Book's question: Map the upper half-plane $\Im z>0$ onto a rhombus in the $w$-plane with angle $\alpha\pi$ at the vertex $A=0$ and side $d$. The correspondence of the points is given by $A=0\to z=0$, $B=d\to z=1$, $C=d(1+e^{i\alpha\pi})\to z=\infty$ and $D=de^{i\alpha\pi}\to z=-1$.
By using the Schwarz-Christoffel Mapping's formula, I have shown that $$w=\frac{2d}{B(\frac\alpha 2,1-\alpha)}\int_0^zt^{\alpha-1}(1-t^2)^{-\alpha}dt$$ where $B(.,.)$ is beta function.
My question: Is there another way to find such a mapping? I thought about Weierstrass P-function and elliptic curves.
Also, is conformal-geometry tag appropriate here?
Thanks in advance.