Good day everyone. I was reading the more advanced lectures on complex analysis and encountered a lot of questions, concerning the determination of complex logarithm. As far I don't even understand the concept of it, but I'll do provide you with several practical questions concerning the topic.
First of all, the determination of complex logarithm on open $\Omega \subset \mathbb{C}$ is continuous $f(w)$ such that:
$\forall w \in \Omega \text{ }\exp(f(w))=w$
So then it starts. Maybe someone could explain what presumptions does the following statement contradicts.
There is no continuous determination for complex logarithm in $\mathbb{C}$ \ ${0}$.
Second part is more practical.
We say that determination of complex logarithm is called principal if it given as a complement of $\mathbb{C}$ and semi axis of negative or zero reals $\Omega_{\pi}=\mathbb{C}$ \ $\{z\in \mathbb{C} : \Re(z) \leq 0\}$. Such as $ f(z) = \log(|z|)+ i\begin{cases} \arcsin{(y/|z|)} & x \geq 0, \\ \pi - \arcsin{(y/|z|)} & x \leq 0,\, y \geq 0, \\ -\pi - \arcsin{(y/|z|)} & x \leq 0,\, y \leq 0. \end{cases}$
We can see that the argument belongs to $(-\pi,\pi]$, but I don't understand neither why a set without a negative reals define such argument neither how does it happens. After this there is an example saying that if we take $\Omega_0=\mathbb{C}$ \ $\{z \in \mathbb{C} : \Re \geq 0\}$ then the argument will be $(0,2\pi]$ but I also didn't get how this happens. What will happen if we take out non positive imaginary part? What kind of argument we will have then?
There is no such topic explained in wiki so maybe the deep answer will help others who encountered the same problem.