I'm new to contour integral involving branch point and stuck on this particular integration. Here is the problem:
$$\int_{\mathcal{C}}\log z\,\mathrm{d}z,$$
where $\mathcal{C}$ is a closed square contour connecting the points $-0.5+i$, $-1.5+i$, $-1.5-i$, $-0.5-i$. The branch cut of $\log z$ is chosen as the positive real axis. As far as I understand, the integrand within the contour is analytic so the result is zero. But Mathematica gives me $-2\pi i$.
Could anybody point it out for me where the problem is?
You are correct. Mathematica probably uses another branch (and treats the integral as an improper one).