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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25 views

How do these Möbius transforms arise?

I am reading Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables. Below is exercise 10 (p.111) in chapter 3: Let $f$ be a holomorphic function in the disc $|z|...
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52 views

Show any involutive Möbius transformation must have two distinct fixed points.

Let $T$ be a Möbius transformation such that $T(T(z)) = z$ for all $z \in \mathbb{\hat{C}}$. That is, $T = T^{-1}$. I want to show that $T$ must have two distinct fixed points. I have tried using a ...
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31 views

The normal form of $T^n$.

Let $T$ be a Möbius transformation, with fixed points $p$ and $q$. Hence the normal form of $T$ is $$ \frac{T(z) - p}{T(z) - q} = \lambda \frac{z - p}{z - q} $$ I want to show that the normal form of ...
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29 views

Finding a Möbius transformation given three points including $\infty$.

I know how to find a transformation given three "regular" points, but I am getting confused as to how to handle this setup. I am given $T(1) = 4$, $T(0) = i$, and $T(\infty) = -1$. I am using the ...
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10 views

Mapping extended set of real numbers under fractional linear transformations??

I have a function $f(z)=\frac{6z}{(1+3i)z-4i}$ and I want to map the set $R^*$ using the function $f(z)$; how can I do it? I have written $f(z)$ into $x,y$ form as $$w=\frac{6(x+iy)}{(1+3i)(x+iy)-4i}$$...
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28 views

Möbius Transformations by action of $\mathrm{SL}_{2}(\mathbb{R})$.

Consider the group $\mathrm{SL}_{2}(\mathbb{R})$ acting on the set $T=\{z\in \mathbb{C}|\text{Im}\hspace{0.1cm}(z)>0\}$. The action is defined by $$ \begin{pmatrix} a & b\\ c & d \end{...
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Is the Schwarzian Derivative a connection?

The Schwarzian derivative is the quadratic differential $$ S(f) = \Bigg( \frac{f'''}{f'} - \frac{3}{2} \Big( \frac{f''}{f'} \Big)^2 \Bigg) (dz)^2 .$$ Bill Thurston, in his Math overflow answer here ...
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27 views

which domain does this Möbius map to $\Re(w) > 0$?

Given the following Möbius: $$ w = T(z) = \frac{1+z}{1-z} $$ How could I find the domain of $Z$ which $T$ maps to $\{\Re(w)>0\} $? I tried to inverse $T$ and got: $$ z = T^{-1}=\frac{w-1}{w+1} $$ ...
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Isometries of ideal triangles in Poincaré disk model

An ideal triangle in the Poincaré disk model is a triangle with vertices in $\mathbb{R} \cup \{\infty\}$. I have to prove that there exists an isometry between every two ideal triangles. I know that ...
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33 views

Moebius transform that maps imaginary axis to itself

I need to describe all moebius transforms $$\phi_A $$ from the extended complex plane in itself, with $$A \in SL(2, \mathbb{R})$$ that map points from the imaginary axis to the imaginary axis. so ...
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How do I show this Mobius transformation maps the unit disk onto itself? [duplicate]

I am struggling to show that, for any $\alpha, z \in \mathbb C$ such that $|\alpha|<1$ and $|z|<1$: $$\left | \frac{z - \alpha}{1-\bar\alpha z} \right | < 1$$ This should be doable with ...
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43 views

An example of a discrete, abellian and not cyclic group in $PSL(2,\mathbb{C})$

There is a theorem which says that: if $\Gamma$ is a abelian discrete subgroup of $PSL(2,\mathbb{R})$, then $\Gamma$ is cyclic. Nevertheless, we do not get it if the group is $PSL(2,\mathbb{C})$, I ...
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Question on Möbius Transformations

If $z_{1}, z_{2}, z_{3}$ are distinct points in $C \cup \infty$ then there is only one Möbius Transformation which sends these points to $1,0,\infty$ Write out the formula for this Möbius ...
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54 views

What is the image of D under the map $f(z)=bz+\frac{1}{z}$?

Let $f:\mathbb C_\infty\to \mathbb C_\infty$ be a sum of Mobius transformations defined by $f(z)=bz+\frac{1}{z}$, where $-1<b<1$. I've succeeded to find the image of the unit closed disc $D$...
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Find such an analytic isomorphism

The question is to find an analytic isomorphism from the open region between $x=1$ to $x=3$ to the upper half unit disk: $\{|z|<1, \text{Im}(z)>0\}$. I would know how to deal with this if it ...
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43 views

$AB$ is a diameter of $D(0,1)$. Find $C\in \overline{D(0,1)}$ s.t. $|AC|\cdot|BC|$ is maximal

Let $AB$ be a diameter of the circle $S(0,1)$. Find all the dots $C$ in $\overline{D(0,1)}$ such that $|AC|\cdot|BC|$ is maximal. With very basic geometry calculus I have figured out that $C$ is the ...
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30 views

$T$ is a Mobius tranformations with $T(z_1)=z_1,T(z_2)=z_2$. Find $T(z_3)$ where $z_3$ is the center of $[z_1,z_2]$

Let $T$ be a Mobius transformation where $T(z_1)=z_1$ and $T(z_2)=z_2$ for some $z_1,z_2\in\mathbb{C}$. Suppose $z_3$ is the center of the segment $[z_1,z_2]$. Describe the possible image of $z_3$ ...
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53 views

Mapping circular lunes $|z|<1,|z+i|<1$

I want to map the circular lunes $|z|<1,|z+i|<1$ onto the upper half-plane Since intersection points are $z_1=\sqrt 3/2-i/2$ and $z_2=-\sqrt 3/2-i/2$ we have $$w_1=\frac {2z-\sqrt 3+i}{2z+\...
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Logic behind Möbius transformation

So I am attempting to get a grasp on Möbius transformations. Specifically, I'm trying to understand the logic and reasoning behind using them to map certain subsets of the complex plane into others. ...
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58 views

An exercise on Möbius transformations

I had an exam on Complex Analysis and I could not solve the following exercise on Möbius transformations: Let $f(z)=\frac{z+1}{z-1}$ and $A=\{z : \operatorname{Im}(z) >0\}\setminus \{|z|<1\}$. ...
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Verifying properties of given conformal mapping

Let's look at $\phi:\ z\mapsto -i\frac{z-1}{z+1}$ and try to verify in an elementary way that it maps the unit disk onto the upper half-plane: My problem is the following: I can easily get $\phi^{-1}:...
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1answer
64 views

Transform Name for $(1-z)/(1+z)$

I am trying to find the actual name of the transform $f(z) = \frac{1-z}{1+z}$; the transform from the open unit disk to the right half plane. Another variant; the transform from the upper half plane ...
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Möbius transformations and groups

I have two questions regarding Möbius transformations and groups. In my notes there are two statements, which I can't prove/understand why they hold. The subgroup of Möbius Transformations which ...
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Homeomorphism of unit disc onto itself interchanging two points. [duplicate]

I've searched this topic already and found similar question. But it was not fully answered. Actually I know the Möbius transformation of form $\frac{z-a}{1-\bar a z}$ is the automorphism of unit ...
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Why is the map analytic?

The map $$z\mapsto \frac{z-i}{z+i}$$ is an analytic isomorphism of the upper half-plane $\mathbb{H}$ and the unit disc $D=\lbrace w \in \mathbb{C}||w|<1 \rbrace$ . So , what I do not undertand is ...
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60 views

Can one replace the Riemann sphere with other objects? What happens?

When doing one complex variable I think we talked about the Riemann Sphere, and how it relates to Möbius transformations, the complex "point at infinity" and other things. Is it possible (and useful) ...
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Mobius transformations from intersection of circles to two straight lines

My textbook is pretty useless on Mobius transformations, and I just wanted to check this reasoning was correct! I want to find a Mobius mapping from the region $$\{z:|z-1|\lt\sqrt{2}, |z+1|\lt \sqrt{...
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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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1answer
53 views

If $z_{1,2,3}\in S(a,R)$ then $z^*={R\over \bar{z}-\bar{a}}+a$

Recall: $z^*\in\mathbb{C}$ is symetric to $z\in\mathbb{C}$ in a relation to a generalized circle $C$ if $\overline{(z_1,z_2,z_3,z)}=(z_1,z_2,z_3,z^*)$ for some $z_1,z_2,z_3\in C$. $(z_1,z_2,...
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Confirm my answer: Find $P\in\mathbb{C}$ s.t. ${|AP|\over|BP|}=r$

The question: Find all points $P\in\mathbb{C}$ such that ${|AP|\over |BP|}=r$ for given $A,B\in\mathbb{C}$. I want to varify my answer, I hope that's allowed: Let $T(z)={z-A\over z-B}$ be a ...
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Mobius Transformation maps extended complex plane to extended complex plane

I have read in quite a few books the statement that "A mobius transformation gives a bijection from the extended complex plane to itself." How do I prove this? I know that you can find a unique Mobius ...
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1answer
44 views

Find mobius transformations that send the upper halfplane to itself

Task: Find all $A\in GL(2,\mathbb{C})$, such that the corresponding mobius transformation maps $\mathbb{H}=\{z\in\mathbb{C}\,|\,\Im(z)>0\}$ to itself. I know that this statement has been asked ...
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1answer
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Find Möbius (conformal) transformation with critiera and still the same boundary?

I have trouble finding a Möbius (conformal) transformation $w=f(z)$ that fulfills the criteria that $w=f(0)=\frac{1}{2}$ for the following case: Find $w=f(z)$ so that $|z|<1$ maps onto $|w|<1$...
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Mobius transformation with fixed points α,β ∈ C

Let $T$ a Mobius transformation with fixed points $\alpha, \beta \in \mathbb{C}$ Show that exist a Mobius transfotmation $S$ such as $STS^{-1}$ fix $0, \infty$, that is unique?
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How to determine if three distinct points $a,b,c \in \Bbb c$ are collinear using Mobius Transformation?

Given three points $\frac{3}{2} + i , 2i,-6+6i$. I have the mobius transformation that maps these three points to $0,1,\infty$ respectively as $M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$ ...
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77 views

Trouble in mapping of möbius transformation

Question:- Show that the transformation $$ w = \frac{2z+3}{z-4}$$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$ My attempt:- The circle $x^2+y^2-4x=0$ is $|z-2|=2$ . . .$(1)$ So ...
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1answer
42 views

Mobius transforms - inversion sends line/circle -> line/circle

I'm trying to prove that the inversion mapping $f(z) = \frac{1}{z}$ sends circles or lines to circles or lines. Apparently the set $$\{z \in \mathbb{C}: |z-a|^2 = r^2 \}$$ describes either a circle ...
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Prove transformation $z \mapsto \frac{az+b}{cz+d}, z = x+ iy, ad-bc = 1$ is isometry of Hyperbolic plane.

Prove transformation $$f: z \mapsto \frac{az+b}{cz+d},\ z = x+ iy,\ ad-bc = 1$$ is isometry of Hyperbolic plane $$M=\{(x,y)\in \Bbb R^2:y>0\} \text{ with Riemannian metric } g= \frac{1}{y^2}(dx \...
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Find Möbius transformation for half-plane to unit disk $|w|<1$?

Consider the half-plane depicted in the following figure How can a Möbius transformation that takes that half-plane onto the unit disk $|w|<1$ be found? What are the steps and things to think ...
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1answer
37 views

Möbius transformation: area outside of two circles mapped onto the interior of a circular ring

In my current homework, there's the following task: "The area outside of these two circular discs $K_1=\{z{\in}\mathbb{C}:|z-\frac{5}{2}|\le\frac{3}{2}\}$, and $K_2=\{z{\in}\mathbb{C}:|z+\frac{5}{2}...
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60 views

Stereographic projection combined with rotation is a Möbius transformation

So for my Complex Analysis class, I need to prove the following question: Consider the function that maps a point from $\mathbb{C} \cup \{\infty\}$ to the sphere via inverse stereographic projection, ...
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26 views

Finding the image of the unit disc under a Mobius transform

I am working through an example of mapping the unit disc under the following Mobius transform: $$f(z)=\frac{iz+3}{iz-1}=1+\frac{4}{iz-1}$$ This can be written as a composition of elementary Mobius ...
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number theory and geometry question-Möbius transformation [duplicate]

I am trying to justfiy the distinct points z1, z2, z3, z4, w1, w2, w3, w4 ∈ C ∪ {∞} such that there is no Möbius transformation T with T(zi) = wi for all i = 1, 2, 3, 4. I am having a real hard time ...
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1answer
42 views

Mobius transformations between two sets

I am doing some revision of complex analysis, and am stuck on this question. I am looking for A mobius mapping sending the set {z: |z+1|<$\sqrt{2}$}, |z-1|<$\sqrt2$} onto the sector {z:3pi/4< ...
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2answers
43 views

Möbius transformations mapping non-unit circle to non-unit circle

I have a problem in which I need to find a möbius transformation which has as one of the criterion to map the circle $|z−2+i| = \sqrt5$ onto the circle $|w+2| = 2$, I dont really understand how to ...
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1answer
38 views

Determine $z\in\mathbb{C}$, $r>0$ so that $0,1,2+i\in \partial B_r(z)$

I'm struggling to see for a method to start this question. It looks like a question related with mobius transforms. We have studied about determining the mobius transform when points from the domain ...
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2answers
44 views

Mobius Transformation viewed as a mapping on $\bar {\mathbb C}$

The question I have on hand is as follows: Suppose that a Mobius Transformation z $\to \frac{az + b}{cz + d}$ (viewed as a mapping on $\bar {\mathbb C}$) maps $\infty \to \infty$. What information ...
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1answer
27 views

Vertical Line through z=0 in complex plane mapped with f(z)=(1+z)/(1-z)

I have the vague notion that the imaginary axis maps to a circle with f(z)=(1+z)/(1-z). $$ \begin{array}{lll} f(\infty) & = & -1\\ f(i) & = & e^{\pi/4}\\ f(-i) & = & e^{-\pi/4}\...
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1answer
56 views

Conformal property of transformation [closed]

I want to know if a LFT, $F$, is conformal on the hyperbolic plane $\mathbb H^2$ , that is if we have the curves $\Gamma_1$ and $\Gamma_2$ that intersect at a point $P$ making the an angle $X$, then $...
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1answer
40 views

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 ...