## Coursera: Analysis of a Complex Kind. 4.4 Mobius Transformations II

### Jul 27, 2016

## Slide 2 summary of Mobius transformation

Plus final result from previous lesson 4.3,

The **particular** Mobius transformation

maps

## Slide 3 Properties

Composition two Mobius xform is a Mobius xform

Inverse mobius is a Mobius xform

<< proof >>

## Slide 4, 5, 6 Deducing mobius transformation that maps a,b,c to d,e,f via 0,1,infinity

Direct Mobius transformation

determine

when

f:0 → -1

f:i → 0

f:∞ → 1

let a = 1

Consider mapping to zero. f(i) = 0 therefore numerator = 0

i + b = 0, b = – i

Consider mapping from infinity. f(∞) = 1

. As z increases then z dominates over constant and function tends to and since f(∞) = 1, then c = 1.

. Consider final mapping f(0) = -1. Substitute into latest expression.

, therefore d = i.

Finally,

## Slide 7

z -> az (rotation and dilation)

z -> z + b (translation)

z -> 1/z Inversion