

 
bn 
1
1



 

f   sin n d
n  1, 2 ,
n  1, 2 ,
The integrations can be performed from
0
to
2
1
a0 
2

2
0
an 

2
bn 
1
2
1
0


0
instead.
f   d
f   cos n d
f   sin n d
n  1, 2 ,
n  1, 2 ,
Example 1. Find the Fourier series of
the following periodic function.
f  
A
0
-A


2
3
4
5
f   A when 0    
  A when     2
f   2   f  
1 2


a0 
f

d


2 0
2
1  






f

d


f

d





0


2
2
1  


A
d



A
d





0


2
0
an 
1


2
0
f   cos n d
2
1 




A
cos
n

d



A
cos
n

d





0




2
1  sin n 
1
sin n 
 A
  A
0



n 0  
n 
bn 
1

2
0
f   sin n d
2
1 




A
sin
n

d



A
sin
n

d



  0

2
1
cos n 
1  cos n 
  A

A

n  0  
n  
A
 cos n  cos 0  cos 2n  cos n 

n
A
 cos n  cos 0  cos 2n  cos n 
bn 
n
A
1  1  1  1

n
4A

when n is odd
n
A
 cos n  cos 0  cos 2n  cos n 
bn 
n
A
 1  1  1  1

n
 0 when n is even
Even and Odd Functions
Even Functions
f(
The value of the
function would
be the same
when we walk
equal distances
along the X-axis

in opposite
directions.
Mathematically speaking -
f     f  
Odd Functions
f(

Mathematically speaking -
f      f  
The value of the
function would
change its sign
but with the
same magnitude
when we walk
equal distances
along the X-axis
in opposite
directions.
The Fourier series of an even function f  
is expressed in terms of a cosine series.

f    a0   an cos n
n 1
The Fourier series of an odd function f  
is expressed in terms of a sine series.

f     bn sin n
n 1
Example 2. Find the Fourier series of
the following periodic function.
f(x)

0

f x  x
3
2
5
7
9
when    x  
f   2   f  
x
1
a0 
2
1

2

1
f  x  dx 
2


x 
x 


3
3
  x  
3
2



2
x dx
an 
1


 

f  x  cos nx dx
1  2


x
cos
nxdx



  
Use integration by parts. Details are shown
in your class note.
4
an  2 cos n
n
4
an   2
n
4
an  2
n
when n is odd
when n is even
This is an even function.
Therefore,
bn  0
The corresponding Fourier series is

cos 2 x cos 3 x cos 4 x


 4 cos x 


 
2
2
2
3
2
3
4


2
THANK YOU