Finding fixed points of mobius transform

[elimenohpee]

Homework Statement

What is the procedure to find fixed points of a mobius transform?

I don't really have an example, how about: f(z)= (z-i)/(z+i)

Homework Equations

From what I understand, fixed points are points that when attempting to transform get mapped back to themselves. So one would need to solve the equation:

f(z) = (z-i)/(z+i) = z

If you solve for z, you should get 2 fixed points I believe, but I'm not sure. When trying to work out an example the arithmetic gets a little hairy.

The Attempt at a Solution

z^2 + (i-1)z + i = 0

use quadratic formula to find roots?

 

[HallsofIvy]

Yes, or (much the same thing) complete the square.

You will probably have to use DeMoivre's theorem to find the square root of a complex number.