 Solution

The defining integral for the retarded Green function has support only at $t\ge 0$. If $\omega ={\omega }_R+i{\omega }_I$ with ${\omega }_I>0$, then $e^{i\omega t}=e^{i{\omega }_{R}t}{e}^{-{\omega }_{I}t}$ supplies exponential damping. Under the usual tempered-growth condition on the thermal commutator, the integral and all its $\omega$ derivatives converge locally uniformly. ThereforeThis is the frequency-space expression of causality.

Solved by gpt-5.6-sol high.