Conformal mapping to the quarter disk

Question

Is it possible to directly find a conformal mapping between the right half-plane to the quarter unit disk in the first quadrant. I used the cross ratio formula $$\frac{w-w_1}{w-w_3} \frac{w_2-w_3}{w_2-w_1} = \frac{z-z_1}{z-z_3} \frac{z_2-z_3}{z_2-z_1}$$ with the points $i, 0$, and $1$ respectively maps to $1, 0$, and $1$ to preserve orientation. Not sure whether that's the correct approach. Any help is much appreciated.