Toronto Math Forum
MAT3342018F => MAT334Lectures & Home Assignments => Topic started by: Ende Jin on December 05, 2018, 08:28:32 PM

I have tried (b) and had some clue about it. Basically I choose the intersection of the original to map to the intersection in the codomain. That means choosing $\{0, \infty\} \mapsto \{0, 1\}$. After that, I can get only a form like $z \mapsto \frac{az}{az+d}$. And then I give it $x \in \mathbb{R}$ and $iy, y \in \mathbb{R}$ to ensure the two condition is satisfied. With that, I can get $ad \in \mathbb{R},\bar{a}d \in \mathbb{R}$. And I arbitrarily chose a solution and it works.
My question is if there is a general way to do this kind of question? Because in the above, there are too many guessing.

Also, is it true that if a mobius transformation maps a circle/a line to a circle/line, then the points in one area in the domain (because there are totally two areas) that is on one side of the circle/line will always maps onto the area in the same side of the circle/line in the codomain?