# Find a conformal mapping that maps $U$ to the unit disk.

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