Finding fixed points of mobius transform

In summary, to find fixed points of a Mobius transform, one must solve the equation f(z) = z for z. This can be done by using the quadratic formula or completing the square. DeMoivre's theorem may also be necessary in finding the square root of a complex number.
  • #1
elimenohpee
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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?
 
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  • #2
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.
 

1. What is a fixed point in a Mobius transform?

A fixed point in a Mobius transform is a point in the complex plane that remains unchanged after applying the transform. In other words, if z is a fixed point, then f(z) = z, where f(z) is the Mobius transform.

2. How do you find fixed points in a Mobius transform?

To find the fixed points of a Mobius transform, you can set the equation f(z) = z and solve for z. This will give you the values of z that remain unchanged after applying the transform, which are the fixed points.

3. Can a Mobius transform have more than one fixed point?

Yes, a Mobius transform can have multiple fixed points. In fact, the maximum number of fixed points a Mobius transform can have is two, since it is a one-to-one mapping in the complex plane.

4. What is the significance of fixed points in a Mobius transform?

Fixed points in a Mobius transform are important because they represent points in the complex plane that do not change under the transformation. This can help in visualizing and understanding the behavior of the transform.

5. Can a Mobius transform have no fixed points?

Yes, it is possible for a Mobius transform to have no fixed points. This can happen if the transform has a rotation or translation component, which would cause all points in the complex plane to move and therefore not remain fixed.

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