Let $G = \mathbb{C}\setminus [-1,1]$. I wan't to find an analytic function $f:G\rightarrow \mathbb{D}$ where $\mathbb{D}$ denotes the unit disk such that $f$ is onto, and preferably if possible one-to-one.
Now I've seen that $g(z) = \frac{1}{2}(z+1/z)$ maps the open unit disk to $\mathbb{C}\setminus [-1,1]$ in a one-to-one fashion and therefore the given candidate would be its inverse.
$$g(z) = w\Leftrightarrow z^2-2zw+1 = 0\Leftrightarrow (z-w)^2 = w^2-1.$$
Here is my problem: I want to show that $g$ is invertible and find a concrete formula for $g^{-1}$ however I am not sure how to do this using square roots. Say we want to use the square root defined from the branch of the logarithm which satisfies $\log re^{it} = \log r+it$ for $0<t<2\pi$. Then we want $w^2-1$ to stay away from $(-\infty,0]$. However writing $w = x+iy$ it is clear that we then need to restrict $w$ to lie in $\mathbb{C}\setminus \Big((-\infty,-1]\cup(1,\infty]\Big)$ which defeats the purpose since we want $w\in \mathbb{C}\setminus [-1,1]$. How should I go about choosing a branch in this case?
Clearly $g$ is undefined at $0$ so this only gives a bijective map from $G$ to $\mathbb{D}\setminus\{0\}$. Also by simple connectivity injectivity is not possible however can we find a map which is onto?
Best Answer
Using the principal branch of the logarithm (or equivalently the unique square root with positive real part for any complex $z$ that is not a negative real) we can take $G$ to the open right half plane minus $1$ by $\sqrt{\frac{z+1}{z-1}}$. Then using the standard Mobius transform $\frac{w-1}{w+1}$ we continue to the punctured unit disc in a bijective fashion from $G$. But now take a Blaschke product of order $2$ with precisely one zero at the origin, eg $z\frac{z-.5}{1-.5z}$ will do. Since this is a $2:1$ map of the unit disc onto itself, it is then surjective from the punctured unit disc onto the unit disc, hence putting these $3$ together we get a holomorphic map from $G$ onto the unit disc which is locally conformal ( non zero derivative) except at one point (which comes from the unique critical point of the Blaschke product inside the unit disc)
With more care one can actually get a locally conformal map from $G$ onto the unit disc, but it is tricky (though doable) to construct explicit such, though they follow easily from covering theory and using a Blaschke product of order $3$ with distinct critical points.