Conformal map from quarter plane minus a ray onto upper half plane

Question

Let $A = \{z: 0 < arg(z) < \pi/2\}/\{w: |w| \ge 1, arg(w) = \pi/4\}$. Can someone give an example of a conformal map from $A$ onto the upper half plane? I'm terrible at these problems, so if you could explain your process for finding the map also I would really appreciate it.

Answer 1

First do $z \mapsto 1/z$, which maps $A$ to the lower right quadrant without the line segment $\{w: |w|<1, \arg w = -\pi/4\}$.

Then do $z \mapsto z^4$, and you will get the whole plane, minus the horizontal ray from $-1$ to $\infty$ which goes through $0$.

Then do $z \mapsto z+1$ and you will get the whole plane, without the horizontal ray from $0$ to $\infty$ which goes through $1$.

Then do $z \mapsto \sqrt z$ and you get the upper half plane.