Let $D = \{|z| < 1\}$ be the unit disk and define $U = D - \{(x,0): \text{$-1 < x \leq 0$}\}$. Find a conformal mapping from $U$ to $D$.
Here $U$ is essentially the unit disk minus the negative real axis. I think this is a relatively simple problem, but I am unable to solve it. I know that with Mobius transformations this can be done, we can probably construct a transformation to solve this. My idea was to map the interval $(-1,0]$ to the non-positive real line. From here I would need to keep track of where $U$ went to and adjust accordingly.
Any help is appreciated.
Edit: After some time looking for similar problems this is what I came up with. First, use the transformation $T(z) = i \frac{z+1}{1-z}$ to map our $U$ to the upper half-plane with a slit going from $(0,i]$. Next, we will extend this slit, so we use the transformation $S(z) = -1/z$. This maps our new domain to the upper half-plane where the slit now is from $[i,\infty)$ (imaginary axis). Then from here, we can follow the solution provided here: Conformal Mappings dealing with Slits.
Keep in mind that the $\sqrt{z}$ function provided in the link is not the typical one, this one is defined for when $z \notin [0,\infty)$.
Is this correct?
First, map your domain $U$ to a half-disk, by $z\to \sqrt{z}.$ Then use the method described here: Find a conformal map from semi-disc onto unit disc