Attached is a line from my book about the spectral function for free electrons. How do they conclude that it is a delta function? I can see it from a handwaving argument since δ is infinitesimal but that does not explain the factor of 2pi. Rather I think that the equation really only make sense if set up as integral identity, but I don't see how exactly how. Also, what is the physical interpretation of the spectral function? My book relates it to how a particular energy can be excited but, I don't understand this. What does the imaginary part of the Greens function tell us?