I am trying to find the most general Mobius transformation that maps the right half-plane to the unit disc carrying the point 5 to the origin. I find a particular Mobius transformation is $\frac{z-5}{z+5}$. But how do I find the most general one?
I am trying to find the most general Mobius transformation that maps the right half-plane to the unit disc carrying the point 5 to the origin. I find a particular Mobius transformation is $\frac{z-5}{z+5}$. But how do I find the most general one?
You compose that transformation with a Möbius transformation of the unit disc that fixes the origin. But the only such transformations are rotations about the origin. So the most general transformation will be $\lambda\frac{z-5}{z+5}$, where $\lambda$ is any complex number of absolute value 1.