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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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Show that $ad-bc \ne 0$ for a composition of two Mobius transformations.

I could prove that a composition of two Mobius transformations is again a Mobius transformation. Let be $T(z)=\dfrac{a_1z+b_1}{c_1z+d_1}, \ (a_1d_1-c_1b_1 \ne 0)$ and $S(z)=\dfrac{a_2z+b_2}{c_2z+d_2},...
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Existance of an analytic function on unit disc

Is there an analytic function $f:B_1(0)\to B_1(0)$ such that $f(0)=1/2$ and $f^{\prime}(0)=3/4$? If it exists, is it unique? The answer to the first part of the question is affirmative. We can use ...
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Explicit calculation of the center of a circle, image of a circle by a Möbius transformation

It's a warm up calculation I decided to carry out while reading "PCT,Spin and statistics, and all that" by Streater and Wightmann. However I do not find what they have. p.79 within the proof of Thm 2-...
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Constructing Möbius transformation

In general my approach to construct a Möbius transformation $\varphi$ between two simply connected domains $G_1$ and $G_2$ is to take 3 points on each boundary and map them onto each other. The cross-...
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Are fractional linear transformations continuous?

I was reading, some answers about fractional linear transformations and find this old question that was never answer and I think is a nice question. How do you prove it? We define a Möbius ...
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Finding a Particular Möbius Transformation from $D \to D$

I'm well aware that the Möbius transformations that take the unit disk to itself, $f:D \to D$, are given by $$ f(z) = \frac{e^{i \theta}(z-\alpha)}{1-\bar{\alpha}z}, $$ where $\theta\in [0,2\pi)$ ...
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Determine the image of $S$ under $f(z)=\frac{z^2+2}{z^2+1}.$

Let $S$ be the region $\{z:0<|z|<\sqrt{2}, \ 0 < \text{arg}(z) < \pi/4\}$. Determine the image of $S$ under the transformation $$f(z)=\frac{z^2+2}{z^2+1}.$$ I'm facing some ...
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Fractional Linear Transformation of the Image of the Line $y=4-x$

I am trying to find the image of the line $y=4-x$ under the fractional linear transformation $$w=\frac{8}{z-2-2i}.$$ My method is as follows: Rearranging yields $$z=\frac{8}{w}+2+2i.$$ Now, \begin{...
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All Möbius transformations that take the unit disk onto itself [duplicate]

I wish to prove that all Möbius transformation raking the unit disk into itself are of the form $k\frac{z-l}{1-z\bar{l}}$ where $|k| = 1$. More specifically, I ask, in addition to the main question ...
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linear fractional transformation with $w(1)=i$ and $\arg w'(1)=\frac{\pi}3$.

Does there exist a linear fractional transformation $w$ such that maps the region $\{z;\Re z>0\}$ onto the region $\{w;\Im w>0\}$ in such a way that $w(1)=i$ and $\arg w'(1)=\frac{\pi}3$? I ...
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What are the transformations that preserve cross ratios on a sphere in higher dimensions?

If we have four points $x,y,z,w$ on a sphere, then the cross ratio is $\frac{|x-z|}{|x-w|}\frac{|y-w|}{|y-z|}$. If we consider $S^1 \subseteq \mathbb{C}$, then the transformations of $\mathbb{C}$ ...
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The cross ratio $ (z_1,z_2,z_3,z_4)$ is real iff the four points lie on a circle or a straight line

It's written in Alfors Complex Analysis that, for a proof of the above, "we need only show that the image of the real axis under any linear transformation us either a circle or a straight line. Indeed,...
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Reflection about a line as a möbius transformation

I am trying to find a matrix representation in Mat$_{2×2}(\mathbb C)$ for a reflection about a line $z=z(t) = a+bt$ where only $t$ is restricted to be in $\mathbb R$ as a parameter. I am thinking ...
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What are all the functions that preserve the cross ratio?

Suppose a function $f:\mathbb {RP}^1\to \mathbb {RP}^1$ satisfy: $$ \left[f(a),f(b);f(c),f(d)\right]=\left[a,b;c,d\right] $$ for all $a,b,c,d \in \mathbb {RP}^1$. What can the function be in general? ...
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How to construct Mobius transformation fixing real axis and mapping imaginary axis to circle

I am working on an exercise from Fisher's Complex Variables text. In Exercise 7 part b on page 205 we are asked to find a fractional linear tranformation $T$ (Mobius Transformation) which maps the ...
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Mobius transformation maps the real axis to the unit circle

Show that any Mobius transformation which takes the real axis (with $\infty$) to the unit circle can be written in the form $$M(z)= \alpha \dfrac{z-\beta}{z-\overline{\beta}}$$ where $|\alpha|...
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Mapping using mobius transformations

I had a fundamental question regarding mobius transforms. Suppose I want to map the unit circle to the upper half plane ($Im$ $z \geq 0$) on the complex plane. I know mapping any three points on the ...
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Evaluate the image of complex function

Given the function $f:\mathbb{C}\setminus\{-i\}\rightarrow \mathbb{C}\setminus \{1\}$, defined by $f(z)=\frac{z-i}{z+i}$. I'm supposed to find the image for $f(\{z\mid\Im (z) > 0\})$. However I'm ...
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From the unit disk to the right half plane and $T(0)=3$

Find a Mobius transformation $T$ from the unit disk to the right half plane with condition $T(0)=3$. First, the transformation from the unit circle to the upper half plane is $T_1(z)=(1-i)\frac{z-i}{...
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Map from the upper half plane to a circle $\vert w-w_0\vert<R$ such that $T'(u_0+v_0i)>0$ and $T(i)=u_0+iv_0$

Transform the upper half plane $\mathop{\mathrm{Im}} z>0$ into the circle $\vert w-w_0\vert<R$ so that the point $i$ correspond to the center of the circle and the derivative in this point is ...
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How to find transformation from the upper half plane into the right half plane?

Find the general form of the linear transformation which transforms the upper half plane into the right half plane. In my notes I have a Mobius transformation from the upper half plane to the ...
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Fractional Linear Transformation

Suppose that $a, b, c, d\in\mathbb{C}$ and $ad-bc=1$. Let $T$ be the fractional linear transformation $$z\mapsto\frac{az+b}{cz+d}.$$ Show that if $a=i, b=-i, c=1, d=i$, then the corresponding ...
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Transform circle to $\mathbb R$: Will any 3 distinct points on the circle work?

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.15 (Exer 3.15) Using the cross ratio, with different choices of $z_k$, find two ...
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Why is $w$ real if z is on the circle through $z_1,z_2,z_3$?

Question on answer to Show that $z$ is on the circle passing through $z_1$, $z_2$ and $z_3$ if and only if [$z$, $z_1$, $z_2$, $z_3$] is real or ∞*, a duplicate of Four complex numbers $z_1,z_2,z_3,...
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Prove this Möbius function maps unit disc to itself bijectively.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.9 I got $(a)$ and $(b)$. My attempt for $(c)$: First, I interpret that $(c)$ is ...
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Suppose $f$ holomorphic and its image is a subset of the unit circle. Then show f is constant.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.8 Suppose $f$ is holomorphic in region $G$, and $f(G) \subseteq \{ |z|=1 \}$. Prove $f$ ...
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Finding Mobius transformations that maps one set to another

I am having a hard time understanding how we find mobius maps from circles, discs to half planes etc. I know how to find maps that take a set of points to another but not sets. I know about cross ...
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A Möbius map from the unit disc onto itself [duplicate]

I know that this has been asked a few times, but I found no thread that fully derived the results. I want to show that for any Möbius transformation from the unit disc onto itself it has the form $e^{...
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Magnitude of Mobius transformation at a point

If I apply a Möbius transformation to an infinitesimal shape at point z1, that shape will show up at z2 and may be rotated and/or stretched/shrunk. How do I calculate the rotation and amplitude of the ...
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Proof: Mobius Transformation Is Conformal (Poles Included)

I'm studying for an exam in complex functions analysis and I've come across a proof which states that the bi-linear Mobius function: $\omega(z)=\frac{az+b}{cz+d}$ such that $ad-bc \neq 0$ and $c \neq ...
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Find a conformal map 2

$\textbf{Exercise}$ Find a conformal mapping which maps the domain $D$ onto the open unit disc, where $D$ is the intersection of $\vert z \vert <1 $ and $\vert z-1 \vert <1 $. I knew that ...
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Find a Mobius transformation that maps the unit-disk onto unit-disk: $1\to1$ and $1+i\to \infty$

I tried to find the Moebius transformation which maps unit-disk to unit-disk and maps 2 points: $z_1=1$ to $w_1=1$ and $z_2=1+i$ to $w_2=\infty$. What I have: $w_1= \dfrac{a+b}{c+d} = 1$ $w_2= \...
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Mobius transformation that maps the upper half plane conformally onto the open unit disc.

Let $H = \{z=x+iy \in \Bbb C:y>0 \}$ be the upper half plane and $D=\{z \in \Bbb C:|z|<1 \}$ be the open unit disc. Suppose that $f$ is a Mobius transformation, which maps $H$ conformally onto $...
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Need help with $\frac{w-a}{1-\overline{a}w} = \mu\frac{z-a}{1-\overline{a}z}$

I have one exercises that i need help. Let the Möbius transformation $w=T(z)$ defined by $\frac{w-a}{1-\overline{a}w} = \mu\frac{z-a}{1-\overline{a}z},\mu=e^i\alpha, \lvert a\rvert<1$ 1.Put the ...
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Finding certain Mobius tranformation

Let $D$ denote the unit disc (|z| < 1). Let $a \in \mathbb{C}, B \in \mathbb{R}$ and $r > 0.$ I want to find a Mobius map $f$ such that (1) $f(D) = \{z = x+iy : |z-a| < r\}$ $\textbf{Sol}$ ...
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A special Mobius Transformation that maps the right half plane to the unit disc

Find the Mobius Transformation that maps the right half plane to the unit disc carrying the point $z=15 $ to the origin. Since the Mobius transformation takes the point $z=15$ to the origin, so I ...
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Proof that automorphisms of unit disk are of the form $\lambda \dfrac{z-a}{\overline{a}z-1}$

I am trying to show that the most general Möbius map that sends the unit disk to the unit disk is $\lambda \dfrac{z-a}{\overline{a}z-1}$ where $|\lambda|=1$ and $|a|<1$. I have shown that the maps ...
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Möbius transformation of crescent-shape to strip

So, the question I am having trouble with is the following: Find a Möbius transformation which maps the region {z: |z| < 2 and |z-i| > 1} onto the region {w : 0 < Im(w) < $\pi$, and which ...
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Conjugated inversion and mobius maps preserving the unit disk

I understand that the aim is to compute the Mobius maps subgroup preserving the unit disk (and there are other posts on this topic, but with different approaches). Part a) looks a bit like "$f(z)$ ...
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Schwarz- Christoffel Formula Question: Finding a transformation which maps the upper half of the w-plane for the following -

I need to find a transformation which maps the upper half of the w-plane inside the triangle with the vertices at -1, 0, and i using the Schwarz-Christoffel formula... Thus far I have drawn the ...
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Automorphism of $\Pi^+$

Suppose $\psi\in \mathbb{Aut}(\Pi^+)$ show that there are $a,\: b, \:c, \:d\in \mathbb{R}$, with $ad-bc=1,$ such that $$\psi(z)=\frac{az+b}{cz+d}$$ for all $z\in\Pi^+.$ Show that These $a,\: b, \:c, \...
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Sequence of different transformations, of a fuchsian group, that converge uniformly in compacts to limit point.

I'm in the context of Möbius Transformations acting on $\hat{\mathbb{C}}$. Let $\lambda$ be a limit point of a fuchsian group $G$, show the existence of a point $\lambda' \in \hat{\mathbb{C}}$, with ...
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How to deduce general solution of DE using Mobius transform

We are given a differential equation, say of the form $$y''(z)+\frac Azy'(z)+\frac B{z^2}y(z)=0$$ This is a differential equation with regular singular points at $0$ and $\infty$. Then we find the ...
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Construct a Mobius transformation $f$ with the specified effect:

Construct a Mobius transformation $f$ with the specified effect: $f$ maps $K(0,1)$ to itself and $K(1/4,1/4)$ to $K(0,r)$ for some $r<1$. My work: Since $i \rightarrow 1 \leftarrow i$ $ 1/4 \...
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Functions for which the Jacobian has orthogonal columns or rows

What can be said about a function $f : \mathbb{R}^m \to \mathbb{R}^n$ for which all singular values of the Jacobian matrix, $\mathbf{J}$, are all 1? I have found this similar question which covers ...
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Mobius transformation, and showing it maps the unit disk D[0,1] to itself bijectively

Consider $a \in \mathbb{C}$, where $|a|<1$ and $f_a(z)=\frac {z-a}{1-\overline az}$. (a) Show that $f_a(z)$ is a Mobius transformation. (b) Show that $f_a^{-1}(z)=f_{-a}(z)$ (c) Prove that $f_a(...
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Image of $f(z)=\frac{z-3}{z-4}$ on $\{z\in\mathbb{C}|2<Re(z)<3\}$

Let $D=\{z\in\mathbb{C}|2<Re(z)<3\}$. Let $f(z)=\frac{z-3}{z-4}$. Find $f(D)$. I thought of using the decomposition of Mobius transformations, i.e. $$f_1(w)=w-4,\ f_2(w)=1/w,\ f_3(w)=1+w$$ ...
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Find a biholomorphic map from the region $\Omega$ onto the unit punctured disk $D^*$.

Find a biholomorphic map from the region $\Omega=\{z=x+iy:x^2+y^2<4 \text{ and } x+y<2\}\setminus\{0\}$ onto the unit punctured disk $D^*=D(0,1)\setminus\{0\}$. My idea is to map the line x+y=2 ...
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Inversion in a circle as radius goes to infinity.

I am trying to show that the in the limit case as the circle gets very large, inversion in it is equivalent to reflection in a line. I have the transform $z \to c+ \frac{R^2}{ (\overline z -\overline ...
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Find the equation of the image of the line $x+y=1$ by Möbius transformation

Find the equation of the image of the line $x+y=1$ by Möbius transformation $$w=\dfrac{z+1}{z-1}$$ My approach, if $x=\Re(z)$ , and $y=\Im(z)$, then $x+y=(\frac{1}{2}-\frac{i}{2})z+(\frac{1}{2}+\frac{...