"The problem with $\log z$ in the complex plane is that it is a "multi-valued function", so we must specify what range of values we are considering the function to have. Because we must make this choice, the function fails to be continuous in any (punctured) disk about $z=0$, and thus is not (complex) differentiable in any neighborhood about the point."
For me, this raises the question of what if the function were differentiable on a punctured disk about $z=0,\;$ that is, is $\ln(|z|)$ holomorphic, and, if so, what is its Laurent series?
I believe from Owen Biesel's answer here that $\ln(|z|)$ is holomorphic, but I am new to complex analysis, so I could be very wrong. If it is holomorphic, how would I go about calculating its Laurent series? I understand that it involves taking a positively oriented closed contour integral, but I am having trouble understanding how to properly take contour integrals, as I keep obtaining answers of 0. Thank you in advance!