Unit I - Fourier series Fourier series in an interval of length 2 f (x ) 

a0  2



n 1

Fourier series of f(x) in 0,2 

f (x ) 

 a0 n x n x      a n cos  b n sin  2 n 1    

1 a0   an

1  

1 bn  



  a

2

n

cos

n x nx   b n sin    

Fourier series of f(x) in  ,  

 a0  2 n 1

f (x) 

 f ( x ) dx

1 a0  

nx 0 f ( x ) cos  dx

1 an  

nx 0 f ( x ) sin  dx

1 bn  

0

2

2

Even Function  a nx   f (x)  0   a n cos  2 n 1   

n x nx    bn sin  a n cos     





f ( x ) dx



f ( x ) cos

nx dx 

f ( x ) sin

nx dx 





 





Odd Function 



  b

f (x) 

n 1

n

sin

nx    



a0 

2 f ( x ) dx  0 2 n x f ( x ) cos dx  0 

a0  0

an  0



an 

2 nx f (x) sin dx  0  

bn 

bn  0 Convergence of Fourier Series:  At a continuous point x = a, Fourier series converges to f(a)

 At end point c or c+2l in (c, c+2l), Fourier series converges to  At a discontinuous point x = a, Fourier series converges to

f ( c )  f ( c  2 ) 2

f (a  )  f (a  ) 2

Fourier series in the Interval of length 2 f (x) 

a0  2



 a n 1

n

cos nx  b n sin nx 

Fourier Series of f(x) in (-  ,  )

Fourier Series of f(x) in (0,2  )

f (x ) 

 a0   a n cos nx  b n sin nx  2 n 1

1 a0   an

1  

1 bn  

2

 f ( x ) dx 0

2

 f ( x ) cos

nxdx

0

2

 f ( x ) sin

nxdx

0

a0

a0    a n cos nx  2 n 1

2   

an 



 f ( x ) dx 0

2 f ( x ) cos nx dx  0

bn  0

 a0   a n cos nx  b n sin nx  2 n 1

1 a0  

Even Function

f (x) 

f (x ) 

an

1  

1 bn  



 f ( x ) dx

 

 f ( x ) cos nxdx

 

 f ( x ) sin

nxdx



Odd Function

f (x ) 



 b n 1

n

sin nx 

a0  0

an  0 

2 b n   f (x) sin nx dx 0

Half Range Fourier series

Fourier Cosine Series

f (x) 

Fourier Sine Series

a0   nx    an cos  2 n1  

2 a0  

 nx   f (x)   bn sin    n 1



 f ( x ) dx



2 n x bn  f ( x ) sin dx  



0

0

2 nx a n   f ( x ) cos dx 0 

Convergence of Fourier Cosine series:  At a continuous point x = a, Fourier cosine series converges to f(a).  At end point 0 in(0,l), Fourier cosine series converges to f(0+)  At end point l in(0,l), Fourier cosine series converges to f(l-)

Convergence of Fourier Sine series:  At a continuous point x = a, Fourier Sine series converges to f(a).  At both end points Fourier Sine series converges to 0.

Harmonic Analysis: a

0

  2 



y   , N 

an

  nx   y cos     2 N  

   ,  

b

n

   2  



 nx   y sin      N  

Parseval’s Theorem: If f (x) 

a0   nx nx     a n cos  bn sin  2 n 1   

is the Fourier series of f(x) in (c, c+2l),

c 2 2  2 a 2 1  Then y  0   (a n 2  b n 2 ) (or) 1  [ f ( x )] 2 dx  a 0  1  ( a n 2  b n 2 ) 4 2 n 1 2 c 4 2 n 1

Root Mean Square Value:

y 2 is the effective value (or) Root Mean square (RMS) value of the function y = f(x), which is given by c 2

 [ f ( x )]

y 

2

dx

c

2

Some Important Results: 1.

Sin n =0 for all integer values of n

2. Cos n= (–1)n for all integer values of n 3. Cos2n=1 for all integer values of n 4. Sin2n = 0 for all integer values of n 5. If f( –x ) = f( x ) then f(x) is even and If f( –x ) = – f( x ) then f( x ) is odd.  1 ( x ) (   ,0 ) is even if either  1 (  x )   2 ( x ) or  2 (  x )   1 ( x ) 6. f (x )    ( x ) ( 0 ,  )  2

1 ( x ) 7. f ( x )    2 ( x )

(  ,0) ( 0,  )

is odd if either 1 (x)  2 (x) or  2 ( x )  1 ( x )

 x, x  0 8. x    x, x  0 9.

ax  e cos bxdx

10 .

e

ax



e ax a cos bx  b sin bx a2  b2



e ax sin bxdx  2 a sin bx  b cos bx  a  b2

11 .  udv  uv 1  u  v 2  u  v 3  ......... Where u 

du d2u , u   , ........ v 1  dx dx 2

 dv ,

v2 

v

1

dx , v 3 

v

2

dx .......... ..