Why isn't the complex logarithm $\log z$ holomorphic on $\mathbb C -\{0\}$? Why isn't the complex logarithm $\log z$ holomorphic on $\mathbb C -\{0\}$? Why can't you just say take the $\arg z$ to be in $[0,2\pi)$ and then you don't have to worry about it being a multivalued function.

Above axis $\mathbb{R}_{>0}\times\{0\}$, the argument you defined will be close to $0$, whereas under it, it will be close to $2\pi$, so the logarithm you defined that way is not even continuous.
