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INDEX
 Fourier Series
 General Fourier
 Discontinuous Functions
 Change Of Interval Method
 Even And Odd Functions
 Half Range Fourier Cosine & Sine Series

FOURIER SERIES
 A Fourier series is an expansion of a periodic
function in terms of an infinite sum
of sines and cosines.

General Formula For Fourier
Series
Where,

Formulas To Solve Examples
 2SC = S + S
 2CS = S – S
 2CC = C + C
 2SS = cos(α-β) –cos(α+β)
 Even*Odd = Odd
 Even*Even = Even
 Odd*Odd = Even
 Odd*Even = Odd

1 2 3 4
' '' ''' ________
uv uv u v u v u v
    

Where,
u, u’, u”, u’’’,_ _ _ _ are denoted by derivatives.
And
V1,v2,v3,v4,_ _ _ _ _ are denoted by integral.

Discontinuous Type Functions
 In the interval
 The function is discontinuous at x =x0
2
C X C 
  
1 0
2 0
( ),
( ), 2
f x C x x
f x x x C 
  
   
f(x)
0 0
0
( 0) ( 0)
( )
2
f x f x
f x
  


So Fourier series formula is
0
0
2
0 1 2
1
( ) ( )
x C
C x
a f x dx f x dx



 
 
 
 
 
 
 
0
0
2
1 2
1
( )*sin( ) ( )*sin( )
x C
n
C x
b f x nx dx f x nx dx



 
 
 
 
 
 
 
0
0
2
1 2
1
( )*cos( ) ( )*cos( )
x C
n
C x
a f x nx dx f x nx dx



 
 
 
 
 
 
 

Change Of Interval Method
 In this method , function has period P=2L ,
where L is any integer number.
 In interval 0<x<2L Then l= L/2
 When interval starts from 0 then l= L/2
 In the interval –L < X < L Then l= L
 For discontinuous function , Take l = C where
C is constant.

 General Fourier series formula in interval
2
C x C L
  
2
1
( )*sin( )
C L
n
C
n x
b f x dx
l l


 
  
 

2
1
( )*cos( )
C L
n
C
n x
a f x dx
l l


 
  
 

2
0
1
( )
C L
C
a f x dx
l

 
0
1
( ) cos( ) sin( )
2
n n
n
a n x n x
f x a b
l l
 


 
  
 
 

Where,

Even Function
 The graph of even function is symmetrical
about Y – axis.
 Examples :
( ) ( )
f x f x
 
2 2
, ,cos , cos , sin
x x x x x x x

Fourier series for even function
1. In the interval x
 
  
 
0
1
0
0
0
( ) cos( )
2
2
( )
2
( )*cos( )
n
n
n
a
f x a nx
a f x dx
a f x nx dx






 

 
  
 




Fourier series for even function
(conti.)
2. In the interval l x l
  
0
1
0
0
0
( ) cos( )
2
2
( )
2
( )*cos( )
n
n
l
l
n
a n x
f x a
l
a f x dx
l
n x
a f x dx
l l




 
   
 

 
  
 




Odd Function
 The graph of odd function is passing through
origin.
 Examples:-
( ) ( )
f x f x
  
3 3
, , cos ,sin , cos
x x x x x x x

Fourier series for odd function
1. In the interval x
 
  
1
0
( ) sin( )
2
( )sin( )
n
n
n
f x b nx
b f x nx dx









Fourier series for odd function
(conti.)
 In the interval l x l
  
1
0
( ) sin( )
2
( )sin( )
n
n
n
n
f x b x
l
n
b f x x dx
l











Half Range Fourier Cosine Series
 In this method , we have 0 < x < π or 0 < x < l
type interval.
 In this method , we find only a0 and an .
 bn = 0

Half Range Fourier Cosine Series
1.In the interval 0 < x < π
 
0
1
0
0
0
( ) cos( )
2
2
( )
2
( )*cos( )
n
n
n
a
f x a nx
a f x dx
a f x nx dx






 

 
  
 




Half Range Fourier Cosine
Series(conti.)
2. In the interval 0 < x < l
Take l = L
0
1
0
0
0
( ) cos( )
2
2
( )
2
( )*cos( )
n
n
l
l
n
a n x
f x a
l
a f x dx
l
n x
a f x dx
l l




 
   
 

 
  
 




Half Range Fourier Sine Series
 In this method , we find only bn
 an =0
 a0 =0

1. In interval 0 < x < π
1
0
( ) sin( )
2
( )sin( )
n
n
n
f x b nx
b f x nx dx









(conti.)
2. In the interval 0 < x < l
1
0
( ) sin( )
2
( )sin( )
n
n
n
n
f x b x
l
n
b f x x dx
l











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