Multiplying a frequency domain function by an imaginary exponential shifts the input of its Inverse Fourier Transform in the time domain. However, something different seems to happen when the exponential has a real part.
I would expect the inclusion of the $1$ to yield an answer $-i \sqrt\frac{\pi}{2}$Sign$[-i + 2g\pi + y]$. However, I have tried expanding this using the definition of Sign for complex numbers, and I do not get logarithms. How do complex exponentials with both real and imaginary parts in the frequency domain shift a function in the time domain?