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.

The logarithm of a complex number depends on the arg function. If you start following a circle around the origin starting at a real number $r$, the arg function starts growing from zero until it nears $2\pi$ when it is finishing a full turn. In consequence, the arg function cannot be continuous on any circle that surrounds the origin, and neither can the logarithm.
