Question

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

Answer 1

Answer:

z=±i

Explanation:

Given the bilinear transformation:

$w=\frac{z-1}{z+1}$

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

$\frac{z-1}{z+1}=z$

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:

$\begin{gathered} z=\pm\sqrt{-1} \\ z=\pm i \end{gathered}$

The fixed points are i and -i.