Is the meromorphic function $g$ defined by $g(w) := \frac{w-1}{w+1}$ a conformal mapping from the singly slit plane $\mathbb{C}\setminus(-\infty,0]$ onto the doubly slit plane $\mathbb{C}\setminus\big((-\infty,-1]\cup[1,+\infty)\big)$?
What does this imply about the map $h$, where $$h(w):=g(w^2)=\frac{w^2-1}{w^2+1}\,,$$ defined on the right half-plane $\text{Re}(w)>0$?