A conformal structure is one that captures the idea of angles, but not lengths. Conformal geometry is the study of geometries with conformal structures, with special focus on transformations that preserves the angles (while possibly changing lengths). Examples of conformal transformations include ...

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Conformal Map from Domain $D$(where $D$ is bounded by $\{z=x+iy,y=1/x,x>0\}$, $2+2i\in D$) to upper Half Plane

I am trying to find the Conformal Map from Domain $D$(where $D$ is bounded by $\{z=x+iy,y=1/x,x>0\}$, $2+2i\in D$) to upper Half Plane $\mathbb{H}:=\{w:Im(w)>0\}$, I am using $z+1/z$ map but ...
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Circle segment conformal mapping

I have to find conformal mapping from the outer area of the set $$ \{ z \in \mathbb{C} | Imz > 0 \cap |z - i| < \sqrt{2} \} $$ (that is a circle segment) into a unit circle. Any ideas how to ...
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21 views

Finding a conformal map

I'm doing some review for my complex analysis final, and have come across the following. Find a conformal map mapping the half strip : $P=\{x+iy:x<0, 0,<y<\pi\}$ to the upper half plane. I ...
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Extension of conformal mapping ouside unit disc

Suppose I have a conformal mapping $f:D\to \Omega$ which takes a unit disc to a connected blob $\Omega\subset \mathbb{C}$. There should exist a conformal mapping $g:\mathbb{C}\setminus D\to ...
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40 views

Are holomorphic maps that “almost” preserve norm “almost” rotations?

Let's say I have a sequence of injective holomorphic maps $f_n \colon \mathbb{D} \to \mathbb{D}$ such that $f_n(0) = 0$. The main thing is that $f$ "almost preserves norms" in the sense that for all ...
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ALL Orthogonality preserving linear maps from $\mathbb R^n$ to $\mathbb R^n$?

That is we have a linear transformation, i.e. an $ n\times n $ matrix $A$, such that for every pair of vectors $ v $ and $ w $ we have $$ \langle v,w\rangle=0 \ \ \ \implies \ \ \ \ \ \langle ...
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15 views

Why does a sequence of increasing expansions converge?

I'm working on a problem from Stein and Shakarchi's Complex Analysis about proving the Riemann mapping theorem. Their general strategy is as follows: an injective function $f \colon K \to \mathbb{D}$ ...
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1answer
67 views

Finding a conformal map to the upper half-plane

Find a conformal map from the set $$\{z \in \mathbb{C}: |\operatorname{Im}z| < \pi \}\setminus \left[-\pi i; 0 \right]$$ to the upper half-plane. I have used a composition of the following maps: ...
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23 views

Is a map that preserves the hyperbolic distance biholomorphic?

Let $\lVert z \rVert_w = \frac{|z|}{1 - |w|^2}$ be the hyperbolic distance in $\mathbb{D}$, and let the hyperbolic metric be $d(z, w) = \inf_\gamma \int_0^1 \lVert \gamma'(t) \rVert_{\gamma(t)} \, ...
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55 views

Conformal map from the union of two disks onto half-plane

Let $U=D_2(-1)\cup D_2(1)$. Find a conformal equivalence from $U$ onto $\mathbb{H}$. We tried many things, like inversion thru one of the circles, and Möbius transformations, but none of that stuff ...
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27 views

Find a conformal mapping from the quarter-disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ onto the upper half plane set $U=\{im z>0\}$

Find a conformal mapping from the quarter-disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ onto the upper half plane set $U=\{im z>0\}$ I'm guided through this problem: First I need to find the ...
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1answer
16 views

Regular parametrization of a surface is conformal iff it preserves angles.

Can anyone give me some hints of how to start the proof, because I have no idea where to start. I know if a parametrization is conformal, then $E=G$ and $F=0$, where E,F,G are values in the first ...
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59 views

Motivating the Cross-Ratio and 'the ratio of ratio's' in $\mathbb{R}\mathbb{P}^2$

Trying to come across the idea of the cross ratio naturally by thinking about the projective plane $\mathbb{R} \mathbb{P}^2$, using ideas from Brannan's Geometry book: given 4 collinear points ...
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35 views

Upper half-plane $\overline{\mathbb{H}}$ with two boundary punctures

Consider $\overline{\mathbb H}$ with two puncture $P_1$ and $P_2$ on the real line, with coordinates $z = x_1$ and $z = x_2$, respectively. Consider another copy of $\overline{\mathbb H}$ with two ...
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73 views

Inverse of the von Kármán-Trefftz transform

I'm having troubles finding the inverse of the Von Kármán-Trefftz transform (it's a conformal map) \begin{equation} z(x;b,k)=k\,b\,\frac{(x+b)^k+(x-b)^k}{(x+b)^k-(x-b)^k}\;,\quad b,k \in ...
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Images of the map $f(z)=\frac{2z-1}{2-z}$

What is the image of the real line the imaginary line the unit circle Under the map $f(z)=\frac{2z-1}{2-z}$ Assume $z=x+iy$. Then setting ...
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Splitting of the 2-forms on a 4-dimensional vector space specifies unique conformal class

The problem I'm facing is the following. Let $V$ be a 4-dimensional real vector space with orientation $\omega\in\Lambda^4V\setminus\{0\}$. Suppose that $U_-,U_+\subset \Lambda^2V$ are linear ...
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150 views

Conformal mapping explanation?

Well I have a question that is "identify the two interesting points in the picture for the function $f(z) = z + 1/z$. Explain your answer." where it refers to the tool here: ...
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23 views

Conformal mapping in terms of first fundamental form

I'm trying to understand conformal mappings in terms of the first fundamental form. I believe that two surfaces with fundamental forms $I_1$, $I_2$ are conformal if $\exists \lambda \not= 0$ such that ...
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17 views

Can a Riemann conformal map be extended to the whole plane?

Let $\Omega$ be a simply connected region of $\mathbb C$. By the Riemann Mapping Theorem, there is a biholomorphism $\Omega\to D$. Under the additional condition that the boundary of $D$ is a Jordan ...
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20 views

Let $ f(w)=\frac{w(1-i)-(i-1)}{w-1} $, where $w$ is the left hand plane. What is the image of this map?

Let $$ f(w)=\frac{w(1-i)-(i-1)}{w-1} $$, where $w$ is the left hand plane. What is the image of this map? The answer should be $|z|^2<2$ if I did everything before correctly. This is a ...
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16 views

Let $f(z)=\frac{z+1-i}{z-1+i}$ be a map. What is the image of $f(S)$ where $S=\{z\in \mathbb C | im z>rez\}$

Let $$f(z)=\frac{z+1-i}{z-1+i}$$ and $S=\{z\in \mathbb C | im z>rez\}$.What is the image of $f(S)$? I sketched the region, and it corresponds to a halfplane, from $\pi/4$ to $5\pi /4$. I tried ...
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Convergence on only some non-tangential limits

I am interested in the boundary behavior of Riemann maps. Is there a nice example of a region $G$ with Riemann map $\phi:\mathbb{D}\to G$ such that the limit of $\phi$ exists along some ...
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89 views

Analytic function on unit disk has finitely many zeros

I am studying complex analysis from Theodore Gamelin's text and Exercise 1 of chapter IX.2 says that if $f$ is analytic inside the open unit disk and continuous on its boundary that satisfies $|f(z)| ...
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1answer
21 views

Image of lines under the Cayley Transform $z \mapsto \frac{z-1}{z+1}$

I am having trouble compute the image of latitude and logitude lines under the Cayley transform $z \mapsto \frac{z-1}{z+1}$. So a horizontal line might be $\mathrm{Re}(z) = k \in \mathbb{R}$, then ...
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39 views

Characterizing all mobius transformations from unit disk to itself.

All answers to this problem involve the following process: Pick a point $a$ in the unit disk such that $T(a)=0$. Then, $T(a^*)=\infty$ if $a^*$ is the symmetric point $a^*=\frac{1}{\bar{a}}$ of a. ...
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Formula for degree 1 finite Blaschke product mapping $a_1$ and $a_2$ to $\pm r$

For a given choice of distinct numbers $a_1,a_2\in\mathbb{D}$, there is a unique choice of real number $r\in(0,1)$ such that some degree $1$ Blaschke product maps $a_1$ to $r$ and $a_2$ to $-r$ (and ...
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19 views

What is Extremal Length?

The question is as the title asks. My background is in computer science, and recently I'm trying to read a paper that involves using extremal length to prove certain properties of planar graphs. I ...
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Is infinity allowed in the definition of a fractional linear map?

I need to create a fractional linear map of the form $$ F(z) = \frac {az+b}{cz+d}, $$ where a,b,c, and d are complex numbers such that $ad-bc \neq 0$ and $F(0) = 1$, $F(1)=\infty$, and $F(\infty)=0.$ ...
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28 views

Proof that Möbius tranformations are conformal

It is easy to understand (even without calculations) that a system of orthogonal circles like in this picture: transforms into another similar system under a Möbius transformation in the complex ...
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Can a conformal map be turned into an isometry?

Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with $$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, ...
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18 views

harmonic measure circle

I'm trying to compare the probability of a particle (performing Brownian Motion), starting a large distance away from a circle, passing through a specific section of that circle is approximately the ...
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1answer
21 views

On conformal metrics notation

A simple question, just for clarifying: suppose we have two riemannian metrics $g$ and $\tilde{g}$ in a differentiable manifold $M$, and assume they are conformal say, with $\tilde{g} = \mu g$ for ...
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1answer
34 views

conformal maps between vector spaces?

let $X,Y$ be two vector spaces furnished with inner products. Now consider a linear map $L:X\rightarrow Y$ with the following additional property: there exists a constant $C>0$ with $$\langle ...
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1answer
52 views

Explicit formula for conformal map from ellipse to unit disc (interior to interior)

I was originally looking for a conformal map that maps a punctured unit disc to unit disc. The only answer I can find lead to this resource. The final step of the answer given rely on a conformal map ...
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1answer
22 views

Findig Fractional Linear Transformation

I am trying to find the fractional linear transformation that sends $0$ to $1$ and maps $-1$ to $1+i$. My approach is to setup the equation $w(z) = \frac{az+b}{cz+d}$ and plug in the values $0$ and ...
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1answer
28 views

Unsure how to prove Conformal Mapping.

I'm asked to show that a function takes the upper half disk to the upper half plane. $$ f(z) = \left(\frac{1+z}{1-z}\right)^2 $$ I have many ways to go about this, such as, I could define the ...
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98 views

How do conformal maps affect curvature?

Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
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25 views

3D mesh segmentation simple algorithm

I am developing the algorithm reported in this article: Lest square conformal mapping Here is presented an algorithm to flat a 3d mesh on the parametric space, but i don't understand the ...
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43 views

Is there any generalization of Riemann Mapping theorem?

Given any two regions in complex plane when can we say they are conformally equivalent? I mean does there exists some "complex-geometric" invariant which determines whether two regions are conformally ...
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35 views

Is any smooth deformation of a metric in dimension 1 conformal?

Consider $(S^1, g)$ where $S^1$ is the unit circle and g is a metric. Now consider the metric $$ \tilde g := f g $$ where f is a smooth positive function. Since in 1 dimension this is the only smooth ...
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115 views

Mapping in the complex plane

I have the following two circles in the complex plane, $z = x + iy$, which bound a region, $R$. The equations for the circles and a sketch of the region is given as follows: $$ x^2 + (y-1)^2 = 1\\ x^2 ...
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21 views

when can a surface conformally equivalent to the sphere be isometrically immersed?

Given a scalar function $s:S^2\to \mathbb{R}$, and the induced Euclidean metric $g$ on $S^2$, when can the sphere, equipped with metric $e^sg$, be isometrically immersed in $\mathbb{R}^3$? Is there a ...
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17 views

Conformal map and linear independence over $\mathbb{Q}$

I would like to know whether the imaginary parts of $n$ complex numbers $z_1,\dots z_n$ sharing the same real part $r$ are linearly independent over $\mathbb{Q}$ if and only if no conformal mapping ...
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1answer
24 views

Conformal mapping, which geometric objects are these?

Given the mapping $$w = \frac{z + i}{z - 1} $$ find the images on the $w$ plane of $$i) |z - i| = |z - 1|$$ $$ii) x^2 + y^2 = 1$$ $$i) |z + i| = |1 + i|$$ $$ii) ???$$ Polar form, maybe? Which ...
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1answer
46 views

$\{z : 0 < |z| < 1\}$ and $\{z : r < |z| < R\}$ are not conformally equivalent.

Prove that $D=\{z : 0 < |z| < 1\}$ and $A=\{z : r < |z| < R\}$ are not conformally equivalent if $r > 0$. I am trying to apply the Riemann mapping theorem to show but am not able to ...
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46 views

Condition to be conformal

I am looking at the following exercise: Let $\Phi : U \rightarrow V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$. Write $$\Phi (u, v)=(f(u, v), g(u, v))$$ where $f$ and $g$ are ...
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Find an open connected set $G \subset \mathbb{C}$ and two continuous functions $f$ and $g$ defined on $G$ such that $f(z)^2=g(z)^2=1-z^2$…

Find and open connected set $G \subset \mathbb{C}$ and two continuous functions $f$ and $g$ defined on $G$ such that $f(z)^2=g(z)^2=1-z^2$ for all $z \in G$. Can you make $G$ maximal? Are $f$ and $g$ ...
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1answer
27 views

Quick question on the roots and poles of a meromorphic function,

Does the degree of the polynomial in the numerator always equal the degree of the polynomial in the denominator? In other words, the number of zeros, counting multiplicity, equals the number of ...
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2answers
38 views

Image of circle under linear fractional transform

Given the LFT for a complex $z$, \begin{align*} \phi:z\mapsto \frac{2z+1}{z+2}. \end{align*} I'm asked about the image under $\phi$ of $C:=\left\{\left\lvert z+\frac25\right\rvert = \frac25\right\}$. ...