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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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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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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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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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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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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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If $\Omega$ is a symmetric domain of $\mathbb{C}$ respect to $\mathbb{R}$ and $f: \Omega\longrightarrow D(0,1)$ is an isomorphism. If $\exists a\in \Omega \cap \mathbb{R}$ such that $f(a)=0$ and $f'(... 2answers 83 views What is the dimension of the set generated by$z \mapsto z + 1$and$z \mapsto \frac{z}{1+z}$? Let's say I have subgroup of$SL_2(\mathbb{Z})generated by two elements, I wanted to compute the limit set. So I wrote down two matrices: A = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \... 1answer 75 views Diagonalising Invertible Mobius Transformation For invertible M \in \mathbb C^{2\times2} we can find invertible S \in \mathbb C^{2\times2} so that SMS^{-1} is either diagonal, or triangular with two equal diagonal elements. There is an ... 1answer 78 views Free discrete action vs discrete subgroup Suppose a group \Gamma \leq PSL(2,\mathbb{R}) is acting fixed point freely on the upper half plane \mathbb{H} via Moebius transformations. I am looking for a quick argument showing that the action ... 1answer 176 views finding all the mobius transformation.? find all the mobius transformation that map the units disc D onto the left half plane H^- = {w \in C : Re w <0 } My attempts : I know that all the Möbius transformations can be ... 1answer 160 views Find direct isometries In hyperbolic space. Let H be the upper half of the complex plane model for hyperbolic geometry. let p,q be points in the plane. show how to find a direct Isometry F of H such that F(p)=q. Now suppose p\prime,q\prime... 0answers 110 views How does this algorithm get the limit set of “kissing” Schottky group? I'm having difficulty understanding why the algorithm presented in this paper works. If I understand correctly, to construct a Schottky group start with 2n circles: A_1...A_n and B_1...B_n, ... 0answers 146 views Sketch the image under the function w = \log z of the following set \{z : Im z > 0\} I would love some help with this question please: Sketch the image under the function w = \log z of the following set \{z : Im z > 0\} What I have done so far is as follows: R= \{x+iy: y&... 2answers 44 views Determining the matrix representations of functions. Show that each function\begin{align}f(x)=\dfrac{ax+b}{cx+d}, g(x)=\dfrac{ex+f}{jx+h}\end{align} can be represented by matrices such that the matrix representation of the composition g(f(x)) is the ... 1answer 40 views Sphere reflection property (geometric proof). I need some help with this exercise I found in chapter 3 of the book "The Geometry of Discrete Groups" by Beardon. Prove (analitically and geometrically) that for all non-zero x,y \in \mathbb{R} ^n ... 1answer 24 views Showing that if T is a Möbius transformation of the disc, \frac{|T(z)-T(w)|}{|1-T(z)\overline{T(w)}|} = \frac{|z-w|}{|1-z\overline{w}|} ... where z, w \in \mathbb{D}, \mathbb{D} the open unit disc. I realise that the way to start working on this is to express T as e^{i\alpha}S_a for some \alpha \in [0, 2\pi] and a \in \... 1answer 163 views Books and references for Möbius transformation, hyperbolic Riemann surface and covering transformation I am reading the book Computational Conformal Geometry by Xianfeng David Gu and Shing-Tung Yau. There is a part in the book which I don't understand and I would like to ask for books and references ... 1answer 70 views Determine the most general Mobius transform that… Determine the most general Mobius transform that satisfies: i) two fixed points z=0 and z=1; ii) maps the unit circle |z|=1 and the bisector of first and third quarter to lines. My guess is:\... 0answers 487 views How to find the “interior boundary” for a set of points? LetS$be a set of points in$\mathbf{R}^n$: $$S=\{ (x_1,\dots,x_n) \ | \ x_i \in \mathbf{R} \ ; i=1,\dots, n \}$$ Is there a way to find the interior boundary of such set of points? Since there ... 2answers 144 views Holomorphic function mapping unit disc to the “pacman”$U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]\}$Find an injective and surjective holomorphic function, that transfers the unit disc$\Bbb D = \{|z|<1\}$to the domain$U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]$}. I ... 0answers 69 views Is a perspective projection a Möbius transformation? Both perspective projection and Möbius transformations exhibit cross-ratio invariance, but can a perspective projection be defined as a subgroup of the Möbius group? In other words, is a perspective ... 1answer 206 views prove Mobius Transformation can be extended to a meromorphic function I have been reading all the notes I could find about Riemann surfaces but still have no clue how to start these 2 proofs... I know they should be easy proofs... 1 first one being, prove/show that ... 0answers 138 views Program that shows action of Möbius transformations This is not directly a mathematical question, but I think this is the best place to ask it. I discovered that I'd find it very helpful to have some sort of a program where I can plot the coefficients ... 1answer 94 views Determining a Mobius transformation from a tiling Suppose we have a tiling of the upper half plane by ideal quadrilaterals. Suppose that the principal quadrilateral consists of perpendicular lines at$Re(z)=0$and$Re(z)=1$, and two circles joining ... 1answer 70 views Proving that$\exists a,b \in \mathbb{C}$such that$\forall z\in \mathbb{C}$either$m(z)=az+b$or$m(z)=a\bar{z}+b$. Let$m \in \text{Homeo}^{C}(\bar{\mathbb{C}})$(the group of homeomorphisms of$\bar{\mathbb{C}}$taking circles in$\bar{\mathbb{C}}$to circles in$\bar{\mathbb{C}}$, where$\bar{\mathbb{C}}=\mathbb{...
I'm looking at a group of transformations from $X \cup \lbrace \infty \rbrace$ to itself, where $X$ is a Hilbert Space, and ultimately I'm looking for an appropriate name for this group. The elements ...
If f and g are Mobius Transformations, why does this hold? $$gf^{n}g^{-1}(z)=z+n$$ implies $$f^{n}(z)=g^{-1}(g(z)+n)$$