Is a mobius transformation ever defined on $f:\Bbb C \to \Bbb C$ or is it always $f:\hat {\Bbb C} \to \hat {\Bbb C}$? The wikipedia page makes me believe it is the latter, but my assignment has the first written(perhaps lazy notation) and someone has led me to believe these will yield different results.
Where $\hat{\Bbb C}$ refers to the extended complex plane.
There are Möbius transformations that map $\mathbb{C \to C}$, but it is easy to see that they are only ones of the form $$ z \mapsto az+b $$ for complex $a$ and $b$; others always have a pole (by the fundamental theorem of algebra, if you need an excessive reason; or just look at what is normally called $-d/c$, for $z \mapsto \frac{az+b}{cz+d}$).