conformal map, slit plane

Find a conformal map of the slit plane $\mathbb{C} \backslash (-\infty, 0]$ onto the open unit disk satisfying $w(0)=i, w(-1+0i)=+1, w(-1-0i)=-1$. What are the images of circles centered at 0 under the map? Sketch them.
The back of the book says that $w=\frac{-i(\sqrt{z}-1)}{\sqrt{z}+1}$ works. I think for the second part of the question I can consider three points on the circles to find the images. However, I don't see how they found this map. Thanks.
Hint : the map $z \mapsto \frac{z-i}{z+i}$ maps the upper half-plane $\mathbb{H}$ to the unit disc $K$, and the map $z \mapsto z^{1/2}$ maps the slit plane to the upper half-plane, provided you pick the appropriate branch of the function.