What is the motivation behind studying Fractional Linear Transformations?

Question

I have little knowledge about complex analysis. I see the topic Fractional Linear Transformations arises in several text, but it does not explain elaborately the application of Fractional Linear Transformations, so my question is What is the motivation behind studying Fractional Linear Transformation? What does it do?

Note:

1. In mathematics, a conformal (or angle-preserving) map is a function that locally preserves angles, but not necessarily lengths.

Answer 1

Explaining someone's motivation is more of a psycholical question than a mathematical one but here are some remarks. If we write a linear transformation in the projective line over a field, any field not necessarily the complex numbers, as a matrix eqation and then pull back the projective points (x:y) and (x':y') to the affine line as x/y and x'/y', we have x'/y'=(a(x/y) + b)/(c(x/y) + d).The complex projective line is the same as the Argand plane,which is the real Euclidean plane with one extra point corresponding to the point(1:0) on the projective line. It is also the same, by stereographic projection,as the surface of the sphere in real Euclidean 3-space. Getting back to the projective line, this transformation preserves cross-ratio, from which flow many other results. Hope this helps with the motivation!

Answer 2

For a simply connected domain $\Omega$, due to the Riemann mapping theorem it is biholomorphic (or conformally equivalent) to either;

1. The Riemann sphere - $\hat{\mathbb{C}}$
2. The complex plane - $\mathbb{C}$
3. The unit disk - $\mathbb{D}$

Fractional linear transformations are the elements in the automorphism groups; $$\mathrm{Aut}(\hat{\mathbb{C}}) = \Bigg\{ \frac{az+b}{cz+b} \; : \; ad-bc \neq 0 \Bigg\} $$ $$ \mathrm{Aut}(\mathbb{C}) = \Bigg\{ {az+b} \; : \; a\neq 0 \Bigg\} $$ $$\mathrm{Aut}(\mathbb{D}) = \Bigg\{ \frac{az+b}{\bar{b}z+ \bar{a}} \; : \; |a|^2 - |b|^2 = 1 \Bigg\} $$

Answer 3

Our mathematical ancestors discovered that the map $z\to 1/z$ has an interesting property: It preserves the family of lines and circles. Since any map of the form $z\to az + b$ also has the same property, compositions of such maps preserve this family. Such compositions are precisely the so-called fractional linear transformations.