Tagged Questions

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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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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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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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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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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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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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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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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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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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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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 ...
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 ...
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 ...