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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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Möbius Transformation on Riemann Sphere

Just started learning about Möbius Transformations myself, and I wanted to know what kind of transformation on the Riemann Sphere would preserve $S^1$ (unit circle) as a set? What would the conditions ...
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How to show this property of Mobius Transformations

How can I show the following property of complex FLTs? $$ F_{XY} = F_X ◦ F_Y$$ where, $X, Y ∈ SL_2(\Bbb R)$ I know the inverse map of $F_X$ exists. This map is also a continuous map of $\Bbb ...
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Showing complex FLT to be continous

If I have the complex Mobius Transformation, $$ F_X (z) = \frac {az + b}{cz + d} $$ how can I show that the transformation is continous w.r.t the metric on $\Bbb C \cup \infty $ which is ...
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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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Fuchsian groups in $\text{SL}(2,\mathbb{R})$ and commensurability in $\text{GL}(2,\mathbb{R})$

Let $\Gamma_1,\Gamma_2 \subset \text{SL}(2,\mathbb{R})$ be two Fuchsian groups. Assume that they are commensurable as subgroups of $\text{GL}(2,\mathbb{R})$, that is, there exists $g \in \text{GL}2,\...
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Mobius transformation maps $\Bbb R_∞$ onto itself iff we can choose its coefficients to be real

I have seen many solutions which are very intricate and/or long. The solution I had was much shorter which lead me to believe that it was incorrect. It went like this : $T$ be a Mobius transformation ...
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Variant of Schwarz-Pick for Different Bound/Disk

All, I'm looking to prove an alternate version of the Schwarz-Pick Lemma: Let $f:D(0,r) \rightarrow \mathbb{C}$ be holomorphic, and suppose that $|f(z)| \leq U \quad \forall z \in D(0,r)$. Then, $\...
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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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Find the map from $\{ z: - \pi/2 < Im(z)<\pi/2\}$ to the vertical strip $\{ z: 0 < Re(z)<\log 2\}$.

Find the map from $\{ z: - \pi/2 < Im(z)<\pi/2\}$ to the vertical strip $\{ z: 0 < Re(z)<\log 2\}$. Using the map $f(z)=i (2/\pi)(\log 2) z$ we get the image of $f$ as $\{ z: \log 1/2 &...
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Describing curves of complex valued functions

I wish to describe the curves $|f|$=constant and arg$f$=constant for the following functions: 1.$f(z)=exp(z^2)$ 2.$f(z)=exp\left(\cfrac{z+1}{z-1}\right)$ My thoughts: I can write down what the ...
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Does a Möbius transformation having fixed points not on the object return the entire complex plane as image?

This arises from a (very early) exercise in H. A. Priestley's Introduction to complex analysis. Given the transformation $\frac{z-1}{z+1}$ we were to find the invariant (=fixed?) points, which are $\...
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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 ...
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Mobius transform |z|<1 to the right half plane

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 ...
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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 ...
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Are $\Bbb R^2\backslash\{(x,y) : x\le 0,y=0\}$ and the unit disc homeomorphic? Difference between conformal map an homeomorphism.

The Riemann mapping theorem says that there exsists a (bijective) conformal map $f$ between $\Omega =\Bbb C\backslash \{z\in\Bbb C: Im(z)=0, Re(z)\le0\}$ and the unit disc $D_1$. $f$ is the ...
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Finding the inverse of $\phi_{\lambda}^2$ where $\phi_{\lambda}$ is the Mobius transform on $\mathbb{D}$

Let $\mathbb{D}$ denote the open unit disk. Fix $\lambda \in \mathbb{D}$. Define the Mobius transform $\phi_{\lambda}:\mathbb{D}\rightarrow\mathbb{D}$ by $$\phi_{\lambda}(z) = \frac{z-\lambda}{1-\...
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Image of unit disk under inversion - Mobius transformations

I am trying to find the image of $\{z \in \mathbb{C} : |z| < 1\}$ under $f(z) = \frac{1}{z}$ Let $$w = \frac{1}{z} \Rightarrow z = \frac{1}{w} = \frac{1}{u+iv} = \frac{u-iv}{u^2 + v^2}$$ if $w = ...
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What is actually the geometry or analysis behind the fact that $Mob(\hat{\Bbb C})$ is simple?

Let, $Mob(\hat{\Bbb C})$ be the group of all Mobius transformations from the extended complex plane to itself i.e. from $\hat{\Bbb C} \to \hat{\Bbb C}$ . I have been able to prove that (i) $Mob(\hat{...
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mobius transformation given points

Let P,Q,R ∈ ˆ C be the points P = − √2 + i√2 , Q = 2i , R = √2 + i√2 . Let M : ˆ C→ ˆ C be the Mobius transformation with M(P) = Q , M(Q) = R The points P,Q,R lie on a common hyperbolic line (you do ...
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What is the radius and center of the image of $|z|=1$ under $ f(z) = \frac{3z+2}{4z+3}$?

I would like to compute the image of the circle $|z|=1$ about the fractional linear transformation: $$ f(z) = \frac{3z+2}{4z+3} $$ In particular, I'd like to compute the new center and radius. ...
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Using conformal maps to solve the Dirichlet problem on $U = \{z : \text{Im}z \geq 0 \}$

I'm trying to solve a question which asks me to solve the Dirichlet problem on $U = \{z : \text{Im}z \geq 0 \}$ on the condition that (where we let $z = x+iy$) $u(x,0) = 0 $ when $ |x| >1 $ and $ u(...
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Solving the Dirichlet problem on $ U = \{z: \text{Im} z \geq 0\}$ using a conformal map

I'm trying to solve the Dirichlet problem on $ U = \{z: \text{Im} z \geq 0\}$ with the conditions $u(x,0) = 0$ when $x>0$, $u(x,0)=1$ when $x<0$. To do so, I'm supposed to use conformal maps ...
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Subgroup of Möbius transformations which are isometries with respect to the standard metric on the Riemann sphere

I'm trying to find which subgroup of Mobius transformations are isometries with respect to the standard metric on the Riemann sphere (the one induced from the Euclidean metric on $\mathbb{R}^3$). The ...
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Finding a conformal map from the intersection of two disks to the unit disk.

I'm trying to solve a problem which asks me to find a conformal mapping from $\{z\in \mathbb{C}: |z-i|< \sqrt2$ and $|z+i|<\sqrt2\}$ onto the open unit disk. I'm really new to these and I'm a ...
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Möbius transformation/biholomophic funtion

I have to show, that the Möbius transformation $$ T(z) = \frac{z-z_0}{1-\bar{z_0}z}$$ is an biholomorphic function on $ \mathbb{D}$. $ \mathbb{D}:=\{ z \in \mathbb{C}: |z|<1 \}$ and $z_0 \in \...
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Finding bilinear transformation which maps $|z|=1$ on to $|w|=1$ [closed]

How can I show that every bilinear transformation which maps $|z|=1$ on to $|w|=1$ must be of the form $$w=K\frac{z-\alpha}{\overline{\alpha}z-1}$$ where $|K|=1$? Please help me. Thanks.
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Mobius transformation producing a curved triangle with 3 intersecting circles

Let $ABC$ be a curved triangle on a plane, whose side $AB$, $BC$ and $CD$ are arcs of circles $S_1$, $S_2$ and $S_3$ passing though a point $D$ (i.e. $S_1∩S_2∩S_3 = D$, $D≠A$, $D≠B$, $D≠C$). Assume ...
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Möbius transformation mapping

I would like to understand how to choose the right Mobius transformation. For example, the Mobius transformation that maps upper half plane onto the unit disk is: $z \rightarrow \frac{z-i}{z+i}$ ...
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$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H$. What kind of points have non-trivial stabilizer? And how many orbits are there?

$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H= \{z \in \mathbb{C} | \Im(z) > 0 \}$ via Mobius transformation. $$ \text{ For } \gamma =\begin{bmatrix} a &b \\c&d \end{...
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Finding the norm of $w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$, where $w$ and $z$ are in $\mathbb{R}^n$

I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function $$M_w(z) = w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z \in \mathbb{R}^n$ and $|w| < 1$. (...
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Image of square under Mobius Transformation

Am i right that there is no Mobius transformation $$ h(z) $$ that sends a square to rectangle "A" with vertices in $$ 0, 2, i, i+2 $$because if we inscribe a cricle into square then it has 4 points $$ ...
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Find a center and radius of circle that is an image of Mobius Transformation of real axis

I need to find a center and radius of a circle that is an image of real axis under homography $$ h(z)= \frac{z-z_1}{z-z_2} $$ I found out that homography preserves symetric points, therefore ...
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Holomorphic bijection from intersection of two circles to a region between two rays

What is a holomorphic map from the nonempty intersection of two circles with "tip points" $a$ (below) and $b$ (neither of which are included in one another) to the region $A$ between two rays? Pictue ...
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Show that $ad-bc \ne 0$ for a composition of two Mobius transformations.

I could prove that a composition of two Mobius transformations is again a Mobius transformation. Let be $T(z)=\dfrac{a_1z+b_1}{c_1z+d_1}, \ (a_1d_1-c_1b_1 \ne 0)$ and $S(z)=\dfrac{a_2z+b_2}{c_2z+d_2},...
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Existance of an analytic function on unit disc

Is there an analytic function $f:B_1(0)\to B_1(0)$ such that $f(0)=1/2$ and $f^{\prime}(0)=3/4$? If it exists, is it unique? The answer to the first part of the question is affirmative. We can use ...
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Explicit calculation of the center of a circle, image of a circle by a Möbius transformation

It's a warm up calculation I decided to carry out while reading "PCT,Spin and statistics, and all that" by Streater and Wightmann. However I do not find what they have. p.79 within the proof of Thm 2-...
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Constructing Möbius transformation

In general my approach to construct a Möbius transformation $\varphi$ between two simply connected domains $G_1$ and $G_2$ is to take 3 points on each boundary and map them onto each other. The cross-...
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Are fractional linear transformations continuous?

I was reading, some answers about fractional linear transformations and find this old question that was never answer and I think is a nice question. How do you prove it? We define a Möbius ...
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Finding a Particular Möbius Transformation from $D \to D$

I'm well aware that the Möbius transformations that take the unit disk to itself, $f:D \to D$, are given by $$ f(z) = \frac{e^{i \theta}(z-\alpha)}{1-\bar{\alpha}z}, $$ where $\theta\in [0,2\pi)$ ...
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Determine the image of $S$ under $f(z)=\frac{z^2+2}{z^2+1}.$

Let $S$ be the region $\{z:0<|z|<\sqrt{2}, \ 0 < \text{arg}(z) < \pi/4\}$. Determine the image of $S$ under the transformation $$f(z)=\frac{z^2+2}{z^2+1}.$$ I'm facing some ...
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Fractional Linear Transformation of the Image of the Line $y=4-x$

I am trying to find the image of the line $y=4-x$ under the fractional linear transformation $$w=\frac{8}{z-2-2i}.$$ My method is as follows: Rearranging yields $$z=\frac{8}{w}+2+2i.$$ Now, \begin{...
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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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linear fractional transformation with $w(1)=i$ and $\arg w'(1)=\frac{\pi}3$.

Does there exist a linear fractional transformation $w$ such that maps the region $\{z;\Re z>0\}$ onto the region $\{w;\Im w>0\}$ in such a way that $w(1)=i$ and $\arg w'(1)=\frac{\pi}3$? I ...
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What are the transformations that preserve cross ratios on a sphere in higher dimensions?

If we have four points $x,y,z,w$ on a sphere, then the cross ratio is $\frac{|x-z|}{|x-w|}\frac{|y-w|}{|y-z|}$. If we consider $S^1 \subseteq \mathbb{C}$, then the transformations of $\mathbb{C}$ ...
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The cross ratio $ (z_1,z_2,z_3,z_4)$ is real iff the four points lie on a circle or a straight line

It's written in Alfors Complex Analysis that, for a proof of the above, "we need only show that the image of the real axis under any linear transformation us either a circle or a straight line. Indeed,...
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Reflection about a line as a möbius transformation

I am trying to find a matrix representation in Mat$_{2×2}(\mathbb C)$ for a reflection about a line $z=z(t) = a+bt$ where only $t$ is restricted to be in $\mathbb R$ as a parameter. I am thinking ...