Consider the following contour on the complex plane
where there is branch cut on the negative real axis. What is the simplest parameterizations to solve the following contour integral $$ \lim_{\varepsilon \rightarrow 0}\lim_{R\rightarrow \infty}\int_{C_{R,\varepsilon}} w^{-z}e^{w}\,\textsf{d}w $$ where $C_{R,\varepsilon} = \ell_1 + \kappa + \ell_2 + K$ and $z\in\mathbb{C}$ of principal value?
EDIT: I have attempted the following but not sure it would give a simple evaluation of the contour integral:
\begin{aligned} \ell_{1} &= \{w\in \mathbb{C}\,:\, w = xe^{-\pi i} + i\varepsilon e^{-i\phi},\, x \in [R,-\varepsilon\cos\phi)\} \\ \kappa &= \{w\in \mathbb{C}\,:\, w = \varepsilon e^{\theta i},\, \theta \in [-\phi,\phi)\} \\ \ell_{2} &= \{w\in \mathbb{C}\,:\, w = xe^{\pi i} + i\varepsilon e^{i\phi},\, x \in [-\varepsilon\cos\phi,R)\} \\ K &= \{w\in \mathbb{C}\,:\, w = R e^{-\theta i},\, \theta \in [-\phi_{K},\phi_{K})\} \end{aligned} where $\phi_{K} = \text{Arg}(Re^{\pi i} + i\varepsilon e^{i\phi})$.
I think my main advice here would be to not get embroiled in formulaic details of the parametrization (and limit). Much as in proofs of continuity, we do not actually need the best delta for a given epsilon, and so on.
Plus, the idea of the thing gives a good guide to what details to ignore. :)
So, first, the idea is that the whole integral is $0$ because there are no poles inside it...
Second, the integral over the large almost-circle and small almost-circle should go to $0$, not because of any cancellation (which would be toooo subtle), but, rather, just based on absolute-value estimates.
Then what is left would be the (limit of) the "incoming" and "outgoing" integrals... along the negative real axis. Because we've had to "slit" the plane to make a well-defined fractional power function, the limits as we approach from above, and from below, differ not only by a sign (due to opposite directions of integration), but also by a $e^{2\pi i z}$.
Perhaps with this qualitative advice you can finish it yourself. :)