I wonder how to solve this integral: $$P(q, \omega) = \frac{-2i}{(2\pi)^4}\times\int\frac{\mathrm d\mathbf{k}~\mathrm d\omega' e^{i\omega'\eta/h}}{\left[\omega' - E(\mathbf k) + i\delta\operatorname{sgn}\left(E(\mathbf k) - \mu\right)\right]\cdot\left[\omega + \omega' - E(\mathbf k + \mathbf q) + i\delta\operatorname{sgn}(E(\mathbf k + \mathbf q) - \mu)\right]}$$
for the poles below (which are not in the contour): $$\begin{align}\omega' &= E(\mathbf k) - i\delta\operatorname{sgn}(E(\mathbf k) - \mu)\\ \omega' &= E(\mathbf k + \mathbf q) - \omega - i\delta\operatorname{sgn}(E(\mathbf k + \mathbf q) - \mu) \end{align}$$
The contour is:
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