For my problem I need to find a conformal map from $\Omega = \{ z \in \mathbb{C} \ | Re(z)>0\} \setminus [0,1]$ to $ \mathbb{D} = \{z \in \mathbb{C} \ | \ |z|<1 \}$. I also need to map two to zero. I'm confused with what I should do with this line segment.
My attempt was trying to get it to the upper half plane (or something like that) and then go to the unit disk. For that last step I can use $f(z) = \frac{z-g(2)}{z+g(2)}$ to make sure that two gets mapped to zero (here $g(z)$ is the composition of the maps I used to get to the upper half plane).
To get to the upper half plane I started with a rotation $f_1(z)=\pi z$ and then I thought maybe I could to a dilatation $f_2(z) = z-i$ to make sure $]0,1]$ was included in my plane. But I don't see what I can do now, because now the real axis without zero is contained in my plane. So if I now compose with $g(z)$ a part of the bound of my disk would we included, but that can't be the case.
I'm not very good with conformal maps, so please help me with constructing a thought proces instead of just giving a solution. I want to understand how I can get to the answer by myself. Thanks in advance!