Fourier Transform: Fourier cosine and Fourier sine integrals

Complex form of the Fourier integrals

The Fourier integral formula for f(x) is

because cos [λ(t‒x)] is an even function of λ.

Also since sin [λ (t‒x)] is an odd function of λ.

which is the complex form of the Fourier integral.

FOURIER SINE AND COSINE INTEGRALS

Fourier sine and cosine integrals

Proof: We know that, the Fourier integral theorem is

Case (i) : When f(t) is odd,

 f(t) cos st is an odd function in (‒∞, ∞)

Case (ii) When f(t) is even,

 f(t) sin st is an odd function in (‒∞, ∞)

Note I:

Equation (2) can be re‒written as

Note II:

Equation (3) can be re‒written as

Problems based on Fourier cosine and Fourier sine integrals

Example 1: Find Fourier cosine integral of the function e^‒ax. Hence deduce the value of the integral .

Solution: We know that, the Fourier Cosine integral of f(x) is given by

Example 2: Using Fourier integral formula, show that

Solution: We know that, the Fourier Cosine integral of f(x) is given by

Example 3: Express f(x) =  as a Fourier sine integral and hence evaluate sin (xλ) dλ.

Solution:

We know that, the Fourier sine integral of f(x) is given by

Example 4: Using the Fourier integral representation show that

Solution: We know that, the Fourier sine integral of f(x) is given by

Example 5: Using Fourier integral formula, prove that

Solution:

We know that, the Fourier sine integral of f(x) is given by

Example 6: Applying the Fourier sine integral formula to the function f(t) = sin x when 0 < x <π and f(t) = 0 when x > π.

Show that 

Solution: We know that, the Fourier sine integral of f(x) is given by