SRI VENKATESWARA COLLEGE OF ENGINEERING (AUTONOMOUS) d

DEPARTMENT OF APPLIED MATHEMATICS 16. Show that FS ( xf ( x))   FC ( s ) .

ds
SUBJECT: MA16351 TRANSFORMS AND PARTIAL 

DIFFERENTIALEQUATIONS 17. Solve the integral equation 

0
f ( x ) cos xdx  e  . [Ans.:
WORKSHEET
UNIT: IV FOURIER TRANSFORMS 2 1 
]
PART- A  1  x 2 
1. State Fourier integral theorem. 18. Define finite Fourier sine and cosine transforms of f(x).
2. Define Fourier Transform Pair. 19. Find the finite Fourier sine and cosine transforms of f’(x).
1 s PART- B
3. If F [ f ( x )]  F ( s ) , prove that F [ f (ax)]  F  .
a a
4. Find the Fourier transform of e  a x , a>0.
5. If F [ f ( x )]  F ( s ) , then show that
a 2  x 2 , x  a
1 1. Find Fourier transform of f(x) where f ( x)   .
F { f ( x) cos ax}  [ F ( s  a )  F ( s  a )] , where F (s ) is the Fourier
2 0 , x  a  0
transform of f (x ). 2
 
x cos x  sin x x  x cos x  sin x 
 ibx  x  
6. If F [ f ( x )]  F ( s ) , show that F e a f     aF (as  b), a  0 .
Hence evaluate (i)  x 3
cos dx
2
(ii)  
x3
 dx.
0 0  
 a

x if x  a [Ans.:
2 sin sa  as cos sa 3 
7. Find the Fourier transform of f ( x)   . F  f ( x )  2 , (i )  , (ii ) ]
 s 3
16 15
0 if x  a 2. Find the Fourier transform of e  a x if a  0 . Deduce that

8. State the Fourier transforms of the derivatives of a function. 1 
9. State and prove the shifting properties on Fourier transform.
 (x
0
2
 a2 )2
dx 
4a 3
if a  0.

1
10. Find Fourier Sine transform of .
x a  x , for x  a
11. If Fs (s ) is the Fourier sine transforms of f(x), show that 3. Find the Fourier transform of f (x) defined as f ( x)   Hence show
1
Fs [ f ( x ) cos ax]  [ Fs ( s  a )  Fs ( s  a )]. 0 , for x  a  0.
2 2 4
 
12. Find the Fourier sine transform of f ( x )  1 in (0, l ).  sin t    sin t  
that    dt  and    dt  .
13. Find the Fourier sine transform of 3e 2 x . 0
t  2 0 
t  3
2 a 2 sin sa  as cos sa  2 3s
[Ans.: (4) (7) (10) (13) ]
 a s
2 2
 s 2
2  4  s2
14. Write down the Fourier cosine transform pair of formulae.
1
15. Show that Fc [ f (t ) cos at ]  [ Fc ( s  a )  Fc ( s  a)].
2
 1 for x  2 sin x 0  x  a
4. Find the Fourier transform of f(x) given by f ( x)   and hence evaluate 12. Find the Fourier sine transform of f ( x)   .
0 for x  2 0 x  a
as
2
x2
  2 13. Find the Fourier cosine transform of e  a for any a>0 and hence prove that
sin x  sin x  2 sin
 x
dx and 0  x  dx. [Ans.: (3) F  f ( x)  2 2 
x2

is self-reciprocal under Fourier cosine transform.

o
 s 2 e 2

2 sin 2 s   cos 2 xdx  e 8
(4) F  f ( x) 
 s
, (i ) , (ii ) ]
2 2
14. Find the Fourier cosine transform of e-4x .Deduce that 0 x 2  16  8 and


x2

s2 x sin 2 xdx  e 8
5. Show that the transform of
e 2 is
e 2 by finding the Fourier transform of

0
x  16
2

2
.
a2 x2
e , a  0. [Ans.:
15. Find the Fourier sine transform and Fourier cosine transform of f ( x )  x n 1
s2
2 2 1  1
F [e  a x
] e 4a 2
] and show that is self reciprocal under both
a 2 x
6. Derive the Parseval’s identity for Fourier transforms. 16. Find the Fourier sine and cosine transform of e  x and hence find the Fourier
x 1
0 , x  0 sine transform of
1 x 2
and Fourier cosine transform of
1 x2
.

Verify Parseval’s theorem for Fourier transform of the function f ( x) 

7.
 x . x, for 0  x  1
e , x  0 
17. Find the Fourier sine and cosine transforms of f ( x)  2  x, for 1  x  2.
1
[Ans.: LHS  RHS  ]
2 0, for x  2
8. State and prove the convolution theorem for Fourier transform. 
e  ax 1
9. Find the Fourier sine transform of where a > 0. 18. Find the Fourier cosine transforms of f ( x)  . Hence derive Fourier sine
x 1 x2
 ax
10. Find the Fourier sine transform of e  ax (a  0). Hence find Fs {xe }. x
transforms of  ( x)  .
2 a 2 s 1 x2
[Ans.: (9) F  f ( x)  tan 1   (10) Fs (e  ax )  ,
 s  a  s2
2

19. If Fc ( s ) and Gc ( s ) are the Fourier cosine transform of f ( x ) and g ( x )

 ax 2 a 2 2as
Fc (e ) , Fs ( xe  ax )  ]  

 a s
2 2
 (a  s 2 ) 2
2
respectively, then prove that 
0
f ( x ) g ( x ) dx  F
0
c ( s ) Gc ( s )ds.
2
x
11. Find the Fourier sine transform of  . 20. Using Parseval’s identities, prove that
xe 2

 x   x 
2 2 2 2
s s
  1 e 2 , F  xe 2  s e 2 ]
   
[Ans.: c  e 2
F dx  sin at 


 2
s



 2
a) 0 (a 2  x 2 )(b 2  x 2 )  2ab(a  b) . b) 0 t (a 2  t 2 ) dt  2 .
 
dx 
c)  (x
0
2
 a2 )2

4a 3


1 ,0    1
21. Solve the integral equation
0 f ()cosd  0 , 1 . Hence evaluate


sin 2 t 2 1  cos x  

0 t2
dt. [Ans.: f ( x) 
  x 2  2
, ]

22. Find the finite Fourier sine and cosine transforms of f(x)= x 2 in (0,1)
23. Find the finite Fourier sine and cosine transforms of
f ( x)  x(  x) in (0,  )
24. Find the finite Fourier sine and cosine transforms of f(x)= 1 in (0,l)
25. Find the finite Fourier sine transform of cosax and finite Fourier cosine transform
of sin ax in (0,  ) .
26. Find f(x), if its finite sine transform is given by
1  cos n
f S ( n)  in 0  x  
n 2 2

2  ( n) n 2  ( n) n
[Ans.: 15) Fs ( x n 1 )  sin , Fc ( x n 1 )  cos ,
 sn 2  sn 2
 1   1  1
Fs    Fc    .
 x  x s

16) Fs  e  x    
2 s 2 1
, Fc e  x  ,
 s 1
2
 s2 1
 x   s  1   s
Fs  2 
 e , Fc  2 
 e .
1  x  2 1  x  2
2  2 sin s sin 2s  2  2 cos s  cos 2s  1
17) Fs  f ( x )   , Fc  f ( x)   .
  s 2 s 2    s2
2  2 sin s sin 2 s  2  2 cos s  cos 2 s  1
18) Fs  f ( x)   , Fc  f ( x) 
  s 2 s 2    s2 
