Verify that the Schwarz-Christoffel mapping of $\mathbb H$ onto the infinite half strip described by $|\Re(z)|<\frac \pi 2$ and $\Im(z)>0$ is given by the arcsine function.
What does that mean and how do I do it?
Schwarz-Christoffel formula for the half-plane $\mathbb H$ to the polygon with exterior angles described by coefficients $\beta_k$ is $f(z)=A_1\int_0^z \frac 1{(w-x_1)^{\beta_1}(w-x_2)^{\beta_2}\cdots(w-x_n)^{\beta_n}}\ dw+A_2,\quad (z \in \mathbb H).$
I realize this isn't the best question, but I'm not even sure what to ask.
Edit: An attempt to add more specific questions:
- The $x_n$ in the integral are supposed to be vertex points of the pre-image. If the pre-image is the upper half plane, how do I find vertex points?
- How do I get the integral to map to the described half plane?
- Where does the arcsine fit in? What does it mean for the mapping created by an integral to be "given by the arcsine function"?