Recovering a function using the inverse fourier transform

[hahashahid]

Homework Statement

The argument of the kernel of the Fourier transform has a different sign for the forward and inverse transform. For a general function, show how the original function isn’t recovered upon inverse transformation if the sign of the argument is the same for both the forward and inverse transform.

Homework Equations

The Fourier and the Inverse Fourier transform integrals

The Attempt at a Solution

Do I need to prove the inverse Fourier theorem or is there a simpler solution?
One attempt that I have seen at recovering the original function using the inverse Fourier transform formula integrates the exponent part to a sin(Mx)/x type function, which is then shown to approach the delta function as M approaches ∞.

 

[vela]

Let
$$F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt.$$ The problem is simply asking you to show that if you calculate
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty F(\omega)e^{-i\omega t}\,d\omega,$$ you do not recover f(t). Just plug the expression for F(ω) into the second integral and calculate away.