This is the paper I am referring to.
(After fighting through page one with success) I don't get the start of page 2 at all. I don't understand the motivation and connection to the previous expression and I also don't fully understand what is going on. He seems to suggest that the curve integral
$$ \int_{\mathcal{C}} \frac{(-x)^{s-1}}{e^{x-1}} \mathrm{d}x =(e^{-\pi s i } - e^{\pi si}) \int_0^\infty \frac{x^{s-1}}{e^{x-1}} \mathrm{d}x $$
For $\mathcal{C}=\lim_{R\to\infty}\mathcal{C}_R$ with $\mathcal{C}_R = \{z|z = R (\cos{\varphi + i \sin{\varphi}), 0\le\varphi<2\pi, R\in\mathbb{R^+}} \}$
- Is this expression the correct interpretation of the text passage?
- How to perform this calculation?
- What is the motivation of this step?