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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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Determine the Mobius transform from $\infty$ to $0$, $0$ to $-i$ and $i$ to $\infty$

Determine the Mobius transform from $\infty$ to $0$, $0$ to $-i$ and $i$ to $\infty$ I have done the following: 1) $$\lim \limits_{z\to \infty} \frac{az+b}{cz+d}=0$$ $$\iff \lim_{z\to 0} \frac{a\...
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Show the Dedekind $\eta(\tau)$ function is a modular form of weight $\frac{1}{2}$ for $\Gamma_0(6)$

Given that $\Gamma_0(6)$ is generated by matrices in the set $$G = \left\{\left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right), \left(\begin{array}{cc} 7 & -3\\ 12 & -5 \end{array}...
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Prove: If f is a bijective meromorphic function from the Riemann sphere to itself, then f is a Mobius transformation. [duplicate]

Knowing that a meromorphic function f requires f to be either analytic or has a pole at every point in its domain, what does bijective meromorphic mean?
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If points of mobius transformation are given, then how to determine the mapping?

A Mobius transformation is a map $$f(x)=\frac{rx+s}{tx+u}$$ where $ru-st \neq 0$. Suppose we have $f(a)=c, f(b)=d, f(c)=a, f(d)=b$, where $a,b,c,d \in \mathbb{R}$. Then from here, the answer given by ...
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4answers
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Constructing a Fractional Linear Map

I am working on a practice prelim question: "Construct a nonlinear fractional map $\phi(z) = \frac{az+b}{cz+d}$ with $c \ne 0$ such that $\phi(\phi(\phi(z))) = z$. I feel like I just need to take ...
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1answer
270 views

Constructing a Mobius transformation that acts on any two points of the upper half complex plane:

I would like to construct a Mobius transformation that sends any two points $z_1$ and $z_2$ from the upper half of the complex plane to i and to $iR^+$, i.e., given any two points $z_1$ and $z_2$, ...
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Action of $SL_2(p)$ on intgers mod p by Möbius transformations

This is simple and may have been asked before, but I couldn't find it. I have been asked to 'Define the action of $SL_2(p)$ (the group of 2 by 2 matrices of determinant 1 with entries in $\mathbb{F_p}...
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1answer
349 views

Is there a measure invariant with respect to the Möbius transformation?

I would like to use a measure ${\rm d} \mu (z)$ on ${\mathbb C}$ so that for any $f(z)$ $$\int_{\mathbb C} f(z) {\rm d} \mu (z)$$ is invariant under Möbius transformations. Taking the ...
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1answer
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Image of Möbius transformation

What's the image of the first quadrant $Rez\ge0$ and $Imz\ge0$ under transformation $f(z)=(i-z)/(i+z)$? I know that real axis is mapped to the unit circle, $f(0+i*0)=1$ and $f(\infty)=-1 $.
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An analytic function on the disk that sends the boundary into the boundary sends the interior onto the interior

Consider $f$ analytic on $B(0; 1) = \{z \in \mathbb{C}\ |\ |z| < 1\}$ and continuous in $D(0; 1) = \{z \in \mathbb{C}\ |\ |z| \leq 1\}$ such that $f(S^1) \subset S^1$. I must show that $f(B(0; 1)) =...
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Finite groups of Möbius Transformations

Let $M_2(\mathbb{C})$ be the group of all Möbius transformations $z\mapsto \frac{az+b}{cz+d}$ from $\mathbb{C}\cup\{ \infty\}$ to itself. Let $PSU(2,\mathbb{C})$ be the group of all Möbius ...
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1answer
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The image of a specific Mobius transformation

Let $f:D\rightarrow \mathbb{C} :f(z)=\frac{z}{z-1}$ and $D= \{ z:|z|=1\}$ ,what is the image of $f$, $f(D)$? Can one elaborate on some general methods of dealing with these kind of questions?
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what is a Möbius transformation with no fixed points?

is there a Mobius transformation with no fixed points? I have the equation $$\frac{az+b}{cz+d}=z\implies cz^2+dz-az-b=0$$ given the fixed points when this is true. So if we set $c\ne 0$ we get two ...
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Is this a Möbius transformation? Line $\mapsto$ hyperbola

I was trying to make a Möbius transformation $M:\Bbb C \to \Bbb C$ with $2$ fixed points. What I did was: Make a degree 2 formula $\frac{az+b}{cz+d}=cz^2+dz-az-b=0 $ so $c\ne 0$ and choose $a,b,c,d$...
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Mobius transformation always defined on $f:\Bbb C \to \Bbb C$ or always $f:\hat {\Bbb C} \to \hat {\Bbb C}$

Is a mobius transformation ever defined on $f:\Bbb C \to \Bbb C$ or is it always $f:\hat {\Bbb C} \to \hat {\Bbb C}$? The wikipedia page makes me believe it is the latter, but my assignment has the ...
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Möbius transform answer check for $0$ to $2$,$-2i$ to $0$, $i$ to $\frac32$

Continuation of this question Is this the correct answer for the möbius transformation corresponding to: $0$ to $2$ $-2i$ to $0$ $i$ to $\frac32$ $$\frac{az+b}{cz+d}\cong \frac{az+2azi}{az+ai}=1+\...
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Mobius transformation $0$ to $2$, $-2i$ to $0$ and $i$ to $\frac32$.

I want to determine the Mobius tranformation mapping $0$ to $2$, $-2i$ to $0$ and $i$ to $\frac32$. I don't know how to do this at all. I am fairly sure what I want is a composition of easier ...
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1answer
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Mobius transformation satisfying certain properties

I'm having some trouble showing that a Mobius transformation $F$ maps $0$ to $\infty$ and $\infty$ to $0$ iff $F(z)=dz^{-1}$ for some $d \in \mathbb{C}.$ Mainly with the "only if" part. Do I need to ...
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2answers
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Mobius transformation from region between two intersecting circles to an annulus

I'm trying to find a Mobius transformation from from the region between the circles $|z|=1$ and $|z+1| = \frac {4}{\sqrt(3)}$ to an annulus. I've tried to find three points in the original region that ...
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643 views

Map that sends a $2x2$ matrix to a Mobius transformation is a homomorphism

The set of all Mobius transformations is a group of homeomorphisms of $\mathbb{C} \cup {\infty}$ onto itself. All Mobius transformations have inverses. How can I show the map that sends the $2x2$ ...
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Finding a Möbius Transformation along a particular contour

Let $C$ be the circle with center $0$ and radius $1$. Find a Möbius transformation which transforms $C$ onto $C$ and transforms $0$ to $1/2$. Notes: consider $h(z)= az+\dfrac{b}{c}z+d$ then $h(0) = 1/...
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1answer
106 views

Unique circle through two points perpendicular to a given line?

Suppose I have a line in the plane, and two points not on the line. How can I prove that there is a unique generalized circle (i.e. circle or line) passing through the two points which intersects the ...
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896 views

Show that Möbius transformation $S$ commute with $T$ if $S$ and $T$ have the same fixed point.

Let $T$ be a Möbius Transformation such that $T$ is not the identity. Show that Möbius transformation $S$ commute with $T$ if $S$ and $T$ have the same fixed point. Here is what I know so far 1) if ...
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1answer
637 views

Möbius Transformation Real Axis to Real Axis.

Prove that a Möbius Transformation maps the real axis to the real axis iff the coefficients of the Möbius Transformation are real. If I assume the coefficients are real and take the $3$ points on the ...
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How do I find a holomorphic function that maps $z \in C : Re(z) > 0$ to $z \in C : Re(z) > 0$ $z\notin [0,1].$

How do I find a holomorphic function that maps $z \in C : Re(z) > 0$ to $z \in C : Re(z) > 0$ $z\notin [0,1].$ I know that I have to use the reverse Cayley transform to work with the unit disk. ...
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1answer
212 views

How to construct Möbius map to tranform so 2D region to another?

I want to know what is the general step-by-step method to find a Möbius map $w(x) = \dfrac{ax+b}{cx+d}$ that transforms a region in a complex plane to another region (2D). I know it takes circle and ...
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1answer
543 views

Show that Möbius transformations that preserve the unit disk are of the matrix form $\tiny \begin{bmatrix}a & b \\ \bar{b} & \bar{a} \end{bmatrix}$

Show that Möbius transformations that preserve the unit disk are of the matrix form $$\begin{bmatrix}a & b \\ \bar{b} & \bar{a} \end{bmatrix},$$ where $|a|^2 - |b|^2 = 1$ and $a,b \in \mathbb{...
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All rotations about a point in the complex plane also Möbius maps

Is my claim true? If so, is my "proof" correct? Claim: All rotations about a point in the complex plane also Möbius maps. Proof: We have shown that the set of Möbius maps is a group. So we can do the ...
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1answer
374 views

Hyperbolic Geometry: Question about the Transitivity of Möbius transformations

I was confronted with this exercise in the book Hyperbolic Geometry by Anderson which states: In each case, find $m \in Möb(\mathbb{H})$ such that the property holds, or prove that no such $m$ ...
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570 views

What is unique about the Möbius transform?

... is it the only map to accomplish a transformation in 2D and keep certain characteristics invariant? Which? What else makes it special to be studied so much?
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1answer
217 views

On the matrix representation of a composition of Möbius transforms

Let the Möbius transform associated to the matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ be defined as $\mu_A:\mathbb C\to\mathbb C:z\mapsto\frac{az+b}{cz+d}$ provided $\det A\neq 0$. It is ...
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Poincare disk and Poincare half plane

My book claims that the Möbius transforms are isometries of the Poincaré half plane model. Thus, the metric is preserved under these maps. But I know that the Poincaré disk can be derived from ...
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Action of Möbius Group

By a circle in $\mathbb{C}\cup \{\infty\}$, we mean a circle in plane $\mathbb{C}$ or a straight in $\mathbb{C}$ union with $\infty$. Given two circles in $\mathbb{C}\cup \{ \infty\}$, does there ...
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What conditions are necessary on $a,b,c,d$ so that the Möbius transformation $w=\frac{az-b}{cz-d}$ has only one fixed point?

Question: What conditions are necessary on $a,b,c,d$ so that the Möbius transformation $w=\frac{az-b}{cz-d}$ has only one fixed point? Attempt: We examine $$ z=\frac{az-b}{cz-d}$$ to find that any ...
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Möbius transformation that preservers the unit circle

How do I show that each Möbius transformation that preservers the open unit circle (maps it to itself) must be of the form: $c \frac{z-z_0}{\bar{z_0}z-1}, |c|=1, |z_0|<1$ ? I've seen previous ...
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Why möbius transformation is isomorphic to projective linear group?

I saw on my complex analysis book that linear fractional transformation is isomorphic to the group of invertable 2x2 matrix such that identify scalar multiplication. Verifying that was easy but I ...
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892 views

Möbius Tranformation: Map two non intersecting circles to concentric circles

Show that two non intersecting circles can always be mapped by a suitable Möbius transformation to two concentric circles. I wanted to map the center of the first circle to the center of the second ...
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240 views

Find a Mobius transformation $f$ that maps $\mathbb{H}=\{z \in \mathbb{C}:Im(z) >0\}$ bijectively to ball $B(0,2)$ such that $f(i)=1, f(1)=-2 \ ?$

Could anyone advise me on this problem: Find a Möbius transformation $f$ that maps $\mathbb{H}=\{z \in \mathbb{C}:\text{Im}(z) >0\}$ bijectively to ball $B(0,2)$ such that $f(i)=1, f(1)=-2 \ ?$ ...
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1answer
784 views

Möbius Transformation interchanging two preassigned points in the upper half plane

Is there a Möbius transformation mapping the upper half plane onto itself that interchange two preassigned points in the upper half plane? If so , how many such Möbius transformations are there? ...
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Möbius transformation of Algebraic curve

I am working on the uniformization of algebraic curve problem. Currently, my adviser gave me a question about building a Möbius transformation between algebraic curves, and then lifting it to the ...
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65 views

Question about a Möbius transformations

Suppose $\text{Im}(f(z)) < M$ for some analytic $f: \Omega - \{a\} \rightarrow \mathbb{C}$. Then why does the Möbius transformation $$ g(z) \mapsto {iMf(z) \over 2iM - f(z)} $$ map $z$ into the ...
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161 views

Möbius transforms with a common fixed point

Let $f,g$ be two Möbius transformations with a common fixed point $z_0$. Show that the Möbius transformation $f \circ g \circ f^{-1} \circ g^{-1} $ is either parabolic or the identity. Möbius ...
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finding fix points of Möbius transformation

I need to find the fixed points of the following: dilation on $\mathbb{C}_{\infty}$ translation on $\mathbb{C}_{\infty}$ inversion on $\mathbb{C}_{\infty}$ I'm thinking using Möbius ...
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1answer
124 views

Soft Question about Möbius Transformations

Very soft question and I may be completely wrong about this, but does it make any sense to think about the Möbius transformation matrix as a change of basis for $\mathbb C$?
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1answer
928 views

Showing that a Möbius transformation exists

I'm trying to show that a specific Möbius transformation exists, where I have some points that map to some other points. (I don't wanna be too specific here about what goes where, as I don't wanna run ...
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94 views

Möbius transformation of the complex plane

Let $\phi_{\alpha}(z)=\frac{z-\alpha}{1-\bar{\alpha}z}$ for $0<|\alpha|<1$ Find all the line $L$ in the complex plane such that $\phi_{\alpha} (L)=L$ Can you help me?
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Progression of a point along geodesics under the action of hyperbolic Möbius transformations

Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model. Suppose also that the angle between ...
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2answers
218 views

Univalent Möbius transformation

Let $f(z)=\frac{(az+b)}{(cz+d)}$ be a Möbius transformation where $c \neq0$. Through the process of actually computing $f^{-1}$ show that $f$ is a univalent function whose domain-set is $\mathbb{C} \...
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1answer
114 views

Coefficients of rational involution.

Question: (Spivak Calulus 3rd, Chapter 3, Problem 8) For which numbers $a,b,c,d$ will the function $$f(x) = \frac{ax + d}{cx + b}$$ satisfy $f(f(x)) = x$ for all $x$? Attempt at an answer: I ...
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1answer
274 views

Counting Fractional Linear Transformations

This is a problem from Ahlfors' Complex analysis, Section 3.5. In an obvious way, which we shall not try to make precise, a family of [Möbius] transformations depends on a certain number of real ...