I have some troubles understanding branch cuts of the complex logarithm, or to be more precise, the shift thereof. So typically, the branch cut is along the negative real axis, connecting the branch points at $0$ and $\infty$. We know, that this choice is not unique – I could, for instance, also choose the positive real axis.
Now, this is where I'm puzzled: is it also possible to shift it to any different spot, say, to the negative imaginary axis? This would lead to a single-valued function on all of the real axis except at the origin, where we still have a singularity – right?
Best Answer
Surely you can do that.
Define $\log \, z=\log|z|+i\theta$ with $-\pi /2 <\theta < 3\pi /2$, where $z=|z|e^{i\theta}$, and you will get an analytic branch of logarithm in $\mathbb C\smallsetminus {\{-it: t\geq 0\}}$.