# Linear Fractional Transformations

### From History of Mathematics

## Contents |

## Definition and basic facts

Let M be a Möbius transformation. Then

- M can be expressed as the composition of a finite number of rotations, translations, magnifications and inversions
- M maps the extended complex-plane to itself
- M maps the class of circles and lines to circles and lines
- M is conformal at every point besides its pole

## Poles and Fixed Points

**A pole** (regular singularity) of a function is a point *z*_{0} where .

**A fixed point** of a function *f*(*z*) is a point *z*_{0} such that *f*(*z*_{0}) = *z*_{0} That is the point gets mapped to the same spot in the UV-plane.

- 1. Find the poles of . How many poles does it have? How many fixed points does it have? (These answers should depend on a, b, c and d.) Show your work (computations).

- 2. A. If a line or circle passes through the pole of M then it must be mapped to what shape?

- B. If a line or circle does NOT pass thru the pole of M it must get mapped to what kind of shape?

- C. Where does M map the point at infinity?

## Inverses of linear fractional transformations

Since M is a one-to-one mapping on the extended complex plane, it has an inverse.

If you solve w = M then where

Note that the inverse is also an LFT.
In general, if S and T are two LFTs, then S(T(z)) is also an LFT.

- 3. A. Let . Find the inverse of f.
- B. Find the image of the interior of the circle |
*z*− 2 | = 2 under the LFT f. - C. Sketch the image and pre-image of C under w=f(z).

## Some special cases

- 4. Show that every Möbius transformation of the form where |
*p*| > |*q*| can be rewritten in the form where |*a*| < 1.

- 5. Show that Möbius transformation of the form where |
*a*| < 1 map the unit circle and unit disk onto itself. Is this also true for the inverse of M?

- 6. Show that Möbius transformation of the form where |
*a*|^{2}− 2 |*b*|^{2}= 1 map the circle |*z*|^{2}= 2 to itself. Is this also true for the inverse of M? What happens to the circle if we apply two different Möbius transformations of this type?

## Group structure

- 7. We can associate a matrix to a Möbius transformation. In problem 6 we would have the matrix . We would like to show that the collection of matrices forms a group. We would need to show that the collection is closed under multiplication, has an identity element, has an inverse and is associative.
- a.
Show that given two arbitrary elements, the product belongs to the collection. I.e. show that is a matrix that belongs to the collection.**Closed under multiplication:** - b.
Show that belongs to our collection.**Identity element:** - c.
Given , show that the inverse also belongs to the collection.**Inverses:** - d.
Show that**Associativity:**