Fourier transform and its inversion

INVERSION FORMULA FOR FOURIER TRANSFORM

Let f(x) be a function satisfying Dirichlet's conditions in every finite interval (‒l, l). Let F(s) denote the Fourier transform of f(x). Then at every point of continuity of f(x), we have

Proof: By Fourier integral theorem,

Fourier transform and its inversion

Formula:

Problems based on Fourier transform and its inversion

Example 1: Find the Fourier transform of the function f(x) defined by f(x)  = 

Hence, prove that  

Solution: The given function can be written as

Example 2: Find the Fourier Transform of

hence deduce that

Solution: We know that,

Note:

Even function:

 If f(‒x) =  f(x) in (‒l,l) then f(x) is an even function.

Odd function:

 If f(‒x) = ‒f(x) in (−l,l) then f(x) is an odd function

In the above problem, a cos sx is an even function,

 a sin sx is an odd function

|x| is an even function,

|x| cos sx is an even function

|x| sin sx is an odd function.

Example 3: Find Fourier transform of e^‒a|x| and hence deduce that

Solution: We know that,

Example 4: Find the Fourier transform of e^‒|x| and hence find the Fourier transform of e^‒|x| cos 2x.

Solution: We know that,