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 $\{|z|<1\}\setminus \{1/2\}$
Can anybody please help ?

A bounded holomorphic function does not have a pole.
