I deal with the Mobius transformations that maps unit cirlce on itself, $S^1\rightarrow S^1$. These transformations can be parametrized in (at least) two ways, $$\mathcal{M}_1(z)=\frac{e^{i\alpha}z+w}{1+\bar{w}e^{i\alpha}z},\quad \alpha\in\mathbb{R},\,w\in\mathbb{C},|w|<1,$$ $$\mathcal{M}_2(z)=\zeta\frac{z-w}{1-\bar{w}z},\quad \zeta\in\mathbb{C},\,|\zeta|=1,\,w\in\mathbb{C},|w|<1.$$ It is quite easy to notice that these parametrizations are closely related to each other.
Nevertheless, I would like to understand: are these parametrization equal or not? How can I check it?
I try to check actions of $\mathcal{M}_1$ and $\mathcal{M}_2$ on specific points: $z=1$, $z=0$ and $z=\infty$, however it does not make sense.
You have $$ \frac{e^{i\alpha}z+w}{1+\overline{w}e^{i\alpha}z} = e^{i\alpha} \frac{z+e^{-i\alpha}w}{1+\overline{e^{-i\alpha}w} \cdot z} = \zeta \frac{z-w'}{1-\overline{w'}z} $$ where $$ \zeta = e^{i\alpha} \, , \, w' = -e^{-i\alpha}w \, . $$ This shows that the parametrizations are equivalent, only the parameter $w$ is chosen differently.