I was reading this where I encountered the following contour integral as given in equation (2.4) of the same.
$$S = -i\int_{-\infty}^{+\infty} d\omega \log(\omega^2 + m^2 + E)$$
where $m,E \in \mathbb{R}$. When $m^2 \to -E$, the poles $\omega = \pm i \sqrt{m^2 + E}$ pinch the contour (which in our case is the real line). The claim is that this leads to the appearance of singular terms as given in equation (2.5) and consequently $S$ has the following form.
$$S = \sqrt{m^2 + E}.$$
How do I see this?