My question is about page 4 of the pdf of the following paper
, one does $\textbf{not}$ need to read pages 1-3 of the paper to understand my question ( the $\textbf{only}$ part that needs to be read is provided in the image below). The precise part I struggle to understand is highlighted in the below image:
$\textbf{Question}$: Accepting that $P^{\ast}$ maps the sector $G$ univalently onto the described domain, I fail to see how to apply the Schwarz reflection principle (called the "Riemann-Schwarz Symmetry principle" in the paper) to find that the Riemann surface of $(P^{\ast})^{-1}$ has the structure described.
$\textbf{Observations}:$
It seems that the description of the Riemann surface of $(P^{\ast})^{-1}$ suggests the following:
There must be $n$ possible branches of $(P^{\ast})^{-1}$ definable on $\mathbb{C} \setminus (\bigcup_{k=2}^{n} T_k^{\ast}) = \mathbb{C} \setminus \{ |w| \geq |\alpha_{n}| : \arg(w) = \pi + \frac{2 \pi (k-2) }{n-1} , 2 \leq k \leq n \}$,
because this domain is simply connected and omits all the critical values of $P^{\ast}$, here $T_k^{\ast} = \{ |w| \geq |\alpha_{n}| : \arg(w) = \pi + \frac{2 \pi (k-2) }{n-1} \}$ . This holds independently of the proposed description of the Riemann surface.
Now using the description of the Riemann surface, we should have the following:
$(\textbf{1})$: $n-1$ of these $n$ branches, which we may index as $P_{k},k=2,...,n$ are such that each $P_{k}$ is holomorphic on $\mathbb{C} \setminus T_{k}^{\ast}$ and is discontinuous on the corresponding cut $T_k^{\ast}$, and another branch $P_1$ is defined on $\mathbb{C} \setminus (\bigcup_{k=2}^{n} T_k^{\ast})$ and is discontinuous on every cut $T_{k}^{\ast}, k=2,...,n$
$(\textbf{2})$: The branching behaviour is as follows: Crossing the cut $T_{k}^{\ast}$ starting with the branch $P_1$ moves us to the branch $P_k$, and vice versa (i.e. starting from $P_k$ and crossing the cut $T_k^{\ast}$ moves us back to $P_1$), this comes directly from the description of the surface.
In other words, proving (1) and (2) using the method proposed by the author (applying the Riemann-Schwarz symmetry principle) or otherwise, one will also be able to obtain the precise description of the Riemann surface as given in the paper.
$\textbf{Edit}$: I will award the bounty to any answer that explains (in detail) why the Riemann surface has the description proposed by the author regardless of whether the explanation involves the Schwarz reflection principle, unless there is also an answer that explains the description of the Riemann surface using the method proposed by the author (using the Schwarz reflection principle) before the bounty closing time.