# Mobius transformations

• Sep 8th 2009, 04:07 AM
i_never_noticed
Mobius transformations
I need help to find the family of Fractional-Linear (Mobius) transformations that map the unit circle onto itself
a tranformation includes a translation, inversion, magnification, rotation and translation. and takes the form w(z)=(az+b)/(cz+d)
Whenever i work through these i get {a,b,c,d}=1 so this isn't helpful to me. any help would be great
cheers
• Sep 8th 2009, 11:06 AM
ThePerfectHacker
Quote:

Originally Posted by i_never_noticed
I need help to find the family of Fractional-Linear (Mobius) transformations that map the unit circle onto itself
a tranformation includes a translation, inversion, magnification, rotation and translation. and takes the form w(z)=(az+b)/(cz+d)
Whenever i work through these i get {a,b,c,d}=1 so this isn't helpful to me. any help would be great
cheers

Let $f$ be a Mobius trasformation, so $f(z) = \tfrac{az+b}{cz+d}$ with $ad-bc\not = 0$. There are two cases to consider. One case is easy, the other case is more difficult. The first case is when $c=0$, when this happens we just have $f(z) = Az+B$. Thus, $f$ is just a translation, rotation, and magnification. Under these conditions the only way to map a unit circle to a unit circle is when the magnification factor is 1, and the translation is zero. However, you can rotate the circle as much as you like, therefore $f(z) = e^{i\theta} z$.

In the second case $f(z) = \tfrac{az+b}{cz+d},c\not =0$ the Mobius transformation will always have a singularity (at $z=-b/c$). The singularity cannot lie on the circle because otherwise the circle will get mapped into a line. The singularity either lies inside the circle or outside the circle. If it lies outside the circle then the $f$ maps the inside of the disk onto itself and the boundary of the disk onto itself by the maximum modolus theorem. It can now be shown that $f$ must have the form $f(z) = e^{i\theta}\cdot \frac{z-\alpha}{1-\bar \alpha z}$ where $|\alpha|<1$ and conversely. So this is the complete description of Mobius transformations which have their singularity outside the circle. I did not solve the problem yet of what happens if the singularity lies inside of the circle, I am thinking of considering the function $1/f$ but so far I was unable to complete the argument in the second case.