I am reading Ahlfors's Complex Analysis book, chapter $6.2.2$; The Schwarz-Christoffel Formula, and I have some questions in the exercises.
Firstly, the Schwarz-Christoffel formula is given as follows:
Theorem. The functions $z=F(w)$ which map the unit disk $|w|<1$ conformally onto polygons with (interior) angles $\alpha_k \pi$ ($k=1,…,n; 0<\alpha_k<2$) are of the form $F(w)=C \int _0 ^w (w-w_1)^{-\beta_1}…(w-w_n)^{-\beta_n} dw +C'$, where $\beta_k=1-\alpha_k$, the $w_k$ are points on the unit circle, and $C,C'$ are constants.
$(1)$ Show that the $\beta_k$ may be allowed to be $-1$. What is the geometric interpretation?
$(2)$ If the vertex of the polygon is allowed to be at $\infty$, what modification does the formula undergo? If in this context $\beta_k=1$, what is the polygon like?
In $(1)$, if the $\beta_k=-1$, then what do I have to justify? Also, if $\beta_k=1$ then $\alpha_k=2$ so the polygon would have a slit; is this the desired geometric interpretation?
Next, in $(2)$, I don't have any idea of modification. If $\beta_k=1$ in this case, then I see that the polygon would be a half-plane. Am I right?
I am having a hard time with these. Any help will be appreciated. Thanks!
Best Answer
You're correct about the first question. The formula still works for $\alpha_k = 2 \pi$ and gives two fully or partially overlapping segments as part of the boundary.
If $\alpha_k \leq 0$, then $w_k$ is mapped to $\infty$ since $(w - w_k)^{\alpha_k/\pi - 1}$ is a non-integrable singularity. If the angle between two lines at $\infty$ is defined as minus the angle at their finite intersection point, the formula also works for polygons with a vertex (or several vertices) at $\infty$.
Since we have to distinguish between inner and outer angles, the angle at $\infty$ for parallel rays (a $\Pi$-shaped boundary) is either $0$ or $-2 \pi$. The angle at $\infty$ for antiparallel rays (a $Z$-shaped boundary) is $-\pi$.
All this is exactly the same for the mapping from the upper half-plane to a polygon, but in that case we can eliminate one of the factors by choosing $w_k = \infty$.