Question

Find the fixed points of the bilinear transformation w=(z-1)÷(z+1)

Answer 1

z=±i

Explanation:

Given the bilinear transformation:

`w=(z-1)/(z+1)`

To find the invariant or fixed points, set w=z and solve for z:

`(z-1)/(z+1)=z`

Cross multiply:

`z(z+1)=z-1`

Open the bracket and simplify:

`\begin{gathered} z^2+z=z-1 \\ z^2=-1 \end{gathered}`

Take the square root of both sides:

`\begin{gathered} z=\pm√(-1) \\ z=\pm i \end{gathered}`

The fixed points are i and -i.