My lecturer told me that when I use the residue theorem, it does not matter if I sum over the residues of the poles on the inside or the outside of a positively oriented simple closed curve $\gamma$. However, in my opinion the sign should change: \begin{equation} \oint_\gamma f(z)\,\mathrm{d}z = 2\pi i \sum_{a\in D_i}\operatorname{Res}_a f = -2\pi i \sum_{a\in D_o}\operatorname{Res}_a f \end{equation} where $D_i$ and $D_o$ denote the sets of poles inside and outside the region enclosed by $\gamma$. Unfortunately I only found examples online where the residues are $0$, so that does not really help. My reasoning is that on the Riemann sphere, the winding number is $-1$ for the poles outside, which is where the sign comes from. Is that correct?
Thanks in advance!
PS.: As I am new here, I hope I did not break any rules of this forum. ;)