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Use fig 13.2 of [2] as reference. Taking the example Qmechanic uses, the idea is that $I(\omega) = \int_{\zeta_a}^{\zeta_b}\frac{Z(\omega)}{\zeta - \xi(\omega)} d\zeta$ $I(\omega)$ needs to be analytically continued from $\omega_1 \rightarrow \omega_2$. The pole of the integrand travels from $\zeta = \xi(\omega_1) \rightarrow \zeta = \xi(\omega_2)$ in ...
I) Let there be given a meromorphic function $\zeta \mapsto F_{w}(\zeta)$ in the $\zeta$-plane with a single (not necessarily simple) pole at the position $\zeta=\xi(w)$, where $\xi$ is a holomorphic function, and $w\in \mathbb{C}$ is an external parameter. Ref. 1 is considering the contour integral \tag{A}I_{\Gamma,w} ~=~ \int_{\Gamma} \! d\zeta ...