Fourier Transform

Fourier Transform
1 𝑠
1) If 𝐹 𝑓(𝑥) = 𝐹 𝑠 then prove that 𝐹 𝑓(𝑎𝑥) = 𝑎 𝐹 ;𝑎 > 0.
𝑎

2) If 𝐹 𝑓(𝑥) = 𝐹 𝑠 then prove that 𝐹 𝑒 𝑖𝛽𝑥 𝑓(𝑥) = 𝐹(𝑠 + 𝛽) .

3) If 𝐹 𝑓(𝑥) = 𝐹 𝑠 then prove that

𝐹 𝑓′(𝑥) = −𝑖𝑠 . 𝐹(𝑠),

4) If 𝐹 𝑓(𝑥) = 𝐹 𝑠 then prove that

𝑑
𝐹 𝑠 = 𝑖 𝐹 𝑥𝑓(𝑥)
𝑑𝑠

5) If 𝐹 𝑓(𝑥) = 𝐹 𝑠 then prove that 𝐹 𝑓(𝑥 − 𝑎) = 𝑒 𝑖𝑎𝑠 𝐹(𝑠) .

1, 𝑥 <𝑎
6) Find the Fourier transform of 𝑓 𝑥 = . Hence evaluate
0, 𝑥 >𝑎
∞ ∞
sin(𝑎𝑥) sin 𝑥
𝑑𝑥 and 𝑑𝑥 .
0 𝑥 0 𝑥

1 − |𝑥| , |𝑥| < 1

7) Find Fourier transform of 𝑓(𝑥) = . Hence prove that
0, 𝑥 >1
∞ ∞
1 − cos 𝑡 𝜋 1 − cos 𝑡 𝑡 𝜋
𝑖 𝑑𝑡 = and 𝑖𝑖 cos 𝑑𝑡 = .
0 𝑡2 2 0 𝑡2 2 4

8) Find the Fourier transform of the function

𝑎2 − 𝑥 2 , 𝑥 ≤𝑎
𝑓 𝑥 = and hence prove that
0, 𝑥 >𝑎>0

∞
sin 𝑥 − 𝑥 cos(𝑥) 𝑥 3𝜋
𝑖 cos 𝑑𝑥 = .
0 𝑥3 2 16
∞
sin 𝑥 − 𝑥 cos(𝑥) 𝜋
𝑖𝑖 𝑑𝑥 = .
0 𝑥3 4

9) If 𝑓 𝑥 = 𝑒 −𝑎𝑥 , 𝑎 > 0 then find the Fourier sine transform and hence prove that
∞
𝑠 sin 𝑠𝑥 𝜋
2 2
𝑑𝑠 = 𝑒 −𝑎𝑥 .
0 𝑎 +𝑠 2

10) If 𝑓 𝑥 = 𝑒 −𝑎𝑥 , 𝑎 > 0 then find the Fourier cosine transform and hence prove that
∞
cos 𝑠𝑥 𝜋 −𝑎𝑥
𝑑𝑠 = 𝑒 .
0 𝑎2 + 𝑠 2 2𝑎

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 Fourier Transform

1
11) Find the Fourier sine transform of .
𝑥

1, 0<𝑥<𝜋
12) Find the Fourier sine transform of the function 𝑓 𝑥 = and hence
0, 𝑥>𝜋
evaluate
∞
1 − cos(𝜋𝑠)
sin(𝑠𝑥) 𝑑𝑠 .
0 𝑠

13) Find the Fourier cosine transform of 𝑓 𝑥 = 2𝑒 −3𝑥 + 3𝑒 −2𝑥 .

14) Find the Fourier sine transform of 𝑓 𝑥 = 3𝑒 −𝑥 + 5𝑒 −3𝑥 .

15) Find 𝑓(𝑥) if its Fourier sine transform is 𝑒 −𝑎𝑠 .

16) Find 𝑓(𝑥) if its Fourier sine transform is

𝑒 −𝑎𝑠
.
𝑠
17) Find 𝑓(𝑥) from the following
∞
1, 0<𝑠<1
𝑓 𝑥 . cos(𝑠𝑥) 𝑑𝑥 = .
0
−1 , 1<𝑠<2

18) Find 𝑓(𝑥) if its Fourier cosine transform is

1
.
1 + 𝑠2
1, −1 <𝑥 <1
19) Find the Fourier transform of 𝑓(𝑥) = and hence using Parseval’s
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
identity show that
∞
sin2 𝑥 𝜋
2
𝑑𝑥 = .
0 𝑥 2

1 − |𝑥| , |𝑥| < 1

20) Find Fourier transform of 𝑓(𝑥) = . Hence using Parseval”s identity
0, 𝑥 >1
prove that
∞ 2 ∞ 4
1 − cos 𝑡 𝜋 sin 𝑡 𝜋
𝑖 𝑑𝑡 = 𝑖𝑖 𝑑𝑡 = .
0 𝑡2 6 0 𝑡 3

1, 0<𝑥<1
21) Find the Fourier sine transform of 𝑓(𝑥) = and hence using
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Parseval’s identity show that

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 Fourier Transform

∞ 2
1 − cos 𝑥 𝜋
𝑑𝑥 = .
0 𝑥 2

22) Using Fourier cosine transform of 𝑒 −𝑎𝑥 and 𝑒 −𝑏𝑥 show that
∞
1 𝜋
𝑑𝑠 = .
0 𝑎2 + 𝑠2 𝑏2 + 𝑠2 2𝑎𝑏(𝑎 + 𝑏)

23) Find Fourier sine transform of f ( x)  e  x ; x  0 and using Parseval’s identity prove

x 2 dx 
that   .
0 x 2
1 
2
4
∞ cos (𝑠𝑡)
24) Using Fourier transform evaluate 0 1+𝑠 2
𝑑𝑠 .
∞ s. sin (𝑠𝑡)
25) Using Fourier transform evaluate 𝑑𝑠.
0 1+𝑠 2

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