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### All Möbius transformations that take the unit disk onto itself [duplicate]

I wish to prove that all Möbius transformation raking the unit disk into itself are of the form $k\frac{z-l}{1-z\bar{l}}$ where $|k| = 1$. More specifically, I ask, in addition to the main question ...
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### Finding the Mobius Transformation that maps open unit disk onto itself

Hi I am trying to find all the Mobius transformations that map unit open disk onto itself i.e., if $|z|<1$ then $|f(z)|<1$ where $f(z)=\frac{az+b}{cz+d}$. I did so far \begin{align*} &\Big|\...
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### Let $\gamma$ be the unit circle. Find a Möbius transformation that transforms $\gamma$ onto $\gamma$ and transforms 0 to $\frac{1}{2}$.

Let $\gamma$ be the unit circle. Find a Möbius transformation that transforms $\gamma$ onto $\gamma$ and transforms 0 to $\frac{1}{2}$. I think it's quite easy to find a Möbius transformation that ...
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### How to find a möbius transformation mapping $B(p, r)$ to $B(0, 1)$?
I have an open connected $\Omega$ of the complex plane and for a point $p \in \Omega$ and $r > 0$ we have a ball $B(p, r) \subset \Omega$. Is there a general möbius transformation to map this to B(...
### $\varphi_a(x)=a+(1-|a|^2)\frac{x+a}{|x+a|^2}$ is a diffeomorphism from unit ball to unit ball in $\mathbb R^n(|a|>1)$.
I want to prove that $\varphi_a(x)=a+(1-|a|^2)\frac{x+a}{|x+a|^2}$ is a diffeomorphism from unit ball to unit ball in $\mathbb R^2$($a\in \mathbb R^n,|a|>1$, where $||$ is the usual Euclidean norm)....