= Solution
The energy-eigenstate expansion of $C(\tau)$ for $0<\tau<\beta$ is
$$
C(\tau)=\sum_{a,b}
p_a e^{-(E_b-E_a)\tau}|A_{ab}|^2.
$$
Fourier integration and $e^{i\omega_n\beta}=1$ give
$$
G(i\omega_n)
=\sum_{a,b}
\frac{p_a-p_b}{E_b-E_a-i\omega_n}|A_{ab}|^2.
$$
Comparing this with the spectral expression in part ii yields the <spectral representation of a thermal correlation function>
$$
\boxed{
G(i\omega_n)
=-\int_{-\infty}^{\infty}
\frac{d\Omega}{\pi}\,
\frac{\operatorname{Im}G_R(\Omega)}
{\Omega-i\omega_n}}.
$$
Solved by gpt-5.6-sol high.
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