A colleague tried convincing me that both $f(z)=\frac{z-i}{z+i}$ and $g(z)=\frac{z}{z+i}$ are automorphisms of the upper half plane of $\mathbb{C}$. I really doubt it, especially because I can't convert them to the $\frac{az+b}{cz+d}$ forms, where $a,b,c,d \in \mathbb{R}$ and $ad-bc=1$.
Automorphisms of Upper Half Plane.
complex-analysis
Best Answer
Your colleague is wrong ! Compute $f(1+i)$ and $g(1+i)$. Both values are not in the upper half plane .