06/19/2009, 04:53 PM

(06/19/2009, 04:19 PM)Tetratophile Wrote: "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 does not mean that the powerseries at 1 converges at every . It means that is complex differentiable at every which is equivalent that it has a powerseries development (with non-zero convergence radius) at every .

The only singularity of the logarithm is 0. That means you can analytically continue from 1 along every path that not contains 0. However if there are different paths to one point the values of the continuations may differ at by multiples of 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 . Hence there no path can revolve completely around 0 and the continuation of the logarithm to any point of the complex plane except the cut has only one value.