I really have a hard time with conformal mappings onto polygons. This is one of the questions I'm stuck. Hope someone can help.
Let F(z)=\int_0^z z^{-\beta_1} (1-z)^{-\beta_2} dz with 0<\beta_1, \beta_2<1 and 1<\beta_1 + \beta_2 <2. Prove that F maps the upper half-plane to a triangle whose vertices are the images of 0,1, and \infty with angles \alpha_1\pi , \alpha_2\pi, \alpha_3\pi where \alpha_i +\beta_i =1 and \beta_1 +\beta_2 +\beta_3=2 and the length of the side of triangle opposite angle \alpha_i \pi is \frac{sin(\alpha_i \pi)}{\pi} \Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)
What happen if \beta_1 +\beta_2=1? If 0<\beta_1 + \beta_2 <1?