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 $\displaystyle F(z)=\int_0^z z^{-\beta_1} (1-z)^{-\beta_2} dz$ with $\displaystyle 0<\beta_1, \beta_2<1$ and $\displaystyle 1<\beta_1 + \beta_2 <2$. Prove that $\displaystyle F$ maps the upper half-plane to a triangle whose vertices are the images of 0,1, and $\displaystyle \infty$ with angles $\displaystyle \alpha_1\pi , \alpha_2\pi, \alpha_3\pi$ where $\displaystyle \alpha_i +\beta_i =1$ and $\displaystyle \beta_1 +\beta_2 +\beta_3=2$ and the length of the side of triangle opposite angle $\displaystyle \alpha_i \pi$ is $\displaystyle \frac{sin(\alpha_i \pi)}{\pi} \Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)$
What happen if $\displaystyle \beta_1 +\beta_2=1$? If $\displaystyle 0<\beta_1 + \beta_2 <1$?