# Questions tagged [mobius-transformation]

For questions about the geometry, complex-analytic, and group-theoretic properties of the Möbius transformations (linear fractional transformations) $z \mapsto \frac{az + b}{cz + d}$ of the complex plane, which can be identified with the group $PGL(2, \mathbb{C})$, or certain subgroups thereof.

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### Surjectivity of $f_\beta(z)=\frac{z-\beta}{z-\bar{\beta}}$

Let $\mathbb{H}=\{z=x+yi\mid y>0\}$ and $\mathbb{E}=\{z\in\mathbb{C}\mid\mid{z}\mid<1\}$. Let $\beta\in\mathbb{H}$ and let $$f_\beta(z):=\frac{z-\beta}{z-\bar{\beta}}$$ I have already shown ...
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### Find the image under the Mobius transformation $f(z) = 1/z$ of $|z-3i|=1$ [closed]

I am given $|z-3i|=1$ which is a circle with radius $1$ and centre $(0,3i)$ on the complex plane. I want to find the image (to sketch it) under the transformation $1/z$ WITHOUT taking points and ...
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### Conformal Mapping Question about Mobius Maps, mapping 1 region to another

I am trying to map the region G, {|z|<1, |z+i|> (2)^0.5} to the infinite vertical strip at x = +/- pi. I have started by using a Mobius Map which sends the two common points of the circles to 0 ...
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### Finding the Möbius transformation from the unit disk to the half plane $\{Re(z)\geq3\}$

I want to find the Möbius transformation from the disk $\{|z-1|\leq2\}$ to the half-plane $\{Re(z)\geq3\}$ that moves the point $0$ to $4+4i$. I know that by specifying the values at 3 points, the ...
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### If $w:\mathbb{D}\rightarrow\mathbb{D}$ is a Möbius transform and $||f||_{\infty} \leq 1$, why is $||w(f(z))||_{\infty} \leq 1$?

Denote the unit disk by $\mathbb{D}$. Let $w: \mathbb{D} \rightarrow \mathbb{D}$ be a Möbius transformed defined by $w(z) = \frac{z-\lambda}{1-\overline{\lambda}z}$ where $\lambda \in \mathbb{D}$ is ...
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### Is this a simpler proof that all transformations mapping D(1:0) onto D(1;0) are of the form $\;\;e^{i\lambda}\frac{z-\alpha}{\bar{\alpha}z-1}$

The question is from an exercise (2.13) in Introduction to Complex Analysis by H.A. Priestley. Before writing this I did check all the questions that might have the answer, but where proofs were given ...
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### Differential Action of Möbius Transformations

The group $\mathrm{PSL}_2(\mathbb{R})$ acts on $\mathbb{H}$ via Möbius transformations, that is \begin{align*} g=\begin{pmatrix} a & b \\ c & d\end{pmatrix}:z\mapsto \frac{az+b}{cz+d}. \end{...
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### Mobius transformation produces either a circle or a line…

The exercise (from H A Priestley) required a transformation that sent $\:0, 1, {\infty}$ to $1, 1+i, i$. I knew the transformation that sent $z_1, z_2,z_3,$ to $0, 1, {\infty}$ ie \frac{(z-z_1)(z_2-...
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### A homeomorphism $T$ from extended complex plane to itself preserving cross ratio is a Mobius map.

Cross ratio preserving means $(Ta,Tb,Tb,Td)=(a,b,c,d)$ where $(a,b,c,d)=\dfrac{(a-b)(c-d)}{(a-d)(c-b)}$. If we assume $T$ fixes infinity, can we prove $T$ is affine?
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### Mobius Geometry identity

How do you prove that all figures consisting of three distinct points are congruent in Mobius Geometry? I understand it relates to the Fundamental Theorem of Mobius Geometry. The concepts of which ...
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### conformal automorphism $f$ of $D$ that interchanges

Let $a$ and $b$ be distinct points in the unit disk $D$. Show that there exists a conformal automorphism $f$ of $D$ that interchanges $a$ and $b$; that is, $f(a) = b$ and $f(b) = a$. Idea: we know ...
Find a Mobius transformation mapping the unit disk {|z| < 1} into the right half-plane and taking z = −i to the origin. My workings: $\phi(t) = \frac{az+b}{cz+d}$ We map -i to the origin (0) by ...
### Conformal mapping of a domain to $f(z)=z^3$
Let W be the domain ${Im(z) < 0, Re(z) > 0}$. Sketch and describe the image of W under the conformal map $f(z) = z^3$. I have absolutely no idea how to tackle this practice exam question. I ...