Recently I am reading Stein and Shakarchi's Complex Analysis and I find a great difficulty in understanding the proof of theorem 4.6 in Chapter 8 (p.242-244), which talks about conformal mappings onto polygons.
It first introduces the function $h_k$ and shows that it can be analytically continuous to an infinite strip. I understand that why $h_k'\ne 0$ but I don't know why that $$\frac{F''(z)}{F'(z)}=\frac{-\beta_k}{z-A_k}+E_K(z)$$
I don't know why this expression is true and also don't understand the meaning of $\frac{F''(z)}{F'(z)}$ for $\text{Im}(z)<0$, since only $h_k$ is defined for $\text{Im}z<0$ but not $F$ (if I define $F=h_k^{\alpha_k}+a_k$, will it be the same $F$ as the one I will discuss in the next paragraph?). Here I assume $$h_k(z)=e^{i\theta}\overline{h_k(\overline z)}\hspace{.5in} \text{for}\hspace{.1in}\text{Im}(z)>0.$$
Secondly, in the next page (p.244), the writer is going to proof that $F$ is holomorphic outside a large circle so is at the infinity. I don't really understand what he means, only have an idea: since a polygon is bounded, any $\text{Im}z<0$ can correspond to a $\bar z$ which $F(z)$ can be defined by reflecting $F(\bar z)$ along an edge of the polygon. I think I miss something because according to this, $F$ can be extend to become an entire function, which seems that I am wrong.
Can anyone please help me understand the proof of this theorem?