I'm warning you that this post relates of a topic I'm not comfortable with, so in order to solve the problem I will write below, I'm very interested in understanding other cases/general cases.
The question I must answer is the following:
Considering, in $\mathbb{C}\backslash[−1,1]$, the regular branch of $f(z) = ln(\frac{1 − z}{1 + z})$ by $f(0 + i0) = \lim\limits_{ε \to 0}f(iε) = 0$, I want to find the values of analytic continuations of the branch just fixed: $f(0 − i0)$, $f(i)$, $f(∞)$. I also want to expand analytic continuation of the branch fixed above in series at the point $z=∞$.
I'm writing down this post because I need help understanding many things in the branch theory. Indeed, from what I understand so far a branch point is a singular point of a function connecting the different branches of $f$, which are its values on different "floors" of the Riemann surface.
However to find such a point I observed several time during seminars of Complex Analysis or here on Mathstack that people usually find these points (when they are $< \infty$ ) as if they were fixed points. For example I saw that the fixed points of the function $f(z) = \frac{z}{1 + \sqrt{z−3}}$ are $\infty$ and $3$. But for this function the opiont 3 is not a singular point to me.. so I'm confused with this method.
Another method, which seems applied by many people as well, is to find the derivative of $f(z_0)$ at a point $z_0 < \infty$ suspected to be a branch point, and see wether this derivative equals $0$ or not: if it is, then $z_0$ is a branch point of $f$, and if not, it's not, from what I understood. But then emerge two more questions in my mind: what is the link between this method and the one I wrote above, and what about the case $z_0 = \infty$? Concerning the former, I've no idea, but concerning the latter I found some answers on mathstack like here: Questions about branch point of holomorphic map or here: Branch points of rational functions . Of course the easiest way to study a function at $\infty$ would be to study its inverse at $0$, yet I have a feeling that on those posts I linked above, they compute the derivative of the inverse of $f$ at the inverse of $z$, i.e $\frac{d}{dw}f(w)$ where $w = \frac{1}{z}$, and I don't understand why either.
Also, I believe that a regular branch is a singlevalued branch, but I'm not sure to understand exactly how the regluar branch of the question I must answer is defined, why is it defined by $f(0 + i0)$ ?
As for analytic continuation, I know it corresponds to the extension of domain of our initial function, which is usually a multivalued function, but I don't understand how to find it and I believe for that I need to find branch points first.. as is done here: Analytic continuation of $\frac{\log(1-z^2)}{z^2}$ for instance.
Sooo if anyone feels good enough to take the time to explain to me those topics and sometimes details I've a problem with, that would be awesome !
PS: Nothing to do with the subject of this post, but does anyone know how to insert a link in a word like "here"?
