# 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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### 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 ...
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### 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 ...
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### 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 $(f \circ f)(x)$...
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### 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 ...
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### $f$ 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'...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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....
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### 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?
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 $y^2=2px,\,\,(... 1answer 78 views ### Characterization of Fuchsian groups containing hyperbolic elements I want to find the Fuchsian groups that acts on the upper half plane$\mathbb{H}$to give$n$-holed torus$\mathbb{T_n}$. I am following the book Fuschian Groups by Svetlana Katok. There's this ... 1answer 114 views ### Find Mobius transformations$\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$... 0answers 33 views ### Cross ratio on$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$u=2i, v=2$, ... 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 \... 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. ... 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-...
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)(...