2
votes
1answer
134 views

Analytic continuation of zeta is meromorphic on $\mathbb{C}$ with simple pole at 1

We have the following identity: For some contour $\gamma$ and $\forall s \in \mathbb{C} $ Re $s > 1$: $$-2i\sin(\pi s) \Gamma(s)\zeta(s)= \Large\int_{\gamma} \frac{(-z)^{s-1}}{e^z-1}dz$$ The ...
1
vote
1answer
81 views

A theorem about contour integration in $\mathbb C$.

The following theorem is stated in a book: If $f$ is continuous on the arcs $\gamma_r=\{a+re^{i\theta}\,:\,\theta_1\leq\theta\leq\theta_2\}$ where $a,\theta_1,\theta_2$ don't depend on $r$, and if ...