map the UHP to an equilateral triangle

Question

Explain how the upper half-plane can be mapped one-to-one and conformally onto an equilateral triangle.

Thanks,

Answer 1

The Riemann mapping theorem tells you that such a map exists, but does not provide the actual map. Here you can use a Schwarz-Christoffel integral of the form $$S(z)=\int_0^z\frac{d\zeta}{(\zeta-A_1)^{\beta_1}\dots(\zeta-A_n)^{\beta_n}} $$ Can you figure out which parameters $A_1,\dots,A_n$ and $\beta_1,\dots,\beta_n$ are suitable?

Answer 2

Hint: Riemann mapping theorem.