Universal uniqueness criterion?

[bo198214]

"the function log(z) is holomorphic on {the whole complex plane except real numbers less than 0}" :

WRONG! the taylor series doesn't converge for real numbers z>2.

holomorphic on \( G \) does not mean that the powerseries at 1 converges at every \( z\in G \). It means that \( f \) is complex differentiable at every \( z_0\in G \) which is equivalent that it has a powerseries development (with non-zero convergence radius) at every \( z_0\in G \).

The only singularity of the logarithm is 0. That means you can analytically continue \( \log \) from 1 along every path that not contains 0. However if there are different paths to one point \( z_1 \) the values of the continuations may differ at \( z_1 \) by multiples of \( 2\pi i \) depending on how often the path revolves around the singularity 0.
To exclude different values (branches) of the logarithm, one chooses the domain of the logarithm with a cut e.g. at \( \Re(z)<0 \). Hence there no path can revolve completely around 0 and the continuation of the logarithm to any point \( z_1 \) of the complex plane except the cut has only one value.