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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:

  1. 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?
  2. How do I get the integral to map to the described half plane?
  3. Where does the arcsine fit in? What does it mean for the mapping created by an integral to be "given by the arcsine function"?
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So, you're asking why you're supposed to be interested in the mapping $$\int_0^z \frac1{\sqrt{1-w^2}} \mathrm dw$$ I take it? – J. M. May 3 '12 at 5:22
No. I'm wondering how to verify that the mapping of $\mathbb H$ onto the given half strip is given by the arcsine function. – Jeff May 3 '12 at 6:28
Yes, and the integral I gave is supposed to be the result of applying Schwarz-Christoffel to the region you're studying... – J. M. May 3 '12 at 6:30
How did you get that? Note that I edited my question, too. – Jeff May 3 '12 at 8:20
An infinite half-strip is a triangle on the Riemann sphere (with a vertex located at the "point at infinity" $\hat{\infty}$). Perhaps a sequence of triangles that converges to the strip in the limit will help you? – anon May 3 '12 at 8:47

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