# Conformal map from $\mathbb C - [-1, 1]$ onto the exterior of unit disc $\mathbb C - \overline{\mathbb D}$.

I am finding a conformal map from $$\mathbb C - [-1, 1]$$ onto $$\mathbb C - \overline{\mathbb D}$$, where $$\mathbb D$$ denote the open unit disk.

The hint of this exercise says that I could make use of square root $$\sqrt{\quad}$$ to construct such a function, but I can't figure out what's the relation here since I don't think $$\sqrt{\quad}$$ can be defined on the domain.

Could anybody give me some further hints?

Rmk: By conformal we means that the desired function $$f$$ is holomorphic with nowhere vanishing derivative but not necessarily bijective.

## 1 Answer

Hint: what does $$z\mapsto z+z^{-1}$$ do the the boundary of $$\Bbb D$$? What does it map $$\Bbb C-\overline{ \Bbb D}$$ to?