UNIT-II-FOURIER SERIES

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```					                  RAJALAKSHMI ENGINEERING COLLEGE
ENGINEERING MATHEMATICS III MA1201
UNIT – II FOURIER SERIES
PART – A
1)   State Dirichlet’s conditions for a given function to expand in Fourier series.
2)   Does f ( x)  tan x possess a Fourier expansion?
3)   If f (x) is discontinuous at x  a , what does its Fourier series represent at
that point?
1
4)   Determine b n in the Fourier series expansion of f ( x)  (  x) in
2
0  x  2 with period 2 .
0;0  x  
5)   If the Fourier series for the function f ( x)                                     is
sin x;   x  2
1 2  cos 2 x cos 4 x cos 6 x            1
f ( x)                              ......   sin x .      Deduce        that
   1.3           3.5    5.7           2
1      1    1                 2
                  
 ..........       .
1.3 3.5 5.7                      4
6)   Find the constant term in the Fourier series corresponding to f ( x)  cos 2 x
expressed in the interval ( ,  ).
7)   What is the constant term a 0 and the coefficient of cos nx , a n in the Fourier
series expansion of f ( x)  x  x 3 in ( ,  ) ?
8) Find b n in the expansion of x 2 as a Fourier series in ( ,  ).
 2x
1   ,  x  0

9) In the Fourier series expansion of f ( x)                        in ( ,  ) , find
1  2 x ,0  x  
 

the value of b n , the coefficient of sin nx .
10) Find a n in expanding e  x as Fourier series in ( ,  ).
11) Find the Fourier constants b n for xsin x in ( ,  ).
12) If the Fourier series of the function f ( x)  x  x 2 in the interval    x  
2 
4           2       
is     (1) n  2 cos nx  sin nx, then find the value of the infinite series
3 n 1        n           n       
1   1     1
2
 2  2  .........
1    2     3
13) Find a Fourier sine series for the function f ( x)  1 ; 0  x   .
14) To which value , the half range sine series corresponding to f ( x)  x 2
expressed in the interval (0,2) converges at x  2 ?
15) Find a0 in the expansion of | cos x | as a Fourier series in ( ,  ).
PART-B
1; (0,  )
1) Find the Fourier series of period 2 for the function f ( x)                and
2; ( ,2 )
1 1          1
hence find the sum of the series 2  2  2  ........ .   
1      3     5
 x; (0,  )
2) Obtain the Fourier series for f ( x)                     .
2  x; ( ,2 )
sin x;0  x  
3) Expand f ( x)                    as a Fourier series of periodicity 2 and
0;   x  2
1     1     1
hence evaluate                          
 .......... .
1.3 3.5 5.7
4) Determine the Fourier series for the function f ( x)  x 2 of period 2 in
0  x  2 .
5) Obtain the Fourier series for f ( x)  1  x  x 2 in ( ,  ) . Deduce that
1   1   1               2
 2  2  .........     .
12 2    3                6

6) Expand the function f ( x)  x sin x as a Fourier series in the interval
  x   .
7) Determine the Fourier expansion of f ( x)  x in the interval    x   .

8) Find the Fourier series for f ( x)  cos x in the interval ( ,  ) .
9) Expand f ( x)  x 2  x as Fourier series in ( ,  ) .
 1  x,  x  0
10) Determine the Fourier series for the function f ( x)                               .
1  x,0  x  
1 1              
Hence deduce that 1    ......... .
3 5               4
11) Find the half range sine series of f ( x)  x cos x in (0,  ) .
12) Find the half range cosine series of f ( x)  x sin x in (0,  ) .
13) Obtain the half range cosine series for f ( x)  x in (0,  ) .
14) Find the half range sine series for f ( x)  x(  x) in the interval (0,  ) .

15) Find the half range sine series of f ( x)  x 2 in (0,  ) .
Hence find f (x) .

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