# Automorphisms on Punctured Disc

I have to find the automorphism group of the punctured unit disc $D = \{|z| <1\}\setminus \{0\}$.

I understand that if $f$ is an automorphism on $D$, then it will have either a (i) removable singularity or (ii) a pole of order 1 at $z=0$.

If it has a removable singularity at 0, then $f$ is a rotation. I am stuck at case (ii).

Also, using this result, later I also have to find the automorphism group of $\{|z|<1\}\setminus \{1/2\}$

Right. For the second part try conjugating by an automorphism of $D$ that sends $0$ to $\frac12$. –  Jonas Meyer Feb 14 '13 at 4:41