I have been trying to solve the exercise 10 from the third chapter of Conway's Function of One Complex Variable- 'find all the Möbius transformations that maps the unit open disc onto itself.'
I have found that the Möbius transformations that maps the circle $|z|=1$ onto itself is of the form $e^{-i\theta}\frac{az+b}{b\overline{z}+\overline{a}}.$ Can we now use the Orientation principle to arrive at some conditions? There are other answers available but none uses this principle.