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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Homothetic transformation and conformal map

What is the difference between a conformal map and a homothety?I know that they both preserve angles.
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Is it possible to understand the Böttcher map without a deep background in mathematics?

I have a background in electrical engineering, but not mathematics. While reading papers related to fractals, I came upon the Böttcher map. $$ \Phi_c(z) = \lim_{n \to \infty}{\sqrt[2^n]{z_n}} $$ My ...
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+100

Sufficient condition for an holomorphic map to be conformal

Let $U,V\subseteq\Bbb C$ be open sets, let $f:U\to\Bbb C$ be holomorphic. If we want to prove that $f$ is a conformal map $U\to V$, my teacher said that is enough to check that $f$ is locally ...
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conformal structure of a disc

I wonder if the conformal structure of the unit disc $D^2=\{(x,y):x^2+y^2\leq 1\}$ is unique. More precisely, given a Riemannian metric $g$ on $D^2$, is it always true that $g=e^{2u}g_0$, where $g_0$...
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Find a conformal map from disc onto trapezoidal plate

how can conformal mapping a disk into a trapezoidal plate with defining favorite central point?
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21 views

Conformal mapping of a stripe to upper half of a plane

I need to find conformal mapping $W$ which maps half-infinite stripe $Z$, bounded by $2\mathbb{i}$ and $5\mathbb{i}$, to upper half plane. In other words this is what I have: And this is what I ...
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$\mathcal{M}_1$ and conformal structures on $\mathbb{T}$

I'm kind of lost trying to understand both what is usually denoted by $\mathcal{M}_1$ and the moduli space of conformal/complex structures on the 2-torus $\mathbb{T}$ (closed orientable surface of ...
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Prove the existence of a specific conformal mapping

Let $U$ be an open set containing $0$ and $f:U \rightarrow C$ a holomorphic function such that $f(0)=0$ and $f^{'}(0)=2$.Prove that there exists an open neighbourhood $0 \in V \subset U $ and a ...
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Mapping interior points of a Joukowski airfoil onto a unit disk

I have been struggling to find a mapping of points interior to a Joukowski airfoil onto the unit disk. According to the Reiman Mapping Theorem, such map should exist since I am looking a a simply ...
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61 views

Conformal mapping of cardoid $r = \rho ( 1 + \cos \theta )$ [closed]

Where does the cardoid $r = \rho ( 1 + \cos \theta )$ map in the $w$ plane, by function $w = \sqrt z$ ?
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2D Poisson Equation With Mixed Boundary Conditions

I need to solve the Poisson equation with mixed boundary consitions analytically. There are complex maps such as (1+z)/(1-z), exp(z), or sin(z) which seem suitable for transformation of this geometry ...
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65 views

Is topology invariant under conformal transformation?

Can conformal transformation change the topology of a manifold? In other words, if two manifolds are conformal, should they have the same topology?
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21 views

Conformal map from $\{0<Re(z)<\frac{\pi}{2}\}$ to $\{0<Im(z)<\pi\}$

Could anyone help me to think about a conformal map from $\{0<Re(z)<\frac{\pi}{2}\}$ to $\{0<Im(z)<\pi\}$? And how could we approach the question about finding a conformal mapping? I know ...
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1answer
41 views

Area of circle in terms of Gaussian curvature

I am asking about a formula in section 2 of these notes. Let $\rho|dz|$ be a conformal metric on $U\subset\mathbf C$. Then the Gaussian curvature of $\rho|dz|$ at $z\in U$ is defined as $$K_\rho(z)=-\...
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23 views

Laplace operator on a compact riemannian manifold $(M^2,g)$ [duplicate]

I'm studying some things about conformally covariant operators and I found this equation that there is an extensive literature about it, second the author. Let be $\Delta_{g_w}$ the Laplace operator ...
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35 views

Give a conformal map, with certain initial conditions, from the open unit disc to another open set …

... that open set being $$\mathbb{C} - \{x\in\mathbb{R} : x\leq-\frac{1}{4}\}$$ and the boundary conditions being $f(0) = 0$ and $f'(0) = 1$. Here is my first try and only idea so far: $Ci\frac{1 - ...
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23 views

Is it possible to analytically solve Laplace's equation between two rectangles?

I need to solve the heat equation without sources (Laplace’s equation) on the green domain which is bounded by two rectangles shown below: Is it possible to do that analytically? So far I haven’t ...
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36 views

Finding a conformal map taking those values at those points

Let $Q := \{x + iy : x > 0, y > 0\}$. Does there exists a conformal mapping $\phi$ from $Q$ to the unit disk such that $\phi(1+i) = 1/2$ and $\phi(1+2i) = -1/2$ ? Here is what I would do : ...
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Maintaining constant area when mapping a square to the unit circle

If I have an equidissection of a square into various polygons, and I want to map each point on the square to a point on the unit circle such that it each piece (which is not necessarily still a ...
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Mapping of $f(z)=z^{\alpha}$

What happens to the angles at the origin under the mapping $f(z)=z^{\alpha}$ when a) $\alpha>1$ b) for $0<\alpha<1$ ($z\in\mathbb{C}$)? I know that the angles increases resp. decreases but ...
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1answer
20 views

Conformal map from doubly slit plane to the open unit disk.

As stated in the title, what is the starting point in finding a conformal map between doubly-slit domain to the open unit disk? I know how to deal with a single-slit domains, but have trouble trying ...
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Constructing a new conformal map, given two conformal maps.

Let's say I have two conformal maps $f_1, f_2$ such that $f_j:\Omega\to D_j$, where $\Omega, D_j$ are open subsets of $\Bbb{C}$. Then my question is whether there is a common technique to obtain a ...
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38 views

Determining conformal mapping to unit disk with initial conditions

Find a conformal map $f: D\to B$, where $$D = \{z\in\mathbb{C} : \frac{\pi}{4}<\mbox{arg}z<\frac{3\pi}{4}\} $$ and $B$ is the unit disk with conditions: $$f(0)=i\ \ \mbox{and}\ \ f(i)=0$$ from ...
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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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1answer
37 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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19 views

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 \mathbb{C}\...
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45 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 Av,...
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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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75 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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1answer
25 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 $d(z, w) = \inf_\gamma \int_0^1 \lVert \gamma'(t) \rVert_{\gamma(t)} \, dt$...
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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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42 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
19 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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63 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 $A,B,C,...
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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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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 \mathbb{R},\;...
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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 $$w=f(z)=\frac{2(x+iy)-1}{2-iy-x}=\frac{(...
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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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216 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: http://rotormind.com/...
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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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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 show ...
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1answer
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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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22 views

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 non-...
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117 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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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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43 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. ...