# Möbius transformation, 3 points

1. Oct 16, 2013

### usn7564

1. The problem statement, all variables and given/known data
Find the Möbius transformation that maps
0 -> -1
1 -> infinity
infinity -> 1

2. Relevant equations
$$w = f(z) = \frac{az + b}{cz+d}$$

3. The attempt at a solution
My first idea was to attempt to solve it as a normal system of eq's but that quickly falls apart due to infinity being there. Been toying with the idea of using the fact that lines will map to lines or circles but don't quite know how to apply it.

And yes I know there's a formula for these exact types of questions but it's in the next sub chapter, book figures it's possible to do without knowing that. Just can't for the life of me figure it out.

2. Oct 16, 2013

### camillio

Start with 0 -> -1. This gives you relation between b and d. Then simply think in limits, when f(z) -> inf, f(z) -> 1?

3. Oct 16, 2013

### usn7564

Got it to work, was a bit too caught up with the fact that infinity was defined as a point (which I need to read up on more) which just messed with me. Just looking at the limits it wasn't actually bad at all.

Thanks.