# Mobius transformations

You should know the following things about Mobius transformations:

What is a Mobius transformation? (a function of the form f(z)=(az+b)/(cz+d) for complex numbers a,b,c,d with ad-bc =\= 0)

Mobius transformations are invertible mappings of the extended complex plane to itself
The inverse of a Mobius transformation is a Mobius transformation

Composition of Mobius transformations is a Mobius transformation

Mobius transformations preserve angles between curves

Mobius transformations send lines or circles to lines or circles

No circle contains the point at infinity, while every line contains the point at infinity

There is exactly one Mobius transformation that sends three distinct point z_1, z_2, z_3 to three distinct points w_1,w_2,w_3 and it is implicitly given by

$\frac{z_2-z_3}{z_2-z_1}\frac{z-z_1}{z-z_3} = \frac{w_2-w_3}{w_2-w_1}\frac{w-w_1}{w-w_3}$

The final exam will have two questions about Mobius transformations worth 4 points each. The second question is as follows:

Is there a Mobius transformation sending the lines Re z=0,Im z=0 to the circles |z-i|=1, |z+i|=1?

The answer is: no, since the angle between the lines is 90 degrees, while the angle between the circles is 0 and Mobius transformations preserve angles.

Alternative explanation: no, since the lines intersect at two points (z=0, z=infinity), while the circles intersect only at one point (z=0). If there were a transformation sending the lines to the circles, it would have to send both z=0 and z=infinity to the same point z=0. But mobius transformations are always one-to-one.