Hello Everyone!

I'm trying to find the Fourier transform of a sinc function without using the duality property. Here's what I got so far:\

Now let's not condier the complex exponentials out of the sine for now, include them later and make use fo the shifting property of the complex exponential.

. Letting and eliminating the odd integral of we get .

So the integral . When adding the complex exponentials, we get this -1/2 shifted to the right and to the left by 1/2. What does that mean? Am I doing something wrong?

NB: I know the FT of a sinc is a rect