## Slide 2 summary of Mobius transformation

Plus final result from previous lesson 4.3,

The particular Mobius transformation $f(z) = \frac{z - z_1}{z - z_3} . \frac{z_2 - z_3}{z_2 - z_1}$

maps $z_1, z_2, z_3 \textrm{ to } 0, 1, \infty$

## 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 $f(z) = \frac{az+b}{cz+d}$

when

f:0 → -1

f:i → 0

f:∞ → 1

let a = 1 $f(z) = \frac{z+b}{cz+d}$

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

i + b = 0, b = – i $f(z) = \frac{z-i}{cz+d}$

Consider mapping from infinity. f(∞) = 1 $f(z) = \frac{z-i}{cz+d}$. As z increases then z dominates over constant and function tends to $f(z) = \frac{z}{cz}$ and since f(∞) = 1, then c = 1. $f(z) = \frac{z-i}{z+d}$. Consider final mapping f(0) = -1. Substitute into latest expression. $f(z) = \frac{0-i}{0+d} = -1 = \frac{-i}{+d}$, therefore d = i.

Finally, $f(z) = \frac{z-i}{z+i}$

## Slide 7

z -> az   (rotation and dilation)

z -> z + b (translation)

z -> 1/z   Inversion