I'm working on $S^2 \cong \mathbb{C} \cup \{\infty\}$. It's well known that mobius transformations are the only automorphisms of $S^2$.
Suppose I have a holomorphic function $f$ and a mobius transformation $m$, and I define $g = m\circ f \circ m^{-1}$. Then I want to show that if $\beta$ is a critical point of $f$, then $m(\beta)$ will be a critical point of $g$.
I just have one hitch in the (elementary) proof, so far I have this:
For later use let $m = (az + b)/(cz+d)$, then $m' = (cz + d)^{-2}$. Letting mobius transforms have determinant 1 so I'm assuming that $ad - bc = 1$.
I use chain rule on $g$: \begin{align} g'(z) &= (m'fm^{-1}(z)) \times (f'm^{-1}(z)) \times (m^{-1})'(z)\\ g'm &= (m'f(z)) \times (f'(z)) \times (m^{-1})'m(z)\\ &= (cf(z) + d)^{-2} \times f'(z) \times (-cm(z) + a)^{-2}\\ &= \frac{f'(z)(cz + d)^2}{(cf(z) + d)^{2}} \end{align} Now assuming that the denominator isn't $0$, this works. However, I can't see what to do when it is $0$. I've tried using L'hopital, but without much luck.
Alternatively, if someone knows a more insightful proof, I would be very glad.