Properties of Fourier Sine and Fourier Cosine Transform

Properties of Fourier sine transform and Fourier cosine

1. Linear property

 (i) Fs [af(x) + bg (x)] = a Fs [f(x)] + b Fs [g (x) ]

 (ii) Fc [af(x) + bg (x)] = a Fc [f(x)] + bFc[g (x)]

Proof: (i) We know that,

= a Fs [f(x)] + b Fs (g(x)]

 (ii) We know that,

2. Modulation property:

(i) FS[f (x) sin ax ] = 1⁄2 [ Fc(s − a) − Fc(s + a) ]

(ii) FS [f (x) cos ax] = 1/2 [Fc(s + a) + Fc(s − a) ]

 (iii) FC [f(x) sin ax] = 1⁄2 [Fs(a + s) + Fs(a − s) ]

 (iv) FC [f (x) cos ax] = 1/2 [Fc(s + a) + Fc(s − a) ]

Proof:

3. FS [f(ax) ] = 1/a Fs [s/a] [Change of scale property]

Proof:

FS [f(ax) ] = √(2/π) 0ʃ∞ f(ax) sin sx dx

4. Fs [f ‘(x)] = −s Fc(s), if f(x) → 0 as x→ ∞.

 [Transform of derivative]

Proof:

5. FC [f ' (x)] = − √(2/π) f(0) + sFS(s)    if f(x) → 0 as x→ ∞.

 [Transform of derivative]

Proof:

6. Fs [xf(x)] = ‒ d/ds [FC(s)]

[Derivatives of transform]

Proof: We know that,

7. Fc [xf(x)] = d/ds FS(s)

Proof: We know that,

Problems based on properties of F.C.T AND F.S.T.

Example 1: (i) Find the Fourier cosine transform of 1 / 1+x².

(ii) Find the Fourier sine transform of 1 / 1+x².

Solution: We know that,

Example 2: Find the Fourier sine and cosine transformations of xe^‒ax.

Solution:

Example 3: Find Fourier cosine transform of  and hence find Fs [x].

Solution:

Example 4: Find the Fourier sine transform of e^‒ax hence find the Fourier cosine transform of x^‒ax.

Solution: We know that,