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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Inequality derived from $f:\mathbb{D}\to \mathbb{D}$ moving two points to another two points

Consider $f:\mathbb{D}\to \mathbb{D}$ (where $\mathbb{D}$ is the unit disc), which moves two points $z_1, z_2\in \mathbb{D}$ to $w_1, w_2\in \mathbb{D}$. I want to prove that if such an $f$ exists, ...
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26 views

Proving that $\exists f: \mathbb{D}\to\mathbb{D}$, biholomorphic, which maps $z_1$ to $w_1$

Consider a pair of points, $z_1, w_1 \in \mathbb{D}$, where $\mathbb{D}$ is the unit disc centred at the origin. Is it sufficient to argue that $f(z)=z+a$ (for $a\in \mathbb{C}$) is biholomorphic, ...
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Conformity of two concentric annuli

I'm trying to prove that two annuli, $A_1:=\{z:r_1<\left| z \right| < R_1 \}$ and $A_2:=\{z:r_2<\left| z \right| < R_2 \}$ are conformally equivalent if and only if $\frac{R_1}{r_1}=\...
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18 views

Kind of isolated singularity for analytic function between punctured disc and annulus

If $S_1:=\{z: 0<\left| z\right|<R_1 \}$ and $S_2:=\{z: r<\left| z\right|<R_2 \}$, where $r, R_1, R_2 > 0$, and $\exists f:S_1\to S_2$ such that $f$ is analytic, then what kind of an ...
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20 views

Proving two punctured domains are conformally equivalent

Prove that $S_1:=\{z:0<\lvert z \rvert<R_1 \}$ and $S_2:=\{z:0<\lvert z \rvert<R_2 \}$ are conformally equivalent. Proof: We need to find an analytic biholomorphic function $f:S_1\to S_2$...
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Mappings and directional derivatives

In one book on Complex Variables, it is said that if the function $h(u,v) = v+2$, the transformation $w=iz^2=i(x+iy)^2=-2xy+i(x^2-y^2)$ is conformal when $z\ne 0$. It maps $y=x$ (for $x>0$) onto ...
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30 views

Proof of a necessary and sufficient condition between annuli centered at the origin [duplicate]

What is a simple way to prove that two annuli $A_1 = {z: r_1 < |z| < R_1}$ and $A_2 = {z: r_2 < |z| < R_2}$ are conformally equivalent if and only if $R_1/r_1 = R_2/r_2$, using standard ...
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20 views

Inversion of lines and circles using explicit parametrizations

Is there a way to parametrize a line and a circle in the complex plane [by $z = z(t)$], to show that under the inversion function $f(z) = 1/z$, a line is mapped either to a line or a circle, and a ...
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31 views

“Approximate Isometry” in Riemannian Geometry

I apologize if the notion I'm asking about is well known, I'm no expert in geometry (and I did not find an answer via google). Suppose $(X,g_X)$ and $(Y,g_Y)$ are (smooth) Riemannian manifolds. I'm ...
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39 views

Conformal class of $\mathbb S^n$ [on hold]

What can we say about the conformal class of the sphere $\mathbb S^n$?
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Fisheye equidistant projection mapping to fisheye stereographic projection?

I have a set of images captured by a wide-angle (fisheye) lens camera, and the projection is linear-scaled (equidistant). I would like to remap from this projection to fisheye stereographic, which is ...
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45 views

Proof that Harmonic Implies Conformal

How do I show that for some function $u$ that $$\Delta u = 0 \implies u \> \> \text{is analytic}$$ and assuming $u$ has non-vanishing derivative everywhere, how do I show $u$ is conformal? ...
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68 views

How do I draw this picture in squares of discrete $\sqrt{z}$?

From Richard Kenyon's homepage gallery: I want to understand the mathematics of this, and similar/related transformations. ... An explanation in words (1st year uni level maths) would be ideal. I'...
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27 views

Determine domain of analyticity

The error function $\mathrm{erf}$ is holomorphic on the whole complex plane $\mathbb{C}$ and because its derivative $2/\sqrt{\pi} e^{-z^2}$ does not vanish on $\mathbb{C}$ it is a conformal mapping. ...
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Number of disjoint circles in half plane minus a disk that touch both boundary components

Let $\Omega \subset \mathbb{C}$ be right half-plane, with the disc $D$ removed, where $D$ is the disk of radius $r=3$ centered at $z_0=5$. What is the maximum number of disjoint open disks in $\Omega$ ...
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Is a Blaschke product/rational function a covering map for a $n$-sheeted covering of $S^{1}$?

We have a Blaschke product $B(z)$ of order $n$ (you can think of it as a rational function with $n$ zeros and $n$ poles), the zeros are obviously inside $\mathbb{D}$. Why is $B(z) \colon S^{1} \to S^{...
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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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83 views

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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47 views

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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41 views

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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22 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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21 views

$\mathcal{M}_1$ 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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63 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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24 views

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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66 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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22 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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42 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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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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25 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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37 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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30 views

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
21 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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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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17 views

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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45 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 \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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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
77 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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27 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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61 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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49 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 ...