Find biholomorphism between these domains I am pretty stuck on constructing a biholomorphism between:
$G_1 = \{z \in \mathbb{C}: Re(z) >0, Im(z)>0 \}$ and
$G_2 =\{z \in \mathbb{C}:|z|<1,  Re(z) >0, Im(z)>0 \}$
I am able to visualize this as some kind of stretching, so first I thought about something like
$$f(z)=\frac{z}{|z|}$$ Unfortunately, I cant handle this at all and from my lecture notes I strongly suspect that this boils down to somehow using Möbius transformations. The conceptual problem I have is, that I don't see how to choose the Möbius transformation. E.g.
$$g(z) = \frac{z-i}{z+i}$$ for this function I do not see to which angles it maps points. We checked in the lecture that it maps indeed to the unit disk but did not say anything about angles. Any help appreciated, starting from the basics if possible. I suppose I lack intuition for this kind of problem.
Edit: From other questions, I feel like there is a way to "know" how the image will look like just from this 3-point-consideration. I would be glad if someone could explain this to me.

So the map you have given $g(z)$ is the Cayley Transform which maps the upper half plane to the unit disc. Now observe that if you restrict this map, to the first quadrant which is essentially $G_{1}$, then the image is the lower semi-circle. 
Now you take a map from the lower semi circle to the upper semcircle(just rotate it by $e^{i\pi}$.) So call $g:G_{1}\to D_{1}$ as your definition of $g$, where $D_{1}$ is the lower semi-circle. Then take $g_{2}:D_{1}\to D_{2}$ as $g_{1}(z)=e^{i\pi}g(z)$, where $D_{2}$ is the upper semi circle. Then take $g_{3}:D_{2}\to G_{2}$ , the square-root map, which maps $D_{2}$ to $G_{2}$.
Hope it's fine!
