1. ## 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

2. 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 $\displaystyle f$ be a Mobius trasformation, so $\displaystyle f(z) = \tfrac{az+b}{cz+d}$ with $\displaystyle ad-bc\not = 0$. There are two cases to consider. One case is easy, the other case is more difficult. The first case is when $\displaystyle c=0$, when this happens we just have $\displaystyle f(z) = Az+B$. Thus, $\displaystyle f$ 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 $\displaystyle f(z) = e^{i\theta} z$.

In the second case $\displaystyle f(z) = \tfrac{az+b}{cz+d},c\not =0$ the Mobius transformation will always have a singularity (at $\displaystyle z=-b/c$). 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 $\displaystyle f$ 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 $\displaystyle f$ must have the form $\displaystyle f(z) = e^{i\theta}\cdot \frac{z-\alpha}{1-\bar \alpha z}$ where $\displaystyle |\alpha|<1$ 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 $\displaystyle 1/f$ but so far I was unable to complete the argument in the second case.