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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1answer
51 views

Mapping circular lunes $|z|<1,|z+i|<1$

I want to map the circular lunes $|z|<1,|z+i|<1$ onto the upper half-plane Since intersection points are $z_1=\sqrt 3/2-i/2$ and $z_2=-\sqrt 3/2-i/2$ we have $$w_1=\frac {2z-\sqrt 3+i}{2z+\...
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68 views

Logic behind Möbius transformation

So I am attempting to get a grasp on Möbius transformations. Specifically, I'm trying to understand the logic and reasoning behind using them to map certain subsets of the complex plane into others. ...
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49 views

An exercise on Möbius transformations

I had an exam on Complex Analysis and I could not solve the following exercise on Möbius transformations: Let $f(z)=\frac{z+1}{z-1}$ and $A=\{z : \operatorname{Im}(z) >0\}\setminus \{|z|<1\}$. ...
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Verifying properties of given conformal mapping

Let's look at $\phi:\ z\mapsto -i\frac{z-1}{z+1}$ and try to verify in an elementary way that it maps the unit disk onto the upper half-plane: My problem is the following: I can easily get $\phi^{-1}:...
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1answer
54 views

Transform Name for $(1-z)/(1+z)$

I am trying to find the actual name of the transform $f(z) = \frac{1-z}{1+z}$; the transform from the open unit disk to the right half plane. Another variant; the transform from the upper half plane ...
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1answer
86 views

Möbius transformations and groups

I have two questions regarding Möbius transformations and groups. In my notes there are two statements, which I can't prove/understand why they hold. The subgroup of Möbius Transformations which ...
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Homeomorphism of unit disc onto itself interchanging two points. [duplicate]

I've searched this topic already and found similar question. But it was not fully answered. Actually I know the Möbius transformation of form $\frac{z-a}{1-\bar a z}$ is the automorphism of unit ...
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1answer
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Why is the map analytic?

The map $$z\mapsto \frac{z-i}{z+i}$$ is an analytic isomorphism of the upper half-plane $\mathbb{H}$ and the unit disc $D=\lbrace w \in \mathbb{C}||w|<1 \rbrace$ . So , what I do not undertand is ...
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53 views

Can one replace the Riemann sphere with other objects? What happens?

When doing one complex variable I think we talked about the Riemann Sphere, and how it relates to Möbius transformations, the complex "point at infinity" and other things. Is it possible (and useful) ...
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Mobius transformations from intersection of circles to two straight lines

My textbook is pretty useless on Mobius transformations, and I just wanted to check this reasoning was correct! I want to find a Mobius mapping from the region $$\{z:|z-1|\lt\sqrt{2}, |z+1|\lt \sqrt{...
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Surjectivity of $f_\beta(z)=\frac{z-\beta}{z-\bar{\beta}}$

Let $\mathbb{H}=\{z=x+yi\mid y>0\}$ and $\mathbb{E}=\{z\in\mathbb{C}\mid\mid{z}\mid<1\}$. Let $\beta\in\mathbb{H}$ and let $$f_\beta(z):=\frac{z-\beta}{z-\bar{\beta}}$$ I have already shown ...
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If $z_{1,2,3}\in S(a,R)$ then $z^*={R\over \bar{z}-\bar{a}}+a$

Recall: $z^*\in\mathbb{C}$ is symetric to $z\in\mathbb{C}$ in a relation to a generalized circle $C$ if $\overline{(z_1,z_2,z_3,z)}=(z_1,z_2,z_3,z^*)$ for some $z_1,z_2,z_3\in C$. $(z_1,z_2,...
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Confirm my answer: Find $P\in\mathbb{C}$ s.t. ${|AP|\over|BP|}=r$

The question: Find all points $P\in\mathbb{C}$ such that ${|AP|\over |BP|}=r$ for given $A,B\in\mathbb{C}$. I want to varify my answer, I hope that's allowed: Let $T(z)={z-A\over z-B}$ be a ...
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Mobius Transformation maps extended complex plane to extended complex plane

I have read in quite a few books the statement that "A mobius transformation gives a bijection from the extended complex plane to itself." How do I prove this? I know that you can find a unique Mobius ...
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33 views

Find mobius transformations that send the upper halfplane to itself

Task: Find all $A\in GL(2,\mathbb{C})$, such that the corresponding mobius transformation maps $\mathbb{H}=\{z\in\mathbb{C}\,|\,\Im(z)>0\}$ to itself. I know that this statement has been asked ...
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Find Möbius (conformal) transformation with critiera and still the same boundary?

I have trouble finding a Möbius (conformal) transformation $w=f(z)$ that fulfills the criteria that $w=f(0)=\frac{1}{2}$ for the following case: Find $w=f(z)$ so that $|z|<1$ maps onto $|w|<1$...
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Mobius transformation with fixed points α,β ∈ C

Let $T$ a Mobius transformation with fixed points $\alpha, \beta \in \mathbb{C}$ Show that exist a Mobius transfotmation $S$ such as $STS^{-1}$ fix $0, \infty$, that is unique?
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How to determine if three distinct points $a,b,c \in \Bbb c$ are collinear using Mobius Transformation?

Given three points $\frac{3}{2} + i , 2i,-6+6i$. I have the mobius transformation that maps these three points to $0,1,\infty$ respectively as $M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$ ...
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Trouble in mapping of möbius transformation

Question:- Show that the transformation $$ w = \frac{2z+3}{z-4}$$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$ My attempt:- The circle $x^2+y^2-4x=0$ is $|z-2|=2$ . . .$(1)$ So ...
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Mobius transforms - inversion sends line/circle -> line/circle

I'm trying to prove that the inversion mapping $f(z) = \frac{1}{z}$ sends circles or lines to circles or lines. Apparently the set $$\{z \in \mathbb{C}: |z-a|^2 = r^2 \}$$ describes either a circle ...
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Prove transformation $z \mapsto \frac{az+b}{cz+d}, z = x+ iy, ad-bc = 1$ is isometry of Hyperbolic plane.

Prove transformation $$f: z \mapsto \frac{az+b}{cz+d},\ z = x+ iy,\ ad-bc = 1$$ is isometry of Hyperbolic plane $$M=\{(x,y)\in \Bbb R^2:y>0\} \text{ with Riemannian metric } g= \frac{1}{y^2}(dx \...
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Find Möbius transformation for half-plane to unit disk $|w|<1$?

Consider the half-plane depicted in the following figure How can a Möbius transformation that takes that half-plane onto the unit disk $|w|<1$ be found? What are the steps and things to think ...
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Möbius transformation: area outside of two circles mapped onto the interior of a circular ring

In my current homework, there's the following task: "The area outside of these two circular discs $K_1=\{z{\in}\mathbb{C}:|z-\frac{5}{2}|\le\frac{3}{2}\}$, and $K_2=\{z{\in}\mathbb{C}:|z+\frac{5}{2}...
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Stereographic projection combined with rotation is a Möbius transformation

So for my Complex Analysis class, I need to prove the following question: Consider the function that maps a point from $\mathbb{C} \cup \{\infty\}$ to the sphere via inverse stereographic projection, ...
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Finding the image of the unit disc under a Mobius transform

I am working through an example of mapping the unit disc under the following Mobius transform: $$f(z)=\frac{iz+3}{iz-1}=1+\frac{4}{iz-1}$$ This can be written as a composition of elementary Mobius ...
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number theory and geometry question-Möbius transformation [duplicate]

I am trying to justfiy the distinct points z1, z2, z3, z4, w1, w2, w3, w4 ∈ C ∪ {∞} such that there is no Möbius transformation T with T(zi) = wi for all i = 1, 2, 3, 4. I am having a real hard time ...
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Mobius transformations between two sets

I am doing some revision of complex analysis, and am stuck on this question. I am looking for A mobius mapping sending the set {z: |z+1|<$\sqrt{2}$}, |z-1|<$\sqrt2$} onto the sector {z:3pi/4< ...
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Möbius transformations mapping non-unit circle to non-unit circle

I have a problem in which I need to find a möbius transformation which has as one of the criterion to map the circle $|z−2+i| = \sqrt5$ onto the circle $|w+2| = 2$, I dont really understand how to ...
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1answer
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Determine $z\in\mathbb{C}$, $r>0$ so that $0,1,2+i\in \partial B_r(z)$

I'm struggling to see for a method to start this question. It looks like a question related with mobius transforms. We have studied about determining the mobius transform when points from the domain ...
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Mobius Transformation viewed as a mapping on $\bar {\mathbb C}$

The question I have on hand is as follows: Suppose that a Mobius Transformation z $\to \frac{az + b}{cz + d}$ (viewed as a mapping on $\bar {\mathbb C}$) maps $\infty \to \infty$. What information ...
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1answer
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Vertical Line through z=0 in complex plane mapped with f(z)=(1+z)/(1-z)

I have the vague notion that the imaginary axis maps to a circle with f(z)=(1+z)/(1-z). $$ \begin{array}{lll} f(\infty) & = & -1\\ f(i) & = & e^{\pi/4}\\ f(-i) & = & e^{-\pi/4}\...
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Conformal property of transformation [closed]

I want to know if a LFT, $F$, is conformal on the hyperbolic plane $\mathbb H^2$ , that is if we have the curves $\Gamma_1$ and $\Gamma_2$ that intersect at a point $P$ making the an angle $X$, then $...
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1answer
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Find the image under the Mobius transformation $f(z) = 1/z$ of $|z-3i|=1$ [closed]

I am given $|z-3i|=1$ which is a circle with radius $1$ and centre $(0,3i)$ on the complex plane. I want to find the image (to sketch it) under the transformation $1/z$ WITHOUT taking points and ...
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Conformal Mapping Question about Mobius Maps, mapping 1 region to another

I am trying to map the region G, {|z|<1, |z+i|> (2)^0.5} to the infinite vertical strip at x = +/- pi. I have started by using a Mobius Map which sends the two common points of the circles to 0 ...
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1answer
50 views

Finding the Möbius transformation from the unit disk to the half plane $\{Re(z)\geq3\}$

I want to find the Möbius transformation from the disk $\{|z-1|\leq2\}$ to the half-plane $\{Re(z)\geq3\}$ that moves the point $0$ to $4+4i$. I know that by specifying the values at 3 points, the ...
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1answer
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If $w:\mathbb{D}\rightarrow\mathbb{D}$ is a Möbius transform and $||f||_{\infty} \leq 1$, why is $||w(f(z))||_{\infty} \leq 1$?

Denote the unit disk by $\mathbb{D}$. Let $w: \mathbb{D} \rightarrow \mathbb{D}$ be a Möbius transformed defined by $w(z) = \frac{z-\lambda}{1-\overline{\lambda}z}$ where $\lambda \in \mathbb{D}$ is ...
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Fuchsian groups in $\text{SL}(2,\mathbb{R})$ and commensurability in $\text{GL}(2,\mathbb{R})$

Let $\Gamma_1,\Gamma_2 \subset \text{SL}(2,\mathbb{R})$ be two Fuchsian groups. Assume that they are commensurable as subgroups of $\text{GL}(2,\mathbb{R})$, that is, there exists $g \in \text{GL}2,\...
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Mobius transformation maps $\Bbb R_∞$ onto itself iff we can choose its coefficients to be real

I have seen many solutions which are very intricate and/or long. The solution I had was much shorter which lead me to believe that it was incorrect. It went like this : $T$ be a Mobius transformation ...
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Variant of Schwarz-Pick for Different Bound/Disk

All, I'm looking to prove an alternate version of the Schwarz-Pick Lemma: Let $f:D(0,r) \rightarrow \mathbb{C}$ be holomorphic, and suppose that $|f(z)| \leq U \quad \forall z \in D(0,r)$. Then, $\...
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Is this a simpler proof that all transformations mapping D(1:0) onto D(1;0) are of the form $\;\;e^{i\lambda}\frac{z-\alpha}{\bar{\alpha}z-1}$

The question is from an exercise (2.13) in Introduction to Complex Analysis by H.A. Priestley. Before writing this I did check all the questions that might have the answer, but where proofs were given ...
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Differential Action of Möbius Transformations

The group $\mathrm{PSL}_2(\mathbb{R})$ acts on $\mathbb{H}$ via Möbius transformations, that is \begin{align*} g=\begin{pmatrix} a & b \\ c & d\end{pmatrix}:z\mapsto \frac{az+b}{cz+d}. \end{...
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Find the map from $\{ z: - \pi/2 < Im(z)<\pi/2\}$ to the vertical strip $\{ z: 0 < Re(z)<\log 2\}$.

Find the map from $\{ z: - \pi/2 < Im(z)<\pi/2\}$ to the vertical strip $\{ z: 0 < Re(z)<\log 2\}$. Using the map $f(z)=i (2/\pi)(\log 2) z$ we get the image of $f$ as $\{ z: \log 1/2 &...
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Describing curves of complex valued functions

I wish to describe the curves $|f|$=constant and arg$f$=constant for the following functions: 1.$f(z)=exp(z^2)$ 2.$f(z)=exp\left(\cfrac{z+1}{z-1}\right)$ My thoughts: I can write down what the ...
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Does a Möbius transformation having fixed points not on the object return the entire complex plane as image?

This arises from a (very early) exercise in H. A. Priestley's Introduction to complex analysis. Given the transformation $\frac{z-1}{z+1}$ we were to find the invariant (=fixed?) points, which are $\...
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Mobius transformation produces either a circle or a line…

The exercise (from H A Priestley) required a transformation that sent $\:0, 1, {\infty}$ to $1, 1+i, i$. I knew the transformation that sent $z_1, z_2,z_3,$ to $0, 1, {\infty}$ ie $$\frac{(z-z_1)(z_2-...
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1answer
31 views

A homeomorphism $T$ from extended complex plane to itself preserving cross ratio is a Mobius map.

Cross ratio preserving means $(Ta,Tb,Tb,Td)=(a,b,c,d)$ where $(a,b,c,d)=\dfrac{(a-b)(c-d)}{(a-d)(c-b)}$. If we assume $T$ fixes infinity, can we prove $T$ is affine?
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1answer
37 views

Mobius Geometry identity

How do you prove that all figures consisting of three distinct points are congruent in Mobius Geometry? I understand it relates to the Fundamental Theorem of Mobius Geometry. The concepts of which ...
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100 views

conformal automorphism $f$ of $D$ that interchanges

Let $a$ and $b$ be distinct points in the unit disk $D$. Show that there exists a conformal automorphism $f$ of $D$ that interchanges $a$ and $b$; that is, $f(a) = b$ and $f(b) = a$. Idea: we know ...
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2answers
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Mobius transform |z|<1 to the right half plane

Find a Mobius transformation mapping the unit disk {|z| < 1} into the right half-plane and taking z = −i to the origin. My workings: $\phi(t) = \frac{az+b}{cz+d}$ We map -i to the origin (0) by ...
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1answer
29 views

Conformal mapping of a domain to $f(z)=z^3$

Let W be the domain ${Im(z) < 0, Re(z) > 0}$. Sketch and describe the image of W under the conformal map $f(z) = z^3$. I have absolutely no idea how to tackle this practice exam question. I ...