PART-A QUESTIONS AND ANSWERS
1.
State Fourier integral theorem.
See Page 4.1
2.
Show that f(x) = 1, 0 < x < ∞ cannot be represented by a Fourier
integral.
3.
Define Fourier Transform pair. (OR)
Define
Fourier Transform and its inverse transform.
See Page 4.25
4.
What is the Fourier cosine transform of a function ?
See Page 4.78
5.
Find the Fourier cosine transform of f (x) =
See Page 4.87
6.
Find the fourier cosine transform e-ax , a>0
See Page 4.89
7.
Find Fourier Cosine transform of e-x
8.
Find the Fourier cosine transform of e-3x
9.
Find the Fourier Sine-transform of 3 e-2x
Solution:
10. Find the Fourier Sine transform od 1/x
See Page 4.107
11. Define Fourier sine transform
and its inversion formula.
See Page 4.80
12. Fine the Fourier sine transform
of f(x) = e-ax, a>0 and hence deduce that
See Page 4.112
13. If Fourier Transform of f(x) = F(s), then what
is Fourier Transform of f(ax) ?
See Page 4.27
14. If F denotes the Fourier Transform operator,
then show that
15. If Fourier transform of f(x) is F(s), prove that the Fourier transform
of of (x) cos ax is 1/2 [F
(s-a) +F(s+a)
See page 4.29
16. Prove that Fc [f(x)
cos ax] = ½ [Fc (s + a) + Fc(s - a)]
where Fc denotes the Fourier cosine transform f(x).
See Page 4.82
17. If F(s) is the Fourier
transform of f(x), then show that the Fourier transform of eiax f
(x) is F(s+a).
See Page 4.28
18.
Given that e-x2/2 is self
reciprocal under Fourier cosine transform, find (i) Fourier sine transform of
xe x2/2
(ii) Fourier cosine transform of x2 e-x2/2
Solution:
Given
19.
If F (s) is the Fourier transform of the Fourier transform of f (x − a).
See Page 4.28
20.
State the convolution theorem for Fourier transforms.
Solution:
Convolution theorem (or) Faltung theorem :
If F(s) and G(s) are the Fourier
transform of f(x) and g(x) respectively, then the Fourier transform of the
convolution of f(x) and g(x) is the product of their Fourier transform.
21.
State the Fourier transform of the derivatives of a function.
See Page 4.30, Q.No. 6. (i) and
(ii)
22.
Find the Fourier sine transform of f (x)=e ̄-x
Solution:
We know that
23.
Give a function which is self reciprocal under Fourier sine and cosine
transforms.
Solution : 1/√x
24.
State the modulation theorem in Fourier Transform.
See Page 4.29
25.
State the Parseval's identity on Fourier Transform.
See Page 4.34
26.
Define self reciprocal with respect to Fourier transform.
See Page No. 4.38
27.
Does Fourier Sine transform of f (x) = k, 0 ≤ x < ∞, exist?. Justify your answer.
Solution:
Given: f(x) = k, 0 ≤ x < ∞
We know that,
28.
State the condition for the existence of Fourier cosine and sine transform of
derivatives.
Solution
:
Let f (x) be continuous and
absolutely integrable on the x-axis,
Let f'(x) be piecewise continuous
on finite interval, an