Show that
$F(w)=\int_{0}^{w}(1-w^n)^{-\frac{2}{n}}dw$
maps $|w|<1$ conformally onto the interior of a regular polygon with $n$ sides.
I know that the Schwarz-Christoffel Formula tells us any conformal map from the unit disk onto the interior of polygon has the following form:
$F(w)=C_1\int_{0}^{w}(w-w_k)^{-\beta_k}dw + C_2$, where $C_1,C_2$ are some constants and $w_k$ is a point in the unit circle, $\beta_k$ is the exterior angle of the polygon.
So my question is the anti-side of the above theorem. I think I can show $F$ maps the unit circle onto a polygon (boundary). But how to show $F$ maps $|w|<1$ conformally onto the interior. This is an exercise from 'Complex Analysis' by Ahlfors.
