Linked Questions

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3answers
5k views

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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2answers
637 views

Find an analytic function that maps the disk $\{|z|<1\}$ onto the disk $\{|w-1|<1\}$ so that $w(0)=1/2$ and $w(1)=0$

Find an analytic function that maps the disk $\{|z|<1\}$ onto the disk $\{|w-1|<1\}$ so that $w(0)=1/2$ and $w(1)=0$ The 3 points theorem: Given 3 point $z_1, z_2, z_3 $ always map into ...
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2answers
104 views

Find all Möbius transformations that map the circle $|z|=R$ into itself

I wish to find all Möbius transformations $T(z)=(az+b)/(cz+d)$ that map the circle $C=\{z\in\Bbb C:|z|=R\}$ into itself. My attempt: Is it sufficient to find all Möbius transformations $T$ such that $|...
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2answers
1k views

Sufficient conditions for a mobius transformation to map the unit circle to itself.

A question on Conway's Complex Functions of One Variable asks: Find necessary and sufficient conditions for a Mobius transformation $T(z)=\frac{az+b}{cz+d}$ to map the unit circle to itself. So if $\...
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2answers
554 views

Characterization of linear fractional transformations that maps the unit disc into itself

I am reading the paper "ADJOINTS OF COMPOSITION OPERATORS ON HILBERT SPACES OF ANALYTIC FUNCTIONS" by MARIA J. MARTIN AND DRAGAN VUKOTIC. In Section 1.1 they say the linear fractional transformation $\...
3
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1answer
122 views

Showing that $b\bar{d}=a\bar{c}\hspace{3mm}\text{and} \hspace{3mm} a\bar{b}=c\bar{d}$ & $\frac{P(z)}{Q(z)}=\omega \frac{z-\alpha}{1-\bar{\alpha}z}$

Let $P(z)=az-b$ and $Q(z)=cz-d$, where $a,b,c,d$ are nonzero complex numbers such that $bc\neq ad$. Suppose that $|\frac{P(z)}{Q(z)}|=1$ whenever $|z|=1$. Show that $$b\bar{d}=a\bar{c}\hspace{3mm}\...
1
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1answer
68 views

Can we characterize the Möbius transformations that map the unit circle into the unit disk?

The Möbius transformations are the maps of the form $$ f(z)= \frac{az+b}{cz+d}.$$ Can we characterize the Möbius transformations that map the unit circle $\{z\in \mathbb C: |z| = 1\}$ into the (closed)...
1
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1answer
52 views

$\varphi_a(x)=a+(1-|a|^2)\frac{x+a}{|x+a|^2}$ 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)....
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1answer
47 views

Entire function that is a bijection on the unit disk is a rotation

I'm working on this problem "Let $f$ be an entire function. Suppose $f$ restricted to the unit disk is a bijection. Prove that $f$ is a rotation." My attempt: It is tempting to use Schwarz lemma. ...
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1answer
49 views

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(...
0
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1answer
758 views

Can we characterize the Möbius transformations that maps the unit sphere onto itself?

Related: Can we characterize the Möbius transformations that maps the unit circle into itself? The Mobius transformation is of the form $$f(z)=\frac{az+b}{cz+d}$$ In the 3D case, all the ...
2
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0answers
345 views

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 ...
0
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0answers
212 views

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 ...