I know this is the most typical example of branches and I think I don't get the concept... Could you help me by giving a detailed development leading to all the required branches? It'd help me understanding more complicated examples...
Thank you very much, this concept really is hard to understand for me...
The purpose of a branch cut or cuts is take a multivalued function $f(z)$ into a single analytic branch of $f(z)$. In the example of $f(z) = \log(z)$, lets consider moving around $z = 0$, from a point on the unit circle. Now, the branch point in this case is at $z=0$.
Let $z = e^{i\theta}$. Now if we start at $z = 1$ so $\theta = 0$, then $f(1) = \log(1) = 0$. Let's traverse around $z = 0$ by $2\pi$; that is, $z = e^{2\pi i}$. $$ f(e^{2\pi i}) = \log(e^{2\pi i}) = \log(1) + 2\pi i = 2\pi i $$ Therefore, as we encircle $z = 0$ by $f(1) = 0\mapsto f(e^{2\pi i}) = 2\pi$. Therefore, $f(z)$ is multivalued. In order to prevent this, we must stop the wrapping around $z = 0$ by a full revolution or $2\pi$. In my encounter, the principal value is general defined as $\theta\in(-\pi, \pi]$ but I have seen it defined as $\theta\in(0, 2\pi]$. If we use the first definition of principal value, we would define our branch cut as $(-\infty, 0]$ as you did in your question but we could also define it as $[0, \infty)$ or any ray zero to infinity.