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

Question

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.

Answer 1

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?