mark
Show that the Mobius transformation $T(z)=-z/(z-1)$ pairs the circles $C_1(1)$ and $C_1(-1)$.
An archived instance of discourse for discussion in undergraduate Complex Dynamics.
Pairing circles
Rick
First , we can define 3 points that define $C_1(1)$ such as $0$,$2$, and $1+i.$
And $T(0)=\frac{-(0)}{(0)-1}=0$
$T(2)=\frac{-(2)}{(2)-1}=-2$
$T(1+i)=\frac{-(1+i)}{((1+i)-1)}=\frac{-(1+i)}{i}=-1+i$.
Since these three points lie on $C_1(-1)$, we can conclude that $T(z)$ maps $C_1(1)$ to $C_1(-1)$.