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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Estimate when $f$ is locally one to one

Suppose $f$ is analytic in $|z|\leq 1$ and $|f(z)|<1$, $f(0)=0$ and $f '(0)=a\neq 0$. Show that there exists a disc of radius $\rho$ s.t for any $z_1,z_2$ in the disc, $f(z_1)=f(z_2)$ implies that ...
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Isometries of hyperbolic space

The metric tensor for the Poincaré ball model of hyperbolic geometry is $$ g_{ij} = \frac{\delta_{ij}}{(1 - \lvert \mathbf{r} \rvert^2)^2} $$ where $\mathbf{r}$ is the position in the ambient ...
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36 views

Looking for a biholomorphism [on hold]

Is there a biholomorphism between $\mathbb{H}_2 \times S^1$ and a half-space of $\mathbb{R}^3$?
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Bijective Conformal Mapping onto the Open Unit Disc $\mathbb{D}$

What is the explicit bijective conformal mapping $f(z):G_n\to\mathbb{D}$, $z\in\mathbb{C}$ for the following domain transformations: $G_1=\{x+iy~|~x>1/2,y>0\}$ is the open region of the first ...
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Find a harmonic function in the first quadrant,

Find a harmonic function $\phi$(x,y) in the first quadrant with the boundary values $\phi$(x,0) = -1 for x>0, and $\phi$(0,y) = 1 for y>0. Is this function unique? My attempt was this: Consider ...
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Conformal Mapping onto the Unit Disc

Given the open vertical strip $G=\{x+iy~|~0<x<1,~-\infty<y<\infty\}$, what is the explicit conformal injective map characterizing $w=f(z):G\to\mathbb{D}$? It is noted that if there ...
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Elliptical Bijective Conformal Mapping

Given the conformal mapping $w=T_1(z):R\to\mathbb{D}$ and $v=T_2(w):\mathbb{D}\to E$, where $R=\{x+iy~|~0<x<1,\epsilon_1<y<\epsilon_2\}$ is the open rectangular domain, $\mathbb{D}$ is the ...
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Optimal set of generators of conformal group in 2D

Can we write Lorentz transformations and dilations in terms of translations and special conformal transformations? In V. Kac's book "Vertex algebra for beginners" 2nd edition, on p.7, Kac writes that ...
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is the Buddhabrot well-defined?

Define the Mandelbrot set $M = \{ c \in \mathbb{C} : P_c^n(0) \not\to \infty \text{ as } n \to \infty \}$ where $P_c(z) = z^2 + c$. Define the complement of the Mandelbrot set $\overline{M} = ...
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Find a harmonic function in the interior of the disk, taking values +1 and -1

Consider the Mobius transformation $$f(z) = \frac{1+z}{1-z}$$ Use this map to find a function $f(x,y)$ which is harmonic in $x^2+y^2<1$ and on the boundary $x^2+y^2=1$ takes values $+1$ when ...
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Biholomorphism of circular region onto the disc

Let $H = \{a+bi \in \mathbb{C} : b > \frac{1}{\sqrt{2}} \}$. I'm trying to find a biholomorphism from $H \cap D$ onto $D$, where $D$ is the unit disc. But I don't know how to go about doing it. ...
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47 views

Riemann mapping under which uncountably many boundary points correspond to a single point

I am interested in the following question, which is 10.4 from this list: Give an example of a domain conformally equivalent to the disc where uncountably many points on the unit circle ...
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Do conformal mappings other than the Mobius transformations preserve symmetry?

Linear fractional transformations preserve symmetric points, e.g., if the real axis in the z-plane gets mapped to the imaginary axis in the w-plane, then points symmetric with respect to the real axis ...
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Why does the upper half plane get mapped inside of the polygon?

If a conformal mapping, e.g., a Schwarz-Christoffel mapping, maps the real line (from left to right) to a polygon, which is traced out from left to right, why is the upper half plane mapped to the ...
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Conformal mapping from upper half-plane to unit disk

I have run into some confusion while reading Newman Bak's Complex Analysis. The text states that if we wish to determine a comformal mapping $h$ of the upper half-plane onto the unit disk, assuming ...
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Automorphisms of the unit disk

I ran into a bit of a hurdle as I was reading a proof in Newman Bak's Complex Analysis regarding the form of automorphisms of the unit disk. The proof begins by showing that ...
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38 views

Why are the exponents in the Schwarz-Christoffel mapping of the form (1- alpha/pi)?

Here is the Wikipedia article on it: https://en.wikipedia.org/wiki/Schwarz%E2%80%93Christoffel_mapping I feel it doesn't make sense. The integration produces real values (on the real line), so f(x) ...
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Why does a conformal mapping create a full tiling of semi-infinite strips in the w-plane?

I know that, specifically for linear fractional transformations, symmetric points get mapped to symmetric points. So, if the real line gets mapped to a circle, then under a LFT, points symmetric ...
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101 views

map the UHP to an equilateral triangle

Explain how the upper half-plane can be mapped one-to-one and conformally onto an equilateral triangle. Thanks,
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Show that Riemann Theorem does not hold when set is not simply connected

Riemann Theorem states that for any simply-connected domain in $\mathbb{C}$ (which is not whole $\mathbb{C}$) there exists biholomorphic map onto the open unit disk. I find it hard to show that we ...
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65 views

Describe the Riemann surface:

$$W = \sqrt{1-z^2}$$ I would like hints only. Using @Dr.MV's hint, I get two factors: the first is $$\sqrt{(x-1)+y^2}^{\frac{1}{2}}e^{i\frac{\theta}{2}}$$, which, when we let theta range from 0 to ...
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Why is $z^2$ a conformal mapping?

It's not a one-to-one mapping, by the Fundamental Theorem of Algebra. $e^z$ is one-to-one, when restricted to a horizontal strip of width = $2\pi i$. Is it a similar argument for $z^2$? Thanks, ...
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When proving that the most general conformal mapping of the UHP is a LFT, can I start the proof with a linear polynomial?

I'm having some difficulty with starting a proof that proves the most general one-to-one conformal map from the UHP onto itself is of the form az+b/cz+d, with a,b,c,d real and ad-bc =1. My idea is to ...
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help verify a conformal map between regions

Find a conformal map from the set $\{z \in \mathbb{C}: |z|>1\}\setminus (-\infty,-1)$ onto the set $\mathbb{C}\setminus(-\infty,0]$. Here is my thought, but I'm not sure if it is correct, can ...
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Is my proof of linear fractional transformations correct?

a) Prove that the most general $1-1$ conformal map of the upper half-plane onto itself is of the form $$z \to \frac{az+b}{cz+d}$$ where $a,b,c,d \in \mathbb{R}$ and $ad-bc =1$. b) Let $f$ be a $1-1$ ...
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Schwarz's Lemma Type Application

So I have the following question: Let f be analytic in an open set which contains the closed unit disc $\overline{\mathbb{D}},$ and assume $M:=\max\{\textrm{Re}(f):|z|=1\}\geq0.$ Prove that for ...
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Why are conformal mappings necessarily 1 to 1?

Say, by the Riemann Mapping Theorem, there exists a biholomorphic, conformal mapping from the upper half plane to the (open) unit disk (since the UHP is simply connected and is not the entire complex ...
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Where do these Mobius transformations map the coordinate half-planes?

They are $$\frac{z-1}{z+1}, \frac{z+1}{z-1},\frac{z-i}{z+i},\frac{z+i}{z-i}.$$ All four look virtually identical, so I would like to know how to best distinguish between them. For example, the ...
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67 views

Using the complex logarithm as a conformal mapping,

I want to map the upper half plane, y>0, conformally onto the semi-infinite strip u>0, $-\pi < v < \pi$ in the w-plane. I then studied the complex logarithm, and noticed that the principal ...
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Mapping the upper half plane conformally onto a semi-infinite strip,

Map the upper half y>0 of the z-plane conformally onto the semi-infinite strip u>0, $-\pi<v<\pi$ in the w-plane. I would like some hints for now, please. I'm not sure how to even get started ...
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Map conformally $D(-2,2)\setminus \overline {D (-1,\frac12)}$ to the annulus $\{1 < |z| < 2\}$

This is coming from this question: http://math.stackexchange.com/questions/1332123/moving-around-a-circle-inside-a-different-circle-conformally I will delete that question. This will be the same ...
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All possible flat conformal metrics of dimension greater than 2

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible ...
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Conformal transformation of a region bounded by a curve $y=x^a, a \in \mathbb{R}$

I would like to solve the 2D Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =0$ on the positive upper half plane: $0 <x<\infty$ and $0 < y < ...
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How to solve 2D Laplace Equation over an infinite rectangular strip (bounded on two edges), with Dirichlet boundary conditions

Is it possible to solve Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0$, over an infinite rectangular strip defined by $0 < x < \infty$ and $0 < y ...
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Showing internal angles of a square are unaffected by a mapping

I recently had an exam in complex analysis, and I am slightly confused by one of the questions, so I'd appreciate any clarification: The mapping from the complex $z$ plane to the complex $w$ plane ...
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Complex potential for a fluid

How do I solve this? Find the complex potential for a fluid moving with constant speed $s$ in a direction making an angle $\alpha$ with the positive $x$-axis I am fairly sure this is a problem ...
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A conformal mapping of point A to point B in upper half plane

Given arbitrary A,B in the upper half plane, how do you write a conformal bijection mapping point A to point B? My first thought is to take the inverse of a mapping $f$ that would send A to the ...
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21 views

Construct conformal mapping for distorting a square box

I have a regular, square, cartesian grid. Let us call the bottom and the left hand boundaries of the grid B and L. Then the angle between B and L is obviously 90 degrees, and also between the vertical ...
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(Simple) Conformal mapping

I'm working on a problem and part of it is to map the sector $\{z \in \mathbb{C}| \frac{\pi}{4} < \text{arg} z < \frac{3\pi}{4}\}$ to $\{z \in \mathbb{C}| \frac{-\pi}{2} < \text{arg} z < ...
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Conformal mapping of the intersection of two ellipses

Can the region between intersection of two ellipses or circles (looks like a crescent) be mapped on to a rectangular region or any other simple region? And is there a systematic way to determine the ...
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What is the most general entire function that takes each complex value once and only once in C? [duplicate]

Some thoughts are: The function should be bijective, but I think the only bijective maps from C -> C are the Mobius transformations, z -> AZ+B / CZ+D, with determinant, AD - BC not zero. The ...
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Family of analytic functions from unit disk to the plane minus a line

Let $\mathcal F$ be the family of analytic functions on the unit disk $\,\mathbb D=\{z: \lvert z\rvert<1 \},$ such that $f[\mathbb D] \subset \mathbb C\setminus(-\infty, 0]$. Show that $\mathcal ...
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Is the Fourier transform a conformal map on $L^{2}$?

I read that a conformal map is one that preserves the angles. I know nothing more about conformal maps. I don't know where to find a generalized definition of a conformal map, but I guess that if ...
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Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
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Conformal maps onto open right half plane

On the Big Rudin there is the conformal map $$\varphi(z) = \frac {1+z}{1-z}$$ which sends $\{-1, 0, 1\}$ to $\{0, 1, \infty\}$. The book says: The segment $(-1, 1)$ maps onto the positive real ...
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45 views

Easy application of the Riemann Mapping Theorem

Riemann Mapping theorem Every simply connected region $\Omega \subset \mathbb C$ is conformally equivalent to the open unit disk (except $\Omega = \mathbb C$) What are application of this ...
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59 views

Conformal map from disk with smaller disk removed to upper half plane

I'm working on a problem that was a previous complex qualifying exam at my university. I believe I have a solution, but I'm not entirely confident in it. The problem is as follows: Find a ...
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Inversion map is a Conformal map

I'm studying PDE by Evans book and I need to show that the inversion map $f:\mathbb{R}^n-\{0\}\to \mathbb{R}^n$, defined by $$f(x)=\frac{x}{\|x\|^2}$$ is conformal. So I have a hint, show that ...
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Constructing a Mobius transformation that acts on any two points of the upper half complex plane:

I would like to construct a Mobius transformation that sends any two points $z_1$ and $z_2$ from the upper half of the complex plane to i and to $iR^+$, i.e., given any two points $z_1$ and $z_2$, ...
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conformal map disc with two removed points

I need to find all the bijective conformal maps from $D = \{ |z| < 1, z\neq \pm 1/2 \}$ onto itself. Since this set is not simply connected, I think that the $180°$ degree rotation is the only ...