Are $\{ z: 1<|z|<2\}$ and $\{z: |z|>1\}$ conformally equivalent?

Question

I don't know whether a ring, i.e. the domain $\{z: 1<|z|<2\}$ and the exterior of the closed unit disk $\{z: |z|>1\}$ are conformally equivalent?

I have tried to look on some topological properties, which must be preserved by conformal mapping. They are not simply connected so I can't use Riemann's Theorem. I would like to find a specific conformal mapping between them. I believe that it is possible to map the boundary of $D_2(0)$ to infinity somehow.

Thanks for any help

Answer 1

Your question is equivalent to asking if $A = \{z: 1<|z|<2\}$ and $B = \{w: 0<|w|<1\}$ are conformally equivalent, and they are not:

The conformal mapping $f : B \to A $ would have a removable singularity at $w=0$, and this easily leads to a contradiction (consider both cases $f(0) \in \partial A$ and $f(0) \in A$).

Generally, two annuli are conformally equivalent if and only if the ratio between outer and inner radius is the same, see for example

• Conformity of two concentric annuli
• Conformal map between annulii
• When can we find holomorphic bijections between annuli?

for different proofs.