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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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6answers
173 views

What is the radius and center of the image of $|z|=1$ under $ f(z) = \frac{3z+2}{4z+3}$?

I would like to compute the image of the circle $|z|=1$ about the fractional linear transformation: $$ f(z) = \frac{3z+2}{4z+3} $$ In particular, I'd like to compute the new center and radius. ...
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1answer
73 views

Using conformal maps to solve the Dirichlet problem on $U = \{z : \text{Im}z \geq 0 \}$

I'm trying to solve a question which asks me to solve the Dirichlet problem on $U = \{z : \text{Im}z \geq 0 \}$ on the condition that (where we let $z = x+iy$) $u(x,0) = 0 $ when $ |x| >1 $ and $ u(...
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0answers
74 views

Solving the Dirichlet problem on $ U = \{z: \text{Im} z \geq 0\}$ using a conformal map

I'm trying to solve the Dirichlet problem on $ U = \{z: \text{Im} z \geq 0\}$ with the conditions $u(x,0) = 0$ when $x>0$, $u(x,0)=1$ when $x<0$. To do so, I'm supposed to use conformal maps ...
2
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1answer
185 views

Subgroup of Möbius transformations which are isometries with respect to the standard metric on the Riemann sphere

I'm trying to find which subgroup of Mobius transformations are isometries with respect to the standard metric on the Riemann sphere (the one induced from the Euclidean metric on $\mathbb{R}^3$). The ...
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1answer
132 views

Finding a conformal map from the intersection of two disks to the unit disk.

I'm trying to solve a problem which asks me to find a conformal mapping from $\{z\in \mathbb{C}: |z-i|< \sqrt2$ and $|z+i|<\sqrt2\}$ onto the open unit disk. I'm really new to these and I'm a ...
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2answers
67 views

Möbius transformation/biholomophic funtion

I have to show, that the Möbius transformation $$ T(z) = \frac{z-z_0}{1-\bar{z_0}z}$$ is an biholomorphic function on $ \mathbb{D}$. $ \mathbb{D}:=\{ z \in \mathbb{C}: |z|<1 \}$ and $z_0 \in \...
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1answer
177 views

Finding bilinear transformation which maps $|z|=1$ on to $|w|=1$ [closed]

How can I show that every bilinear transformation which maps $|z|=1$ on to $|w|=1$ must be of the form $$w=K\frac{z-\alpha}{\overline{\alpha}z-1}$$ where $|K|=1$? Please help me. Thanks.
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1answer
42 views

Mobius transformation producing a curved triangle with 3 intersecting circles

Let $ABC$ be a curved triangle on a plane, whose side $AB$, $BC$ and $CD$ are arcs of circles $S_1$, $S_2$ and $S_3$ passing though a point $D$ (i.e. $S_1∩S_2∩S_3 = D$, $D≠A$, $D≠B$, $D≠C$). Assume ...
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2answers
263 views

Möbius transformation mapping

I would like to understand how to choose the right Mobius transformation. For example, the Mobius transformation that maps upper half plane onto the unit disk is: $z \rightarrow \frac{z-i}{z+i}$ ...
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3answers
156 views

$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H$. What kind of points have non-trivial stabilizer? And how many orbits are there?

$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H= \{z \in \mathbb{C} | \Im(z) > 0 \}$ via Mobius transformation. $$ \text{ For } \gamma =\begin{bmatrix} a &b \\c&d \end{...
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1answer
35 views

Finding the norm of $w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$, where $w$ and $z$ are in $\mathbb{R}^n$

I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function $$M_w(z) = w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z \in \mathbb{R}^n$ and $|w| < 1$. (...
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0answers
57 views

Image of square under Mobius Transformation

Am i right that there is no Mobius transformation $$ h(z) $$ that sends a square to rectangle "A" with vertices in $$ 0, 2, i, i+2 $$because if we inscribe a cricle into square then it has 4 points $$ ...
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3answers
70 views

Find a center and radius of circle that is an image of Mobius Transformation of real axis

I need to find a center and radius of a circle that is an image of real axis under homography $$ h(z)= \frac{z-z_1}{z-z_2} $$ I found out that homography preserves symetric points, therefore ...
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0answers
30 views

Holomorphic bijection from intersection of two circles to a region between two rays

What is a holomorphic map from the nonempty intersection of two circles with "tip points" $a$ (below) and $b$ (neither of which are included in one another) to the region $A$ between two rays? Pictue ...
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2answers
71 views

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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3answers
121 views

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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3answers
60 views

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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1answer
93 views

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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1answer
107 views

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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1answer
57 views

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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2answers
101 views

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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1answer
79 views

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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0answers
78 views

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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1answer
82 views

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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1answer
80 views

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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0answers
79 views

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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1answer
117 views

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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1answer
74 views

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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4answers
154 views

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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3answers
215 views

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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1answer
51 views

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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1answer
43 views

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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4answers
514 views

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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2answers
94 views

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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2answers
126 views

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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2answers
85 views

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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3answers
212 views

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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1answer
104 views

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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3answers
88 views

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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3answers
262 views

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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2answers
361 views

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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1answer
125 views

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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0answers
30 views

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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0answers
162 views

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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1answer
116 views

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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1answer
132 views

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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1answer
91 views

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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0answers
32 views

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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1answer
41 views

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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1answer
215 views

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 ...