Let $B=\{z \in \mathbb C:0<\operatorname{Im} z<\pi\}$. Find all holomorphic automorphisms of $B$.
I can see what the set $B$ looks like, but I don't know how to find the automorphisms, and since there is no general method to find them, I am lost on how to solve the proble, in particular because it asks for ALL the automorphisms.
I thought of having some Moebius transformation do the job, or maybe I should be looking at big Picard's theorem?