## Coursera: Analysis of a Complex Kind. 4.3 Mobius Transformations I

### Jul 27, 2016

## Slide 2 Definition: of form f(z)

a, b, c , d

Properties

As z f(z) leading to f(z) if c = 0

Define and

Also, by looking at denominator,

We say f is a mapping from the extended complex plane: to the extended plane

## Slide 3 Properties of Mobius Transformation

Differentiate: Use quotient rule

f ‘ (z) = so ad-dc neq 0 guarantees f not constant.

Mobius is one-one in

Show: Rearrange w = to z =

Therefore Mobius are (the only) conformal mapping from

Special case

c = 0, d = 1, f(z) = az + b (a 0), called **affine** transformations. Since then by excluding this point from domain C maps to C. Therefore **affine** transformations are (only) transforms maps C to C.

f(az) is a rotation and dilation

f(z + b) is a translation

## Slide 5 Inversion f(z) = 1/z

Image a circle under inversion?

. The image is equation of circle centre origin, radius 1/r.

Since and f interchanges the inside and outside of the** unit** circle and circle centre origin, radius r goes to circle centre origin radius 1/r.

Other circles?

## Slide 6. Inversion on general circle *Important example*

K={z: |z- 3| = 1}, circle centre 3, radius 1.

Image under inversion?

w \in f(K) <=> 1/w \in K, so put this point 1/w in definition of K

| 1/w – 3 | = 1

multiply both sides by |w|

square both sides ***

(*)

Since , (*) becomes ***

which is a circle centre 3/8, radius 1/8.

So a circle has been inverted to a circle.

## Slide 8 Reciprocal of circle whose circumference passes through origin

K={z: |z – 1| = 1}, unit circle centre (1,0), cuts through origin

<< diagram >>

w <=> 1/w <=> |1/w – 1| = 1

|1 – w| = |w|

(*)

Since

(*)

Now, imaginary parts of complements cancel, and real parts of complements are same so double

Re(w) = 1/2 (there is no constraint or condition on imaginary part , so can take any value)

Plot is vertical line through Re(w) = 1/2

<< to continue >>

## Slide 12

IMPORTANT RESULT

The **particular** Mobius transformation

maps

i.e. f

f

f