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.

Filter by
Sorted by
Tagged with
2
votes
1answer
47 views

Partial derivatives of a complex function?

In real algebra, if I have a differentiable function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, say $f(x,y)=[u,v]$, I can calculate four different partial derivatives $\frac{\partial u}{\partial x},\...
1
vote
1answer
49 views

Presentation for a hyperbolic group with 2-sphere boundary.

I am looking for examples of hyperbolic groups that have boundary homeomorphic to the 2-sphere, $S^2$. I would like an explicit presentation of such a group so that I can draw its Cayley graph and ...
0
votes
4answers
74 views

If $f(x)=\frac{1-x}{1+x}$, then how do I explain graphically why $f(f(x))=x$?

Let $f(x)=\dfrac{1-x}{1+x}$. Then $f(f(x))=x$. The domain of $f(f(x))$ is $\mathbb{R}$\ ${-1}$ How do I explain graphically why the formula for $f(f(x))$ is the way it is?
2
votes
2answers
52 views

Möbius transformations of finite order

I saw this question, which brought up the question: can we classify all the Möbius transformations (with complex coefficients) of finite order? In particular, do these only consist of the rotations? I ...
1
vote
0answers
33 views

Optimizing a Möbius transformation for more than three points?

Assume that I have a set of $N$ complex numbers $\{z_1,...,z_N\}$ with corresponding images $\{w_1,...,w_N\}$. A Möbius transformation can be defined by specifying three points and their images. If $N&...
1
vote
1answer
28 views

Finding all Möbius transformations that fix $0$ and $1$.

I need to find all Möbius transformations that fix $0$ and $1$. I'd like to know if my proof is correct: I used the fact that for any three points $z_1, z_2, z_3 \in \Bbb C$ there is a unique Möbius ...
2
votes
3answers
63 views

Deriving a Möbius transformation specified by three points

A Möbius transformation is given by $$f(z)=\frac{az+b}{cz+d}$$ with parameters $a$, $b$, $c$, and $d$. The Wikipedia article provides rules for finding these parameters based on three points $z_1$, $...
1
vote
0answers
49 views

Finding the standard form of a Möbius transformation

Consider the Möbius transformation $m(z)=\dfrac{(az+b)}{(cz+d)}$ where $a,b,c,d$ are complex numbers such that $ad-bc$ is nonzero and $m$ is normalized so that $ad-bc=1.$ Assuming that m is elliptic, ...
0
votes
2answers
21 views

Equality between Möbius transformations at the same argument

Let $f(z)=(az+b)/(cz+d)$ and $g(z)=az/(ez+f)$ be two Möbius transformations, with $a,b,c,d,e,f$ real numbers (note the same coefficient $a$ in $f$ and $g$) and with $cz+d$ and $ez+f$ non constant. My ...
1
vote
1answer
50 views

To show that Möbius transformation is holomorphic on $\Bbb C_{\infty}$

A function $f: \Bbb C_{\infty}\to \Bbb C_{\infty}$ is biholomorphic if $f$ is bijective, holomorphic and $f^{-1}$ is holomorphic on $\Bbb C_{\infty}$, where $\Bbb C_{\infty}$ is the one point ...
0
votes
1answer
52 views

What is the Möbius transformation that maps $\mathbb{C}\setminus]-\infty;0]$ onto the unit disk?

I'm studying complex calculus and one conclusion of the Riemann theorem is that the domain where the complex natural logarithm is holomorphic, meaning $\mathbb{C}\setminus\mathbb{R}_-$, can be mapped ...
1
vote
0answers
26 views

Describe the image of a complex function defined by line integral.

I was asked to describe the image of the function $f:\mathbb{H}\to \mathbb{C}$ defined by: $$f(z)=\int_{i}^z\frac{dw}{(w-1)^a(w+1) ^b}$$ where $a,b \in (0,1)$ and $a+b<1$. This is from a past ...
0
votes
0answers
53 views

A problem on Schwarz Lemma [duplicate]

I am currently solving problems on Schwarz's Lemma from Conway's book and stumbled in one problem (prob-3,Page-133). Let $D=\{z\in \mathbb{C} \mid |z|<1\}$ be the open unit disk and $f:D\to \mathbb{...
1
vote
0answers
25 views

Möbius transformation in a boundary value problem

I have a problem understanding the following usage of the Möbius transformation. Given: $ T(z)=\frac{z}{z-i} $ $T$ maps $ \{ z \in \mathbb{c} : |z-\frac{4i}{3}|= \frac{3}{2} \} $ onto the circle $ \{ ...
0
votes
2answers
24 views

Finding the Möbius transformation when $z= \infty$

The following is available: $ T(2i) = \infty $ $ T(0) = -i $ $ T(\infty) = i $ So I've got: $ \frac{a(2i)+b}{c(2i)+d} = \infty \Rightarrow d=-2ic $ $ \frac{b}{d}=i \Rightarrow b = -2c $ $ \frac{a \...
0
votes
0answers
23 views

Showing certain regions are conformal to the unit disk.

I need to show the following regions are conformal to the unit disk, I think i found the maps but I need to make sure they are correct, and if my methods of finding them works. Intersection of $|z-i|&...
2
votes
2answers
74 views

Möbius transformation from disk to itself defined defined by interior points?

We can find a unique Möbius transformation from the unit disk to itself by specifying three points and their images on the unit circle of the $z$ and $w$-plane, respectively, to find the ...
1
vote
1answer
53 views

Can we characterize the Möbius transformations that map the unit circle into the unit disk?

The Möbius transformations are the maps of the form $$ f(z)= \frac{az+b}{cz+d}.$$ Can we characterize the Möbius transformations that map the unit circle $\{z\in \mathbb C: |z| = 1\}$ into the (closed)...
2
votes
3answers
49 views

Reference request: parameters of Möbius transformation by three points and images?

I am interesting in finding the four parameters of a Möbius transformation given three complex points $z_1$, $z_2$, and $z_3$, and their images $w_1$, $w_2$, and $w_3$. Wikipedia provides the ...
0
votes
0answers
26 views

For any $z_0$ in the unit disk $\mathbb{D}$, the Möbius map $\varphi_{z_0}: z \mapsto \frac{z_0 - z}{1 - z\bar{z_0}}$ is a conformal map…

...from $\mathbb{D}$ to $\mathbb{D}$. In the lecture our instructor shows several properties of the map $\varphi_{z_0}$, such as it maps $\mathbb{D}$ to $\mathbb{D}$, it is onto and injective, and $\...
2
votes
2answers
63 views

KAK decomposition of $SL(2,\mathbb{R})$

I am trying to understand the proof of KAK decomposition of $SL(2,\mathbb{R})$ from the following text: I am a bit confused about the rotation around $i$ part. How exactly do we know what amount of ...
9
votes
1answer
204 views

Combinatorial interpretation of the identity $(f \circ f \circ f)(x) = x$ where $f(x) = 1/(1-x)$ for $x\in(-1,1)$

Let $f(x) = 1/(1-x)$. We can interpret this as the generating function of an infinite list of $1$s: $(1, 1, 1, \cdots)$. Now, let's consider $(f \circ f \circ f)(x)$. We first compute $(f \circ f)(x)$...
0
votes
2answers
36 views

The mobious transformation of ${z\over 2z-8}$ from circle $|z-2|=2$

through mobious transformation $f(z)$=${z\over 2z-8}$ a have to know what is the image from circle $K=\{z:|z-2|=2\}$. I know that transformation is injective and onto. So I try this: $w\in f(K) ...
2
votes
1answer
48 views

Is there any way to represent any function on a z=0 plane transform into an arbitrary differentiable continuous non-linear surface?

Question Let's assume that there is a given line or a given arbitrary function defined on a $z=0$ plane. For example, $x^2+y^2=1$ Now I twist the plane into a non-linear 3D surface that can be ...
2
votes
2answers
43 views

Find the image of D on Möbius transformation.

I am not sure how to solve the following exercise on Möbius transformations: Let $D=\{z:|z-1|\le\sqrt{2}\wedge|z+1|\le\sqrt{2}\}$ and $f(z)=\frac{-2}{z+i}$. Find the image of the set D through the ...
1
vote
1answer
86 views

Möbius transformation that maps the unit circle to itself

I need to find necessary and sufficient conditions on the coefficients of a Möbius transform $T(z)=\frac{\tilde a z+\tilde b}{\tilde c z+ \tilde d}$ so that it maps the unit circle $\{z: |z|=1\}$ into ...
0
votes
1answer
25 views

How to write a matrix from $SU(2)$ in terms of one angle and one complex number $z$ , where $z$ is from sphere $S^{2}$

For given a matrix from $SU(2)$ , how can represent it in terms of two parameters: one angle and one complex number $z$ from the sphere $S^{2}$ ? Does this have any links with : $\mathrm{SU}(2)$ axis ...
0
votes
0answers
46 views

Möbius Transformation from a disk into a specific region

I am having a problem with a Complex Analysis question. It goes as follows: Find a general Möbius Transformation $w=Tz$ that maps $\mathbb{A} = \{ z : |z-a|\leq R \}$ into $\mathbb{B} = \{ w : Re(w) ...
0
votes
2answers
52 views

Möbius Transformation from a unit disk to the upper half plane

Consider the unit-disk $\mathbb{D} = \{ z : |z|\leq 1 \}$. I need to find a Möbius Transformation $w=Tz$ that maps $\mathbb{D}$ to the upper-half plane $\mathbb{H} = \{ w : Im(w) \geq 0\}$. I have ...
0
votes
0answers
19 views

$f$ is analytic in $|z|\leq 1$ and $|f|=1$ when $|z|=1$ then $f$ is rational [duplicate]

This is a problem from Ahlfors' Complex Analysis "$f$ is analytic in $|z|\leq 1$ and $|f|=1$ when $|z|=1$ then $f$ is rational" This is in the section of Reflection Principle, but I don't know what'...
0
votes
1answer
93 views

Show u(r, θ) is a solution to the Dirichlet Problem for the unit disk

Show that $u(r,\theta) = \frac{1}{\pi}\arctan\left(\frac{1-x^2-y^2}{(x-1)^2+(y-1)^2-1}\right)\\$ where $\arctan(t) \in [0,\pi]$ is the solution to Dirichlet's problem for a unit disk for the piecewise ...
0
votes
0answers
32 views

A mobius transformation and inversion confusion

Consider the Mobius transformation given by $$f(z)=\frac{z+i}{z-i}$$ I think this maps the unit disc to the left-half plane, since $f(1)=i$, $f(i)= \infty$ , $f(-i)=0$, so the boundary is mapped to ...
0
votes
1answer
22 views

Determining a Möbius transformation image [closed]

so I was attempting a question on Möbius transformations and I've encountered a problem in my workings. The question is "Determine the image of the strip $-1 < \Re(z) < 1$ under $g(z) = (iz+1)/(...
0
votes
1answer
25 views

Where does exponential function $f=e^z$ map lower half plane?

Given exponential function $w=e^z$ , map the lower half plane. Can someone please help me what is the image of this function? I know that I have to write $z=x+iy$ , so for $w=u+iv$ and $w=e^x(cosy+...
0
votes
0answers
36 views

Fixed points of automorphism of unit disk

I'm reading the book 'A Course in Complex Analysis and Riemann Surfaces' by Wilhelm Schlag but I'm stuck at the following statement in section 4.8: Groups of Möbius transformations. We have that an ...
0
votes
0answers
39 views

For given eight(8) points : $a,b,c,d,e,f,g,h$ on unit circle, does always a Möbius transformation w that PRESERVES UNIT DISC described below exist?

For given eight(8) points : $a,b,c,d,e,f,g,h$ on unit circle(boundary of unit disc), does always a Möbius transformation w that PRESERVES UNIT DISC so that $w(a)=e,w(b)=f,w(c)=g,w(d)=h$ exist? I ...
0
votes
1answer
34 views

How to prove uniqueness?

For the question asked in Is a Möbius transformation that PRESERVES UNIT DISC uniquely determined by three distinct points and their images? , how can I prove the UNIQUENESS of those ...
1
vote
1answer
39 views

Is a Möbius transformation that PRESERVES UNIT DISC uniquely determined by three distinct points and their images?

If I have given four different points in a disc : $a,b,c,d$ Does always exist a Möbius transformation $w$ which PRESERVES UNIT DISC , so that $w(a)=c$ and $w(b)=d$? If yes, is it unique? If given 6 ...
1
vote
1answer
44 views

Does there exist any Möbius transformation that preserves upper unit disc?

I know how to find Möbius transformations that preserve unit disc.Can I link my question with that and how? Or maybe Möbius transformations that fix upper unit disc don't exist(exept identity)?Why?
3
votes
1answer
32 views

What is the density of distribution which is obtained by acting with a Mobius transformation on the unit disc with uniform distribuition?

So, I have given a Mobius transformation that preserves unit disc. In unit disc I have uniform distribution (distribution that has constant density). I have to act with this Mobius transformation on ...
0
votes
1answer
15 views

Inverse of Möbius Transform in $\mathbb{R}^n$

I'm having a little trouble computing the inverse of the Möbius transform in $\mathbb{R^n}$, as outlined here in "higher dimensions". I assume it exists because it goes on to say that it forms a group....
0
votes
1answer
57 views

Does function $f(z)=-i\sqrt{z}$ map unit disc to upper disc?

I think that $\sqrt{z}$ is not defined on $[-1,0]$ , but this is the function I got as a result in problem to map conformally unit disc on upper unit disc... Can anybody help me?
1
vote
0answers
36 views

Conformal transformation

This is a problem in Volkovyskii's book. I have a little difficulty finding the right result. Who can show me how to find the conformal mapping how affects the exterior of the parabola $y^2=2px,\,\,(...
1
vote
1answer
78 views

Characterization of Fuchsian groups containing hyperbolic elements

I want to find the Fuchsian groups that acts on the upper half plane $\mathbb{H}$ to give $n$-holed torus $\mathbb{T_n}$. I am following the book Fuschian Groups by Svetlana Katok. There's this ...
2
votes
1answer
114 views

Find Mobius transformations $\varphi$ satisfying $\sum (1-|\varphi_n(z)|)<\infty$ in the unit disc.

Suppose that $\varphi$ is a Mobius transformation which maps the unit disc onto itself. Let $\varphi_n(z)=\varphi(\varphi_{n-1}(z))$, where $n=1,2,\ldots$ and $\varphi_0(z)=z$. Find all $\varphi$ ...
0
votes
0answers
33 views

Cross ratio on $T(z)=(z,2i;2,-2)$ and $S(w)=(w,-1;2i,1+4i)$

Cross ratio on $T(z)=(z,2i;2,-2)$ and $S(w)=(w,-1;2i,1+4i)$. I am using these to find a Mobius transformation that takes the circle $|z|=2$ to the line $2x-y=2$. I want to map the points $u=2i, v=2$, ...
1
vote
1answer
96 views

Bounded analytic function in $D$ invariant under a Mobius transformation

Suppose that $\varphi(z)=\frac{az+b}{cz+d}$ is a Mobius transformation. What is the form of $\varphi$ such that one can have a bounded analytic function $f$ in the unit disc with \begin{equation*} f \...
0
votes
3answers
46 views

Find one moebius transformation which returns a segment for a given arc of a circle

I'm doing some 2d geometry and I think the moebius transformation (as documented here https://en.wikipedia.org/wiki/M%C3%B6bius_transformation) can give me the answer I'm looking for, but I'm stuck. ...
0
votes
2answers
52 views

I need help understand this Möbius transformation

Show $w=\frac{z-i}{z+i}$ maps upper half plane into a unit disk centered at origin. I rewrote the equation as $z=-i(\frac{w+1}{w-1})$ and since $|z|>0$ on upper half plane. I say $|-i(\frac{w+1}{w-...
0
votes
0answers
14 views

Mobius Transformation of General complex binary cubic form

I want to show that there exists an appropriate Mobius Transform which makes General complex binary cubic form into $xy(x+y)$ Here was the idea. Since we can express the general cubic form as $(x-ay)(...

1
2 3 4 5
12