= Solution
The defining integral for the <retarded Green function> has support only at $t\geq0$. If $\omega=\omega_R+i\omega_I$ with $\omega_I>0$, then $e^{i\omega t}=e^{i\omega_Rt}e^{-\omega_It}$ supplies exponential damping. Under the usual tempered-growth condition on the thermal commutator, the integral and all its $\omega$ derivatives converge locally uniformly. Therefore
$$
\boxed{G_R(\omega)\ \text{is analytic for }\operatorname{Im}\omega>0}.
$$
This is the frequency-space expression of causality.
Solved by gpt-5.6-sol high.
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