Teaching note: Möbius Transformations
This short note is written for those taking my second-year analysis course.
Möbius transformations are brilliant. They are conformal (=angle preserving) maps from the extended complex plane to itself.
Definition 1 (The extended complex plane). The extended complex plane is , i.e. together with an entirely new element, denoted .
For now, we won’t extend the domains of and to include ; so the extended complex plane has no more algebraic structure than the field of complex numbers.
Definition 2 (Möbius transformations). A Möbius transformation is a function such that
for some such that .
The second and third cases in the definition above are important, but we specify a Möbius transformation simply by writing e.g.
and leave the other two cases implicit.
Nevertheless, such a Möbius transformation does not determine the quadruple , but does determine the tuple in projective space. For a trivial example: the identity map is a Möbius transformation, equal to .
Example 1. Consider the Möbius transformation defined by Note that really is a Möbius transformation since . A few example evaluations: , , and .
Proposition 3. The composition of two Möbius transformations is a Möbius transformation.
Proof: For , consider a Möbius transformation , where . The composition is which is a Möbius transformation since is not zero.
Proposition 4. Möbius transformations are bijections.
Proof: Let , with . We find the candidate for its inverse, and check it really is. Define
the identity! The composition is the identity by the same calculation.
Definition 5 (Möbius group). The set of Möbius transformations, equipped with composition, forms the Möbius group, denoted .
We are able to represent Möbius transformations as matrices, in the following way.
Definition 6. Define the map
Naturally we wonder whether, under this representation of Möbius transformations as matrices, the composition of functions corresponds to the multiplication of matrices.
Proposition 7. is an isomorphism.
Proof: It is clear that is a bijection. For , consider defined by , where .
The composition is
and we have
which shows that is a homomorphism.
Proposition 8. Möbius transformations act -transitively on .
Proof sketch: To see this, let be three distinct elements of the extended complex plane. We wish to choose such that
Equivalently, such that the three equations
hold. It suffices to satisfy which can indeed be accomplished.