0
$\begingroup$

Given an arbitrary circle in the upper half-plane $$\left|z-z_0\right| = r,$$ how do I find transformations which map it to itself?

What if it touches the real line?

I feel like Mobius transformations need to be used, but I can't understand how.

  • 0
    Except $w=z$.?. – 2017-01-10
  • 0
    Sorry, I don't understand the question? – 2017-01-10
  • 0
    Identiy function $w=z$ maps every set to itself. – 2017-01-11
  • 0
    I see. Yes, other than the identity function. – 2017-01-11
  • 0
    Use Mobius transformations to map unit circle to itself, then shift obtained map to $z_0$. – 2017-01-11

1 Answers 1

1

$w(z)=K\dfrac{z-\gamma}{\bar{\gamma}z-1}$ indicates all functions which map unit circle to itself where $|K|=1$ and $|\gamma|<1$. Now, we map $|z|

  • 0
    @Ziggy It's Done. – 2017-01-11
  • 0
    Thank you for your help :) – 2017-01-22