According to my book, Riemann's Zeta Function, Cauchy's Integral Formula is applicable to the following integral for all negative values of $s$:
$-\frac{\Pi(-s)}{2\pi i}\int_{|z|=\epsilon}(-2\pi in - z)^{s-1}\frac{z}{e^z - 1}\frac{dz}{z} = -\Pi(-s)(-2\pi in)^{s-1}$
where $\Pi(-s) := \Gamma(-s+1)$.
Could someone explain to me how exactly this works, I can't seem to figure it out, thanks.