Fourier transforms and their inverse
I'm having a little bit of trouble that I know should be trivial but for some reason I cannot get it. As a simple example consider . I want to perform a Fourier transform on and then perform an inverse Fourier transform on this transformed function to get back to the original function. Now, the Fourier transform is given by
and the inverse Fourier transform is
What I did was the following: The Fourier transform of is given by
then performing an inverse Fourier transform on this new function yields
Now, the terms become 1 and I'm assuming cancels with , so we are left with
which is clearly not the original function I started with. So, does anyone know what rookie error I made to go so horribly wrong?