Automorphisms of circles in UHP

Question

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.

Answer 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 $|z|<r_1$ to $|w|<r_2$. By replacing $\dfrac{z}{r_1}$ instead of $z$ and $\dfrac{w}{r_2}$ instead of $w$ conclude that $$w(z)=Kr_2\dfrac{z-r_1\gamma}{\bar{\gamma}z-r_1}$$ With The transformation $z\to z_0$ we translate the centers of circles to $z_0$ in pre-image and image also, so all functions $$w(z)-z_0=Kr_2\dfrac{(z-z_0)-r_1\gamma}{\bar{\gamma}(z-z_0)-r_1}$$ or $$w(z)=Kr_2\dfrac{(z-z_0)-r_1\gamma}{\bar{\gamma}(z-z_0)-r_1}+z_0\hspace{0.5cm};\hspace{0.5cm}|K|=1~,~|\gamma|<1$$ map $|z-z_0|=r_1$ to $|w-z_0|=r_2$. For $r_1=r_2=r$ these maps have the shape $$w(z)=Kr\dfrac{(z-z_0)-r\gamma}{\bar{\gamma}(z-z_0)-r}+z_0\hspace{0.5cm};\hspace{0.5cm}|K|=1~,~|\gamma|<1$$ Constant $K$ plays role of rotation. For Example with $K=1$ and $\gamma=\frac12i$ the map $w=\dfrac{(8-4i)z-16-9i}{-4iz+4i-8}$ maps $|z-(1+i)|=\dfrac12$ to itself.