# Möbius transformation, 3 points

## Homework Statement

Find the Möbius transformation that maps
0 -> -1
1 -> infinity
infinity -> 1

## Homework Equations

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

Theorem:

Let f be any Möbius transformation. Then

i, f can be expressed as the composition of a finite sequence of translations, magnifications, rotations and inversions.

ii, f maps the extended complex plane one-to-one onto intself.

iii, f maps the class of circles and lines to itself

iv, f is conformal at every point except its pole

## 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.

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