# 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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### Holomorphic mappings of Riemann Sphere that preserves each hemisphere

What are some examples of holomorphic functions that satisfy the title? Especially for Mobius Transformation $f(z) = \frac{az+b}{cz+d}$ Is there a requirement on $ad-bc>0$ ?
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### Matrix representation of Möbius transformation from unit disc to upper half plane.

The task is as follows: Verify that the Möbius transformation $z \mapsto \frac{iz + i}{-z + 1}$ from the disc model of the hyperbolic plane to the upper half-plane model may be defined by an ...
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### A question about Möbius transformation and Beltrami forms

In the Page 170, Proposition 4.8.19 of John H. Hubbard's book "Teichmuller theory and applications to geometry, topology and dynamics", the author said: Let $A: P^{1}\rightarrow P^{1}$ be a ...
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### Describing Complex transformation

Consider the transformation $z \to \frac{2}{1-z}$ for $z \in \mathbb{C}$. Find and describe the image of the unit circle and the line $\{x = 1\}$ Attempt: For the unit circle, we know it has equation ...
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### Show that a linear fractional transform is a rotation

I have a linear fractional transformation given by: $$F_K (z) = \frac{\cos{\theta}z + i\sin{\theta}}{i\sin{\theta}z + \cos{\theta}}$$ And I am supposed to find the fixed points and then verify that ...
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### Find a conformal mapping $f$ that maps $\{z \in \mathbb{C}: Re(z) > 0\}$ onto itself that maps $f(1)=2$

Find a conformal mapping $f$ that maps $\{z \in \mathbb{C}: Re(z) > 0\}$ onto itself that maps $f(1)=2$. So we need to ensure $f(1)=2$ and that the right half plane maps to the right half plane. ...
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### Elementary proof that a function is a conformal automorphism of the disk

I would like to show that the function $$f=\frac{c-z}{1-\bar c z}$$ with $|c| < 1$ is holomorphic and an automorphism from the complex disk $\{z: |z|<1\}$ to itself? I'm looking for an ...
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### Hyperbolic Reflection of polygon

I'm working on visualizing the reflections of a polygon in the Poincaré disk along each side of it using SageMath. The figures below show the reflections of a polygon (a 4-gon and a 3-gon, the ...
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### Why the name linear fractional map?

Fractional linear transformation is a map from extended complex plane to itself, defined by: $$z\to \frac{az+b}{cz+d}$$ with $ad-bc\ne0$. Wikipedia says that "a linear ...
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### What is the intuition behind the classification of möbius transformation in hyperbolic, elliptic and parabolic transformations?

I am currently reading through Katok on fuchsian groups chapter 2.1. There the author introduces a classification of PSL(2, R) by the trace of the tranformation. It then says that hyperbolas are in ...
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### Plotting regions in complex plane [closed]

Plot the region represented by $\frac{π}{3}\leqslant \arg\left(\frac{z+1}{z-1}\right)\leqslant\frac{2π}{3}$ in the Argand plane. I know that such inequalities usually represent the minor or major arcs ...
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### Fractional Linear Transformations; # of Parameters.

A fractional linear transformation is a function $$w = f(z) = \frac{az+b}{cz+d}$$ where $a,b,c,d$ are complex constants satisfying $ad-bc \not = 0$. In the book Complex Analysis, by Gamelin, the ...
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### How to prove that the following composition of mappings from Riemann sphere to Riemann sphere is a möbius transformation?

I’m reading O.Foster’s «Lectures on Riemann surfaces» and trying to solve the edited version of exercise 1.3. O. Foster’s «Lectures on Riemann surfaces» Instead of proving that the mapping is ...
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### Find all fractional linear transformations

i) Find all the fractional transformations that map the circle $|z|<1$ onto the circle $|w|<1$ for which $w(a)=b, arg(w'(a))=\alpha, (|a|<1, |b|<1).$ ii) Find all the fractional linear ...
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### Finding the image of set under a Möbius transformation

A Möbius transformation maps the points $0$, $-i$, and $\infty$ to $10$, $5-5i$, and $5+5i$, respectively. The question is to find the image of the set $S = \{ z: \operatorname{Re}(z) < 0 \}$. I ...
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### Is this family of functions normal?

I wanna tackle the following exercise: Let $\mathcal{F}$ be the class of all functions $f \in \mathcal{H}(\mathbb{D})$ satisfying $f(0)=1$ and $\Re(f) > 0$. Show that $\mathcal{F}$ is a normal ...
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### Sharp bounding of a sum involving Möbius function

I am trying to bound as sharply as possible the partial sum $$S(n)=\sum_{k=1}^{\frac{n}{p_{\pi\left(\sqrt{n}\right)}}} \mu(k) \left(\pi\left(\frac{n}{k}\right) + f(n,k)\right)$$ Where $\pi(x)$ is the ...
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### Help: Exercises about Types of Mobius transformations

I'm working on the following exercise of Jones, Gareth A., and David Singerman. Complex functions: an algebraic and geometric viewpoint. Cambridge university press, 1987. Exercise: 2D Let $S$ be a ...
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### Exercise about beahviour of Mobius transformation

I'm working on the following exercise 2A of ...
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### Defining a Mobius transformation based on transformed points

If you have a Möbius transformation that is defined by the transformation of 3 points ,by example it maps points 1, −i, −1 to −i, 0, i, respectively, how is it possible to find the exact Möbius map ? ...
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### Quotient of an ideal triangle in the upper half-plane by a cyclic group.

Let $T$ be an ideal triangle in the Poincare upper half-plane $\mathbb{H}$ with the point $i$ as its "circumcenter" (by which I mean that the point $i$ is the center of symmetry of this ...
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### Automorphism of unit disc which maps two points on the boundary to $1$ and $-1$

Let $\mathbb{D}=\{ z \in \mathbb{C} : |z|<1\}$ and its boundary $\mathbb{T}=\{ z \in \mathbb{C} : |z|=1\}$. If $\lambda \ne \mu$ are two points in $\mathbb{T}$, then I need to show that there is an ...
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### Is $f(z)=z+\frac{1}{10}(1-z)^3$ a maps which sends the unit disk to itself?

I was currently viewing the paper Rigidity of Holomorphic Mappings and a New Schwarz Lemma at the Boundary, by Daniel M. Burns and Steven G. Krantz. In this paper they prove the following statement. ...
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### $\langle x,y \rangle, \ x,y \in \mathrm{PSL}(2,\mathbb{C})$ reducible $\iff$ $\text{tr}[x,y] = 2$

I don't understand this statement that I am trying to prove that can be found in this book (p.52): $\langle x,y \rangle, \ x,y \in \mathrm{PSL}(2,\mathbb{C})$ reducible $\iff$ $\text{tr}[x,y] = 2$ ...
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### How to find conformal map from half plane x<-y to circle

I don't quite get the procedure of finding a conformal map. For example the following exercise: Give a conformal map that pictures the half plane x<-y onto the disk with center 2i and radius 1. I ...
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### Conformal map from righthalfplane without line segment onto unit disk

For my problem I need to find a conformal map from $\Omega = \{ z \in \mathbb{C} \ | Re(z)>0\} \setminus [0,1]$ to $\mathbb{D} = \{z \in \mathbb{C} \ | \ |z|<1 \}$. I also need to map two to ...
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### Mobius Map from disk to itself that has no fixed points

If I have a Mobius map $f: \mathbb{D} \rightarrow \mathbb{D}$ that has no fixed points in the disk. Is it true that every orbit of $f$ escapes any compact set $K \subset \mathbb{D}$? If so, I'm a ...
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### Are Möbius transformations homotopic to the identity on the upper half-plane?

probably this question is totally easy and obvious but I am very confused at the moment. So assume we have a matrix $\gamma$ in $SL_2(\mathbb{R})$ acting on the usual upper half-plane $\mathcal{H}$ by ...
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### Looking for a concrete example of Hyperbolic groups

Background On page 232 of [1], the authors mentioned a few examples of Fuchsian groups that could be explicitly written down, namely, four cyclic groups; Hyperbolic, Parabolic, Elliptic, and Modular. ...
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### Decompose a Mobius transformation to understand what it does.

I understood that a Mobius transformation can be decomposed into a composition of elementary operations, e.g., rotation, dilation, and inversion. Here, I'm facing the following transformation to ...
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### Implementation of rotation by Mobius transformation

I've been stuck at implementing the rotation in the form of Mobius transformation (ie., $\frac{a z + b}{c z + d}$). Could somebody point me out where goes wrong?? ...
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Denote the unit circle with $\mathbb{D}$. Let $\zeta_1, \zeta_2 \in \partial\mathbb{D},\, \zeta_1\neq\zeta_2$ and $C=K_r(a)$ (circle radius $r$ around the point $a$), which intersects \partial\mathbb{... • 177 2 votes 1 answer 184 views ### Automorphisms on upper half plane I am supposed to prove that: \begin{align} Aut(\mathbb{H})= \left\{ T:\mathbb{H}\to \mathbb{H}, T(z)=\frac{az+b}{cz+d} \vert a,b,c,d\in\mathbb{R}; ad-bc>0 \right\} \end{align}. where\mathbb{H}$... • 177 1 vote 1 answer 109 views ### Inversion in a circle in the complex plane I'm self-studying a book about complex analysis and Riemann surface and would like to get some help with the following question. Background On page 28 of [2], the authors have derived an equation for ... • 696 2 votes 3 answers 84 views ### Möbius transformation with$B(-i) = i,B(i) = 2i$and maps real line to real line Given$B$Möbius transformation such that$B(-i) = i, B(i) = 2i$and$B(\{z\in \mathbb{C} : Im z = 0 \} \cup \{ \infty\}) = \{z\in \mathbb{C} : Im z = 0 \} \cup \{ \infty\}$Does such transformation ... • 13.5k 3 votes 2 answers 102 views ### Is the complex conjugation map a Mobius transformation? I have been asked to prove whether the complex conjugation map$z\mapsto \bar{z}$is a Mobius transformation. My solution was: Suppose$\bar{z}\in \mathcal{M}$, so that we can write$\bar{z} = \frac{...
I am wondering if any smooth bijection $f \in C^\infty(\mathbb{C})$ on $\mathbb{C}$ preserves the Hausdorff dimension of any given subset $A \subset \mathbb{C}$? In particular, I am working on ...