Fourier Integral Theorem

STATEMENT OF FOURIER INTEGRAL THEOREM

Fourier integral theorem (without proof)

If f(x) is piece‒wise continuously differentiable and absolutely integrable in (‒∞, ∞), then

Equation (1) can be re‒written as

This is known as Fourier integral theorem or Fourier integral formula.

(OR)

Let us assume the following conditions on a function f(x)

1. f(x) is piece‒wise continuous in any finite interval (a, b)

2. _‒∞ʃ^∞ | f(x) | dx is convergent.

Then the Fourier integral theorem states that

The double integral in the right hand side is known as a Fourier integral expansion of f (x)

 (OR)

If f(x) is a function defined in (‒l, l) satisfying Dirichlet's conditions, then

The double integral in the right hand side is known as Fourier integral to represent f(x)

Note: We assume the following conditions on f(x)

(i) f(x) is defined as single‒valued except at finite points in (‒l, l)

(ii) f(x) is periodic outside (‒l, l) with period 2l.

(iii) f(x) and f ‘(x) are sectionally continuous in (‒l, l)

 (iv) _‒∞ʃ^∞ | f(x) | dx converges

i.e., f(x) is absolutely integrable in (‒∞, ∞)

Problems based on Fourier integral theorem :

Example 1: Show that f(x) = 1, 0 < x < ∞ cannot be represented by a Fourier integral.

Solution:

 i.e., ₀ʃ^∞ | f(x) | dx is not convergent.

Hence, f(x) = 1 cannot be represented by a Fourier integral.

Example 2:

Solution: We know that, the Fourier integral formula for f(x) is

Aliter :

We know that, the Fourier integral formula for f(x) is

Example 3: Express the function  as a Fourier integral. Hence evaluate  and find the value .

Solution: We know that, the Fourier integral formula for f(x) is

Example 4: Find the Fourier integral of the function

Solution: We know that, the Fourier integral formula for f(x) is

Example 5: Find the Fourier integral of the function

Verify the representation directly at the point x = 0.

Solution: We know that, the Fourier integral formula for f(x) is

The value of the given function at x= 0 is 1/2.

Hence, verified.