I'm asked to evaluate the following complex integral with C being the unit circle:
$$\oint_{C}^{}{\log(z-z_0)dz}\quad |z_0|>1$$
$\log(z-z_0)$ is multivalued and has a branch point at $z=z_0$. However, this lies exterior to the contour C, hence Cauchy's Theorem is applicable. Therefore,
$$\oint_{C}^{}{\log(z-z_0)dz}=0\quad |z_0|>1$$
Is my approach to this problem valid? I'm a bit rusty and wanted to get some advice.