Linear Fractional Transformations
From History of Mathematics
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 z0 where .
A fixed point of a function f(z) is a point z0 such that f(z0) = z0 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?
- 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. Closed under multiplication: Show that given two arbitrary elements, the product belongs to the collection. I.e. show that is a matrix that belongs to the collection.
- b. Identity element: Show that belongs to our collection.
- c. Inverses: Given , show that the inverse also belongs to the collection.
- d. Associativity: Show that