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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Do smooth bijections always preserve Hausdorff dimension?
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 ...
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Proving the Poisson integral formula and Schwarz formula on the unit disk using a Mobius transformation
I'm supposed to prove the Poisson integral formula $$f(z)=\frac{1}{2\pi i}\int_{\partial\mathbb{D}}\text{Re}\left(\frac{w+z}{w-z}\right)\frac{f(w)}{w}dw,$$ where $U$ is an open set such that $\...
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Biholomorphic mobius transformations preserving hyperbolic distances on the poincare disk
On wikipedia it says that mobius transformations of the form $$f(z)=e^{i\phi}\frac{z+b}{\bar{b}z+1}$$ where $b\in\mathbb{C},|b|<1$ and $\phi\in\mathbb{R}$ is a biholomprhic map $D\rightarrow D$, ...
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Image of interior of a unit disc under a complex transformation
If $w=T(z)=\sqrt{\frac{1-iz}{z-i}}$, then what is the image of $\{z : |z|<1\}$ under $T$ ?
I rewrote $T$ as $T(z)= e^{\frac{1}{2}log(\frac{1-iz}{z-i})}$, and found image of interior of unit disk ...
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Show that the action of PSL$_2(\mathbb Z)$ on a quadratic form by $g \cdot Q = Q(ax + by, cx + dy)$ preserves the set of properly represented numbers
I am trying to show that
the action of PSL$_2(\mathbb Z)$ on a quadratic form by $g \cdot Q = Q(ax + by, cx + dy)$ preserves the set of properly represented numbers, where
$$g = \begin{pmatrix}
...
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When do two translations in the hyperbolic plane commute?
If I'm correct, to define a hyperbolic translation you need an ideal point $p$ as a source, a different ideal point $q$ as a sink and a length $d \ne 0$ along the geodesic that joins $p$ and $q$. Let'...
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How can I draw the image of this fractional linear transformation?
Let me consider the map $$T:\Bbb{C}\setminus \{-i\}\rightarrow \Bbb{C}, ~z\mapsto \frac{z-i}{-iz+1}$$ I want to look at it as a map from the upper half plane to the unit disk, because there I know ...
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For mobius map why we need a invertible matrix of determinant $1$. [closed]
The map $T:\mathbb{C}_\infty\to \mathbb{C}_\infty $ defined by $T(z)=\frac{az+b}{cz+d}$ is called Mobius map . My question is why we may assume $ad-bc=1$ ? If $p\in \mathbb{C}-\{0\},$ then $\frac{...
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Given a Mobius transformation with real coefficients such that $T(0)=0$ and $T(2)=\infty$, what is $T^{-1}(\{iy : y \in \mathbb{R}\})$?
I took a final examination recently in a complex analysis course, and one of the questions still alludes me. I wrote it down after the exam to work on it later, but I'm not seeing how to do this. It ...
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Understanding the attractive fixed point of Mobius transformation
Book: An introduction to Teichmuller Space by Imayoshi & Taniguchi.
Let $\gamma$ be a Mobius transformation on $\Bbb C$ such that $\Bbb H\to\Bbb H$ (Automorphism of upper half plane). Suppose
$$\...
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Two complex numbers are symmetric with respect to a circle iff a certain equation is satisfied
Let
$ \gamma = ${$z \in \mathbb{C} : |z-a| = R$}. Two complex numbers $z_1,z_2$ are said to be symmetric with respect to $\gamma$ iff
$$ (z_1-a)\overline{(z_2-a)} = R^2. $$
I am trying to prove that ...
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Why are the Möbius transformations the conformal automorphisms of the Riemann sphere in the sense of Riemannian geometry?
There are two ways, how one can define a conformal structure on an Riemann surface. Either in terms of a complex structure, or in terms of a Riemannian metric.
I understand, that the Möbius ...
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Family of Moebius transforms with $A(1)=1$ reduces to a single map
I have a two parameter family of moebius transform $A_{q,p}$ with $q,p\in\mathbb{N}$ and
$$A_{q,p}=\begin{pmatrix}0& 2q \\ p&q\end{pmatrix}$$
Additionally, I assume that $A_{q,p}(1)=1$ for all ...
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Biholomorphic mapping of a quarter-disk to the upper half-plane
Can you verify this solution?
The question asks for a biholomorphic mapping of the domain $D=\{z \in \mathbb{C} : |z| < 1, \text{Re}(z) > 0, \text{Im}(z) > 0\}$ to the upper half-plane $P=\{z ...
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Which of the following is the bijective analytic function?
Let $D :=\{z\in \mathbb{C}:|z|<1\}$ and let $f \colon D\to D$ be a bijective analytic function such that $f(\frac{1}{2})=0$ then :
$f(z)=\frac{2z-1}{2-z}$
$f(z)=e^{i\theta}\big({\frac{2z-1}{2-z}}\...
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Characterization of the circles which are orthogonal to all the circles passing through two complex numbers
Let $\hat{\mathbb{C}}=\mathbb{C} \cup \left\{{ \infty}\right\}$ be the extended complex plane, let's define a circle in $\hat{\mathbb{C}}$ as a set $K$ which is whether an ordinary circle in $\mathbb{...
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Involution on triplets
Show that the map $\Phi$ on triplets
$$(a,b,c) \overset{\Phi}{\mapsto} \left(\frac{ (b+c) a - 2 b c}{2 a -(b+c)}, \frac{(a+c)b-2 a c}{2 b-(a+c)}, \frac{(a+b)c- 2 a b}{2 c-(a+b)}\right)$$
is an ...
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Domain and range of Mobius transformation
I read that the Mobius transformation $(z-i)/(z+i)$ is a biholomorphism from $\Re{z}>0$ to $B_1(0)$. How do I see this, and more generally, how do I determine the domains/ranges of general Mobius ...
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Mobius transformations map $\overline{\mathbb{R}}$ to $\overline{\mathbb{R}}$ if and only if we can choose its coefficients to be real (proof check)
I am wanting to prove that a Mobius transformation $T(z)=\frac{az+b}{cz+d}$ with $ad-bc \neq 0$ maps $\overline{\mathbb{R}}$ to $\overline{\mathbb{R}}$ if and only if one can choose the coefficients a,...
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Mobius Transform as functions of u(x,y) and v(x,y)
I am doing some plotting for my own interest of Mobius transforms, but my current system uses $u$ and $v$ axes where $u=u(x,y)$ and $v=v(x,y)$. I want to plot some Mobius transform f(z) as functions ...
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Show that for every $z,w \in \mathbb{C}$ there exist $a,b,c,d \in \mathbb{R}$, s.t. $\frac{az+b}{cz+d}=w$ with $ad-bc=1$ and Im(z)>0 and Im(w)>0
Show that for every $z,w \in \mathcal{H}$ there exist $a,b,c,d \in \mathbb{R}$, s.t. $\frac{az+b}{cz+d}=w$ with $ad-bc=1$.
This is just a small lemma in a bigger proof about a relation between ...
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Möbius transformation composition
Suppose we have
$F(z)=f(\phi(z))$ where $\phi$ is a mobius transformation which maps points of the unit circle to points of the unit circle. Suppose also that there is an interval of length $\pi$ such ...
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Find an example of $f \in PSL(2, \mathbb{C})$ that gives an isometry of $\mathbb{H}$ with Riemannian metric
I started studying on differential and projective geometry and I came across the following question on which I have diffuculties to solve/find examples:
Let $\mathbb{H}$ = { $z \in \mathbb{C}$ | $Im(z)...
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Moebius transformation on circles
I'm working on a program that draws the limit sets of Schottky groups defined by pairs of circles. If circle $C$ with radius $r$ is centered at $P$, then to map that circle to some other circle $C'$ ...
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Is complex conjugation a Möbius transformation?
This question is probably a little confused. I'm teaching myself some geometry of the hyperbolic plane, and this statement from the Wikipedia page on the Poincaré half-plane model is tripping me up:
...
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How to composite multiple Möbius transformations into one Möbius transformation?
We know that in two dimensions, two Möbius transformations performed one after another can be replaced with a single Möbius transformation that is the composite of those two, which can be easily found ...
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Does the QR/Iwasawa decomposition have any geometric meaning for $2 \times 2$ complex matrices?
It's well known that $2 \times 2$ complex matrices act on the complex plane (with a point at infinity) by $\begin{pmatrix}a & b \\ c & d \end{pmatrix}\cdot z = \frac{az + b}{cz + d}$.
Based on ...
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Constraint elements of a complex vector to unity
How could a describe a complex vector, in which the absolute value of the elements is all 1? It is not that the norm of the vector is 1, but the norm of all its elements.
For example:
$x = [ 1 e^{j20};...
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Find all Mobius transforms that preserve the real and imaginary axes
I need to find all Mobius transforms $f(z)=\frac{Az+B}{Cz+D},\text{ } A,B,C,D \in \mathbb{C}$ which map real numbers to real numbers and imaginary numbers to imaginary numbers. That is if $z \in \...
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Möbius transform which fixes real/imaginary axes must also transform $|z| = R_1$ to $|z| = R_2$ for $R_1,R_2 \in \mathbb{R}$
Suppose there exists $f(z) = \frac{az+b}{cz+d}, \quad a,b,c,d \in \mathbb{C}$ such that if $z_0=K \in \mathbb{R}$ then $f(z_0)\in \mathbb{R}$ AND if $z_0 = L \in i\mathbb{R}$ then $f(z_0) \in i\mathbb{...
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Find the Bilinear Transformation [closed]
Find the bilinear transformation which maps the points $2, i, -2$ into the points $1, i, -1$
This is what I did but I couldn't find a get through
$S(z)=\frac{az+c}{cz+d}$
So, $S(2)=1$ , $S(i)=i$ and $...
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Non Conformal Points of Joukowsky Transform
I am studying the use of the Joukowsky transform on the shape of airfoils. The transform is defined as
$$
f(\zeta) = \zeta + \frac{b^2}{\zeta}.
$$
I have found that $f$ is not conformal at $\zeta = \...
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which mobius transformation maps the unit disc onto te right half plane?
Which of the following mobius transformation maps the unit disc onto the right half plane.
a) $f(z)=\frac{z-i}{z+i}$
b)$f(z)=\frac{z-1}{z+1}$
c) $f(z)=\frac{1+z}{1-z}$
d)$f(z)=i\Big(\frac{1+z}{1-z}\...
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Linear transformation on $\mathbb{C}$ fixing the origin and preserving all distances [duplicate]
This is a question about problem 3 from Ahlfors section 3.1, which states
Prove that the most general transformation which leaves the origin
fixed and preserves all distances is either a rotation or ...
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Analytic function that maps upper half plane to upper half plane given two distinct values
Let $f$ be an analytic function that maps upper half plane to itself. Also, let $f(i)=i, \ f(2i)=\frac{i}{2}$. Then, what can be said about $|f(1+i)|$?
Now, by using the Schwarz-pick lemma and its ...
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Confusion about conformality of Möbius maps
I am aware that Möbius maps are conformal maps, that is, they preserve oriented angles. So I was thinking, say I have a circle in $C$. Then I can find a Möbius map that maps it to a a line (a circle ...
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Finding conformal map verification
I would like to know if this reasoning is correct:
I want to find a conformal map from $\mathbb D$ to $\mathbb C$.
My reasoning:
First we seek a conformal map from $\mathbb H$ to $\mathbb D$ which is ...
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Image of a line under Mobius transformation
Let $T$ be a Mobius transformation such that $T(0)=\alpha$, $T(\alpha)=0$ and $T(\infty)=-\alpha$, where $\alpha = (-1+i)/\sqrt 2$. Let $L$ dnotes the straight line passing through origin with slope $-...
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Show that $R(N) = \sum_{d|N} \mu(d) * (N/d)^2$ where $\mu$ denotes the Mobius function
I came across the following advanced question in the textbook:
Let $R(N)$ denote the number of ordered pairs $(n, m) \in [N]^2$ such that $n, m, N$ are relatively prime.
I have to show that $R(N) = \...
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Image of Upper Unit Semi Circle under Joukowsky Transformation
I'm trying to understand how the Joukowski Transformation would map the following region:
$$\{z|0<arg(z)<\pi , |z|<1\}$$
with the Joukowski Transformation being : $w = \frac{1}{2}(z+\frac{1}{...
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How to prove that Linear-Fractional Transformations map circles/straight lines onto circles/straight lines?
There this theorem that states the Linear Fractional Transformations map circles and straight lines to circles and straight lines. How can you prove this?
This is what the theorem states in my lecture ...
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1
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Finding a Möbius transformation, why do I need to expand area?
I'm currently studying for an re-exam in complex analysis, and got a question regarding Möbius transformation.
The exam-question is following:
Find a conform and bijective mapping from $A := ${$ z: 0 &...
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Two non-identity elements in PSL(2,C) commute iff
I know non-identity 2 elements in $PSL(2,\mathbb{R})$ commutes iff they have the same fixed points in $\hat{\mathbb{C}}$.
But for $PSL(2,\mathbb{C})=Isom^{+}(\mathbb{H}^3)$, it seems a bit tricky for ...
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1
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How to Identify Adjoint Orbit of $\mathfrak{s}\mathfrak{u}(2)$ with Riemann Sphere?
The adjoint orbits of $SU(2)$ acting on $\mathfrak{s}\mathfrak{u}(2)$ are given by $$\begin{pmatrix}
ia &z\\
-\bar{z}+iy &-ia\\
\end{pmatrix}$$
where $a^2+|z|^2=\text{constant}\,,$ where $a\in\...
- 6,047
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Show that Möbius functions can be uniquely decomposed as $f=R\circ L \circ T$
A Möbius map is a rational function of the form $$z\mapsto \frac{az+b}{cz+d} \,,$$ where $ad-bc\ne0$ and $a,b,c,d \in \mathbb{C}$.
Show that Möbius functions can be uniquely decomposed as $f=R\circ L \...
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Action of $GL(2,\mathbb{Z})$ on lower half plane of $\mathbb{C}$?
I began reading about modular forms and I had a question. So, I know that $SL(2,\mathbb{Z})$ is mapped to the upper half complex plane using a function. The way I see it, the reason we map it to the ...
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Möbius transformation mapping $\mathfrak{J}(z)>0$ to itself [duplicate]
I got following proof for "Search for all Möbius transformations that map the upper half plane to itself bijectively"
Proof:
Define
\begin{align*}
&M(z) = \frac{az+b}{cz+d} \\
&M(0) ...
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1
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Mobius transformation for circles
Let S1, S2, S3 be three nonintersecting circles whose centers do not belong
to one line. Prove that there is a unique circle S orthogonal to S1, S2, S3.
I wanted to use a definition of the Mobius ...
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1
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How to find conformal maps - Complex analysis [closed]
I would like to know how one can find a conformal map that maps a given set onto another (not $\mathbb D\rightarrow\mathbb D$ or $\mathbb H\rightarrow \mathbb H$ since those are clear).
For example, ...
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1
answer
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How can a general automorphism of the unit disc be parabolic or hyperbolic?
I am trying to work through Needham's Visual Complex Analysis on my own. We are told that the general automorphism of the unit disc is given by the Mobius transformation $M_a^{\phi}(z)=e^{i\phi}\left(\...
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