# 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 ∞.

$$F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt.$$ 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.