Repeat the above exercise for the case where both the \(p= \pm \omega\) poles in the integral are avoided in the complex space from above (via a complex integration contour that is slightly above the real line at the level of both poles).

Data From Previous Exercise:-

Consider the propagator obtained by the replacement \(\omega^{2} \rightarrow \omega^{2}+i \epsilon\) ("anti-Feynman") in the Fourier transform. Calculate the integral, and find the corresponding boundary conditions for the (Hilbert space of) functions over which we invert the kinetic operator.