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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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323 views

Mobius transformation maps the real axis to the unit circle

Show that any Mobius transformation which takes the real axis (with $\infty$) to the unit circle can be written in the form $$M(z)= \alpha \dfrac{z-\beta}{z-\overline{\beta}}$$ where $|\alpha|...
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55 views

Mapping using mobius transformations

I had a fundamental question regarding mobius transforms. Suppose I want to map the unit circle to the upper half plane ($Im$ $z \geq 0$) on the complex plane. I know mapping any three points on the ...
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1answer
46 views

Evaluate the image of complex function

Given the function $f:\mathbb{C}\setminus\{-i\}\rightarrow \mathbb{C}\setminus \{1\}$, defined by $f(z)=\frac{z-i}{z+i}$. I'm supposed to find the image for $f(\{z\mid\Im (z) > 0\})$. However I'm ...
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588 views

From the unit disk to the right half plane and $T(0)=3$

Find a Mobius transformation $T$ from the unit disk to the right half plane with condition $T(0)=3$. First, the transformation from the unit circle to the upper half plane is $T_1(z)=(1-i)\frac{z-i}{...
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95 views

Map from the upper half plane to a circle $\vert w-w_0\vert<R$ such that $T'(u_0+v_0i)>0$ and $T(i)=u_0+iv_0$

Transform the upper half plane $\mathop{\mathrm{Im}} z>0$ into the circle $\vert w-w_0\vert<R$ so that the point $i$ correspond to the center of the circle and the derivative in this point is ...
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160 views

How to find transformation from the upper half plane into the right half plane?

Find the general form of the linear transformation which transforms the upper half plane into the right half plane. In my notes I have a Mobius transformation from the upper half plane to the ...
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144 views

Fractional Linear Transformation

Suppose that $a, b, c, d\in\mathbb{C}$ and $ad-bc=1$. Let $T$ be the fractional linear transformation $$z\mapsto\frac{az+b}{cz+d}.$$ Show that if $a=i, b=-i, c=1, d=i$, then the corresponding ...
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217 views

Transform circle to $\mathbb R$: Will any 3 distinct points on the circle work?

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.15 (Exer 3.15) Using the cross ratio, with different choices of $z_k$, find two ...
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119 views

Why is $w$ real if z is on the circle through $z_1,z_2,z_3$?

Question on answer to Show that $z$ is on the circle passing through $z_1$, $z_2$ and $z_3$ if and only if [$z$, $z_1$, $z_2$, $z_3$] is real or ∞*, a duplicate of Four complex numbers $z_1,z_2,z_3,...
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95 views

Prove this Möbius function maps unit disc to itself bijectively.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.9 I got $(a)$ and $(b)$. My attempt for $(c)$: First, I interpret that $(c)$ is ...
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370 views

Suppose $f$ holomorphic and its image is a subset of the unit circle. Then show f is constant.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.8 Suppose $f$ is holomorphic in region $G$, and $f(G) \subseteq \{ |z|=1 \}$. Prove $f$ ...
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503 views

Finding Mobius transformations that maps one set to another

I am having a hard time understanding how we find mobius maps from circles, discs to half planes etc. I know how to find maps that take a set of points to another but not sets. I know about cross ...
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1answer
139 views

A Möbius map from the unit disc onto itself [duplicate]

I know that this has been asked a few times, but I found no thread that fully derived the results. I want to show that for any Möbius transformation from the unit disc onto itself it has the form $e^{...
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1answer
138 views

Find a conformal map 2

$\textbf{Exercise}$ Find a conformal mapping which maps the domain $D$ onto the open unit disc, where $D$ is the intersection of $\vert z \vert <1 $ and $\vert z-1 \vert <1 $. I knew that ...
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148 views

Find a Mobius transformation that maps the unit-disk onto unit-disk: $1\to1$ and $1+i\to \infty$

I tried to find the Moebius transformation which maps unit-disk to unit-disk and maps 2 points: $z_1=1$ to $w_1=1$ and $z_2=1+i$ to $w_2=\infty$. What I have: $w_1= \dfrac{a+b}{c+d} = 1$ $w_2= \...
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107 views

Mobius transformation that maps the upper half plane conformally onto the open unit disc.

Let $H = \{z=x+iy \in \Bbb C:y>0 \}$ be the upper half plane and $D=\{z \in \Bbb C:|z|<1 \}$ be the open unit disc. Suppose that $f$ is a Mobius transformation, which maps $H$ conformally onto $...
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Need help with $\frac{w-a}{1-\overline{a}w} = \mu\frac{z-a}{1-\overline{a}z}$

I have one exercises that i need help. Let the Möbius transformation $w=T(z)$ defined by $\frac{w-a}{1-\overline{a}w} = \mu\frac{z-a}{1-\overline{a}z},\mu=e^i\alpha, \lvert a\rvert<1$ 1.Put the ...
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1answer
41 views

Finding certain Mobius tranformation

Let $D$ denote the unit disc (|z| < 1). Let $a \in \mathbb{C}, B \in \mathbb{R}$ and $r > 0.$ I want to find a Mobius map $f$ such that (1) $f(D) = \{z = x+iy : |z-a| < r\}$ $\textbf{Sol}$ ...
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233 views

A special Mobius Transformation that maps the right half plane to the unit disc

Find the Mobius Transformation that maps the right half plane to the unit disc carrying the point $z=15 $ to the origin. Since the Mobius transformation takes the point $z=15$ to the origin, so I ...
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191 views

Proof that automorphisms of unit disk are of the form $\lambda \dfrac{z-a}{\overline{a}z-1}$

I am trying to show that the most general Möbius map that sends the unit disk to the unit disk is $\lambda \dfrac{z-a}{\overline{a}z-1}$ where $|\lambda|=1$ and $|a|<1$. I have shown that the maps ...
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How to deduce general solution of DE using Mobius transform

We are given a differential equation, say of the form $$y''(z)+\frac Azy'(z)+\frac B{z^2}y(z)=0$$ This is a differential equation with regular singular points at $0$ and $\infty$. Then we find the ...
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35 views

Construct a Mobius transformation $f$ with the specified effect:

Construct a Mobius transformation $f$ with the specified effect: $f$ maps $K(0,1)$ to itself and $K(1/4,1/4)$ to $K(0,r)$ for some $r<1$. My work: Since $i \rightarrow 1 \leftarrow i$ $ 1/4 \...
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294 views

Functions for which the Jacobian has orthogonal columns or rows

What can be said about a function $f : \mathbb{R}^m \to \mathbb{R}^n$ for which all singular values of the Jacobian matrix, $\mathbf{J}$, are all 1? I have found this similar question which covers ...
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1answer
322 views

Mobius transformation, and showing it maps the unit disk D[0,1] to itself bijectively

Consider $a \in \mathbb{C}$, where $|a|<1$ and $f_a(z)=\frac {z-a}{1-\overline az}$. (a) Show that $f_a(z)$ is a Mobius transformation. (b) Show that $f_a^{-1}(z)=f_{-a}(z)$ (c) Prove that $f_a(...
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1answer
49 views

Image of $f(z)=\frac{z-3}{z-4}$ on $\{z\in\mathbb{C}|2<Re(z)<3\}$

Let $D=\{z\in\mathbb{C}|2<Re(z)<3\}$. Let $f(z)=\frac{z-3}{z-4}$. Find $f(D)$. I thought of using the decomposition of Mobius transformations, i.e. $$f_1(w)=w-4,\ f_2(w)=1/w,\ f_3(w)=1+w$$ ...
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112 views

Find a biholomorphic map from the region $\Omega$ onto the unit punctured disk $D^*$.

Find a biholomorphic map from the region $\Omega=\{z=x+iy:x^2+y^2<4 \text{ and } x+y<2\}\setminus\{0\}$ onto the unit punctured disk $D^*=D(0,1)\setminus\{0\}$. My idea is to map the line x+y=2 ...
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63 views

Inversion in a circle as radius goes to infinity.

I am trying to show that the in the limit case as the circle gets very large, inversion in it is equivalent to reflection in a line. I have the transform $z \to c+ \frac{R^2}{ (\overline z -\overline ...
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Find the equation of the image of the line $x+y=1$ by Möbius transformation

Find the equation of the image of the line $x+y=1$ by Möbius transformation $$w=\dfrac{z+1}{z-1}$$ My approach, if $x=\Re(z)$ , and $y=\Im(z)$, then $x+y=(\frac{1}{2}-\frac{i}{2})z+(\frac{1}{2}+\frac{...
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1answer
41 views

If $\Omega$ is a symmetric domain of $\mathbb{C}$ and $f$ an isomorphism with certain conditions then $f(\overline{z})=\overline{f(z)}$

If $\Omega$ is a symmetric domain of $\mathbb{C}$ respect to $\mathbb{R}$ and $f: \Omega\longrightarrow D(0,1)$ is an isomorphism. If $\exists a\in \Omega \cap \mathbb{R}$ such that $f(a)=0$ and $f'(...
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What is the dimension of the set generated by $z \mapsto z + 1$ and $z \mapsto \frac{z}{1+z}$?

Let's say I have subgroup of $SL_2(\mathbb{Z})$ generated by two elements, I wanted to compute the limit set. So I wrote down two matrices: $$ A = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \...
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75 views

Diagonalising Invertible Mobius Transformation

For invertible $M \in \mathbb C^{2\times2}$ we can find invertible $S \in \mathbb C^{2\times2}$ so that $SMS^{-1}$ is either diagonal, or triangular with two equal diagonal elements. There is an ...
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78 views

Free discrete action vs discrete subgroup

Suppose a group $\Gamma \leq PSL(2,\mathbb{R})$ is acting fixed point freely on the upper half plane $\mathbb{H}$ via Moebius transformations. I am looking for a quick argument showing that the action ...
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176 views

finding all the mobius transformation.?

find all the mobius transformation that map the units disc$ D$ onto the left half plane $H^-$ = {$w \in C : Re w <0 $} My attempts : I know that all the Möbius transformations can be ...
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1answer
160 views

Find direct isometries In hyperbolic space.

Let H be the upper half of the complex plane model for hyperbolic geometry. let $p,q$ be points in the plane. show how to find a direct Isometry F of H such that $F(p)=q$. Now suppose $p\prime,q\prime$...
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How does this algorithm get the limit set of “kissing” Schottky group?

I'm having difficulty understanding why the algorithm presented in this paper works. If I understand correctly, to construct a Schottky group start with $2n$ circles: $A_1...A_n$ and $B_1...B_n$, ...
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Sketch the image under the function $w = \log z$ of the following set $\{z : Im z > 0\}$

I would love some help with this question please: Sketch the image under the function $w = \log z$ of the following set $\{z : Im z > 0\}$ What I have done so far is as follows: $R= \{x+iy: y&...
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Determining the matrix representations of functions.

Show that each function\begin{align}f(x)=\dfrac{ax+b}{cx+d}, g(x)=\dfrac{ex+f}{jx+h}\end{align} can be represented by matrices such that the matrix representation of the composition $g(f(x))$ is the ...
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1answer
40 views

Sphere reflection property (geometric proof).

I need some help with this exercise I found in chapter 3 of the book "The Geometry of Discrete Groups" by Beardon. Prove (analitically and geometrically) that for all non-zero $x,y \in \mathbb{R} ^n $...
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1answer
24 views

Showing that if $T$ is a Möbius transformation of the disc, $\frac{|T(z)-T(w)|}{|1-T(z)\overline{T(w)}|} = \frac{|z-w|}{|1-z\overline{w}|}$

... where $z, w \in \mathbb{D}$, $\mathbb{D}$ the open unit disc. I realise that the way to start working on this is to express $T$ as $e^{i\alpha}S_a$ for some $\alpha \in [0, 2\pi]$ and $a \in \...
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1answer
163 views

Books and references for Möbius transformation, hyperbolic Riemann surface and covering transformation

I am reading the book Computational Conformal Geometry by Xianfeng David Gu and Shing-Tung Yau. There is a part in the book which I don't understand and I would like to ask for books and references ...
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1answer
70 views

Determine the most general Mobius transform that…

Determine the most general Mobius transform that satisfies: i) two fixed points z=0 and z=1; ii) maps the unit circle $|z|=1$ and the bisector of first and third quarter to lines. My guess is: $$\...
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487 views

How to find the “interior boundary” for a set of points?

Let $S$ be a set of points in $\mathbf{R}^n$: $$S=\{ (x_1,\dots,x_n) \ | \ x_i \in \mathbf{R} \ ; i=1,\dots, n \}$$ Is there a way to find the interior boundary of such set of points? Since there ...
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Holomorphic function mapping unit disc to the “pacman” $U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]\}$

Find an injective and surjective holomorphic function, that transfers the unit disc $\Bbb D = \{|z|<1\}$ to the domain $U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]$}. I ...
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69 views

Is a perspective projection a Möbius transformation?

Both perspective projection and Möbius transformations exhibit cross-ratio invariance, but can a perspective projection be defined as a subgroup of the Möbius group? In other words, is a perspective ...
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1answer
206 views

prove Mobius Transformation can be extended to a meromorphic function

I have been reading all the notes I could find about Riemann surfaces but still have no clue how to start these 2 proofs... I know they should be easy proofs... 1 first one being, prove/show that ...
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138 views

Program that shows action of Möbius transformations

This is not directly a mathematical question, but I think this is the best place to ask it. I discovered that I'd find it very helpful to have some sort of a program where I can plot the coefficients ...
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1answer
94 views

Determining a Mobius transformation from a tiling

Suppose we have a tiling of the upper half plane by ideal quadrilaterals. Suppose that the principal quadrilateral consists of perpendicular lines at $Re(z)=0$ and $Re(z)=1$, and two circles joining ...
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1answer
70 views

Proving that $\exists a,b \in \mathbb{C}$ such that $\forall z\in \mathbb{C}$ either $m(z)=az+b$ or $m(z)=a\bar{z}+b$ .

Let $m \in \text{Homeo}^{C}(\bar{\mathbb{C}})$ (the group of homeomorphisms of $\bar{\mathbb{C}}$ taking circles in $\bar{\mathbb{C}}$ to circles in $\bar{\mathbb{C}}$, where $\bar{\mathbb{C}}=\mathbb{...
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75 views

The group of Mobius Transforms and their conjugates/reflections?

I'm looking at a group of transformations from $X \cup \lbrace \infty \rbrace$ to itself, where $X$ is a Hilbert Space, and ultimately I'm looking for an appropriate name for this group. The elements ...
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1answer
56 views

Quick question on Mobius Transformation Iteration [closed]

If f and g are Mobius Transformations, why does this hold? $$gf^{n}g^{-1}(z)=z+n$$ implies $$f^{n}(z)=g^{-1}(g(z)+n) $$