02 FT PbmSheet
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boiroy02 FT PbmSheet
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boiroyDepartment of Mathematics, NIT Warangal
MA 201 :: Mathematics - III (for II B.Tech. EEE, Chem, MME & BioTech)
Problem Sheet - 02 Fourier Transforms
0, t < 0; 1 1. Find the Fourier integral representation of f (x) = , t = 0; Verify the representation directly 2 eat , t > 0. at the point t = 0. sin t, 0 < t ; 2. Applying the Fourier Sine Integral formula to the function f (t) = , show that 0, t > . sin t, 0 < t < ; sin t sin 2 d = 0, 1 2 0 t > . 3. Applying the Fourier Cosine Integral formula to the function f (t) = 1 when 0 t < 1 and f (t) = 0 1 , 0 t < 1; 2 sin cos t 1 d = when t > 1, show that , t = 1; 4 0 0, t > 1. 4. Using appropriate Fourier Integral formulae, show the following: cos x, 0 < |x| < ; cos cos x 2 2 2 (a) d = 2 1 0 0, |x | .
2
(b)
0
( 2
(c) eax
ex cos x +4 2 u sin xu 2(b2 a2 ) du, ebx = 2 (u + a2 )(u2 + b2 ) 0 4 d =
+ 2) cos x
(a, b > 0)
5. Find the Fourier Transform of the function ea|x| , a > 0.
(a) Deduce that
0
cos pt a2 + t2
dt =
ea|p| .
2 2 . (1+ 2 )2
(b) Deduce also that the Fourier Transform of xe|x| is i
6. Find the Fourier Cosine Transform of the function eax , a > 0. (a) Use it to nd the Fourier Transform of ea|x| cos bx. cos px (b) Show also that dx = eap , (p 0) 2 2 a +x 2 0 1 (c) Hence, nd the Fourier Cosine Transform of 2 . a + x2 7. Find the Fourier Sine Transform of the function eax , a > 0.
(a) Using it, show that
0
x sin px a2 + x2
dx =
eap , (p > 0) x + x2 .
(b) Hence, nd the Fourier Sine Transform of
a2 1
8.
a |x|, |x| < a; . Find the Fourier Transform of f (x) = 0, |x| > a > 0.
2
Hence, deduce that
sin2 t t2
dt =
, and that
0
sin4 t t4
2 x2
dt =
.
x2 2
9. Find the Fourier Transform of ea Transform. 10. Find the FCT of ea
2 x2
. Hence, show that e
is self-reciprocal with respect to Fourier
and hence, deduce the FST of xea
2 x 2
2 x2
. Hence, also show that e
x2 2
is
self-reciprocal under FCT and that xe 11. Find the FST of eax x , a > 0.
is self-reciprocal under FST.
12. Solve the integral equation
0
f (x) cos xdx = e . ea
13. Find the function f (x) whose Sine Transform is
, a > 0.
3( 2 ) . 14. Find the function f (x) whose FT is 2 sin ( 2 )
1, |x| < a; 15. Find the FT of f (x) = . Hence, nd FT of f (x) 1 + cos 0, |x| > a. 16. Using the FCT and FST of eax , nd FST and FCT of xeax . 17. Find the FST and FCT of xa1 , 0 < a < 1. Hence, deduce that transforms. Also, nd the FT of 1 .
|x| 1 x
x a
is self-reciprocal under both the
18. Using FT/FCT/FST and Parsevals Identities, show the following:
(a)
0
1 (x2 + a2 )2 x2
dx =
4a3 4a = 2ab(a + b) 2(a + b)
(b)
0
(c)
0
(x2 + a2 ) dx (x2 + x2
dx = 2 +
a2 )(x2
b2 )
(d)
0
(x2 + a2 )(x2 + b2 )
dx =
Formulas considered: 1 FT of f (x) is F (f (x)) = f ( ) = 2 2 2
t=
f (t)e
it
1 dt and IFT is f (x) = 2 2 2
f ( )e
ix
FCT of f (x) is Fc (f (x)) = fc ( ) =
f (t) cos tdt and IFCT is f (x) =
t=0
fc ( ) cos xd
=0
FST of f (x) is Fs (f (x)) = fs ( ) =
f (t) sin tdt and IFCT is f (x) =
t=0
fs ( ) sin xd
=0