I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{1}{\sqrt{3}}x \}$. The regions $A$ and $B$ are the right-angled triangles with angles $\{\frac{\pi}{4},\frac{\pi}{2},\frac{\pi}{4}\}$ and $\{\frac{\pi}{6},\frac{\pi}{2},\frac{\pi}{3}\}$, respectively.

Can anyone help me derive an explicit equation for $f(x,y)$? I am vaguely familiar with the Schwartz-Christoffel mapping and the Schwartz triangle mapping, but I do not have a rigorous enough understanding of complex analysis to apply these to the above case. Any advice would be much appreciated!