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 transformation of complement of disk in upper half plane

Let $U$ be the complement in the half-plane $\operatorname{Im} z > 0$ of a disk of radius $a<1$ centered at $i$. I am looking for a conformal transformation that maps $U$ onto an annulus. Since ...
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54 views

Showing that stereographic projection is a homeomorphism

For any $n\geq 0$,the unit $n$-sphere is the space $S^{n}\subset \mathbb{R^{n+1}}$ defined by $$S^{n}=S^{n}(1) :=\left\{ (x_{1},...,x_{n+1}) \left\vert\,\sum_{i=1}^{n+1} x_{i}^{2}=1\right.\right\}$$ ...
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29 views

Does there exists known special cases of a zero Riemann tensor for 3D metrics?

In two dimensions, if one has a flat metric $g_{ab}$, then one can transform $g_{ab}$ to another flat metric $h_{ab}=e^{2\varphi}g_{ab}$, when $\nabla^2 \varphi =0$ and the Riemann tensor remains ...
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How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
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57 views

Dirichlet Problem using conformal mapping

Using appropriate conformal maps, solve the Dirichlet problem (for Laplace's equation) for the following region and boundary condition: $U=\{\text{Im}(z)>0\cup \text{Im}(z)=0\}$, with boundary ...
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35 views

Harmonic maps and angle preservation

I have the following question: What are the angle preservation properties of harmonic maps? Conformal maps preserve angles exactly, but they distort lengths. In this sense a conformal map is the ...
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15 views

Conformal map from the inside of the unit disk to the inside of an ellipse [duplicate]

I lack intuition when it comes to some conformal mappings and I'm presently looking for a conformal map taking the inside of a disk, let's say the unit disk and sending it to the inside of an ellipse. ...
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33 views

Conformal/Biholomorphism equivalence classes in $\mathbb{C}^n$

Recently I have got interested in the topic of conformal equivalence classes of complex domains, mostly one-dimensional ones. Here by conformal map $f: U \rightarrow V$ I mean a complex holomorphic ...
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35 views

Clifford Algebras for Projective and Conformal Geometry

According to Clifford Algebra: A Visual Introduction, A Clifford Algebra over $\mathbb{R}^3$ may describe the rigid motions in space (namely, conjugation acts as a reflection by a plane). A ...
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38 views

Existence of such function

So we know that if $g(z)=\frac{z-c}{1-\overline{c}z}$ $(c\in\mathbb{C})$ $|g(z)|=1$ for $|z|=1$. Does there exist a function $f(z)$ satisfies the following properties: (1) $f$ is analytic in some ...
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34 views

Find a Harmonic Function which is $1$ on $|z-5i| = 4$ and $0$ on $Im(z) = 0$

I've been trying to solve the following problem: Find a function $\varphi$ which is harmonic in the upper half-plane exterior to the circle $|z-5i| = 4$, is $1$ on $|z-5i| = 4$, and $0$ on the real ...
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43 views

Does a conformal map take boundaries to boundaries?

I think it is a well-known result that conformal maps between sets in $\mathbb{C}$ take boundaries to boundaries. However, I looked around a little and I had trouble finding this result. Is it true? ...
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29 views

Enumerating Automorphisms of Upper Half Plane

I'm trying to find all conformal automorphisms of the upper half plane $\{\Im[z] \gt 0\}$, known to be $f(z) = \frac{az + b}{cz + d}$ where $a, b, c, d$ are real and $ad - bc \gt 0$. The main work ...
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32 views

Plotting the region $ -1 < Re(z) \le 1$ and $ -\pi/2 < Im(z) \le \pi/2$ before and after being transformed by $w=e^z$

Can someone please verify that my diagram plot based on the calculations below is correct -Thanks. $$w=e^z=e^x\cos(y)+ie^x\sin(y)$$ $$\implies u=e^x\cos(y)\space and \space v=e^x\sin(y) \tag{1}$$ ...
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42 views

Transform $\Re(z)=1 \space , \Re(z)=\Im(z) \space and \space \Re(z)=-\Im(z)$ using the mapping $w=iz^2$

Can someone please verify whether i am doing this the right way -Thanks. $$w=iz^2=i(x^2+2xiy-y^2)=-2xy+i(x^2-y^2)$$ $$\color{green}{u=-2xy \tag{1}}$$ and $$\color{green}{v=x^2-y^2 \tag{2}}$$ ...
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35 views

Transformation of line $y=k=constant$ under the mapping $w=cos(z)$

I have been going over this post and found myself confused by the calculations given by the OP. $$\color{red}{ shouldn't \space this \space be \space done \space as \space follows:}$$ ...
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19 views

Conformal Mapping defined by Cubic Polynomial

Ahlfors studies the mapping defined by $\omega = a_0 z^3 + a_1 z^2 + a_2 z + a_3$ in Complex Analysis 3rd Edition at the bottom of page 95. First he says we can get rid of the quadratic term by the ...
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43 views

Determine the image of the strip $S$ consisting of all points $z$ with $\frac{-\pi}{2}\lt Re(z) \lt \frac{\pi}{2}$ and $Im(z)>0$ under $w=i\sin z$

$\color{green}{\text{transformation is}\space w=i\sin z}$ $$w=i\sin z = i\sin(x+iy)=\frac{1}{2}\left(e^{ix-y}-e^{-(ix-y)}\right)=-\cos(x)\sinh(y)+i\sin(x)\cosh(y)$$ $\therefore u = -\cos(x)\sinh(y) ...
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36 views

The action of the conformal mapping -(1/z)

I know that the mapping -1/z is conformal away from the origin, since the mapping would then be analytic and have a non-zero derivative everywhere in C. It apparently also maps the upper half plane ...
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44 views

Find the region in the w-plane to which the line y = 1 is transformed by $\frac{1}{z}$

I tried to do the following: $$w=\frac{1}{z}=\frac{x-iy}{x^2+y^2}$$ $\implies u = \frac{x}{x^2+y^2} and\space v = \frac{-y}{x^2+y^2}$ $\color{green}{need\space to\space transform\space the\space ...
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59 views

Conformal mapping for region between a square and a circle

I was wondering if anyone knew a mapping that would take the region between a square of length 4 and a unit circle, and map it into the region between two circles. It would be great if the mapping is ...
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26 views

Conformal transformation of the divergence

Let $f$ be a smooth function on a $n$-dimensional Riemannian mainfold $(M, g)$, so that $\tilde{g} = e^{2f} g$ is a conformal transformation of $g$. I'm trying to show that the divergence operator ...
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42 views

Schwarz Lemma/Conformal mapping problem

Let $F:\mathbb{H}\rightarrow \mathbb{D}$ be holomorphic, where $\mathbb{H}$ is the upper half plane and $\mathbb{D}$ is the unit disc. Show that if $F(i)=0$, then $$|F(z)|\leq ...
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38 views

The image of a Joukowsky transform,

How can I show that the Joukowsky transform, $J(z)= z+\frac{1}{z}$ maps the set $R =\{(x,y)\in R^2: x^2+y^2>1, y>0\}$ conformally onto the upper half-plane? It's clear that the parts of the ...
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45 views

Compute $f''(0)$ for a holomorphic function on a square given $f'(0)$ and $f(0)$

Let $S$ be the square $\{x + iy: |x| < 1, |y| < 1\}$ and $f:S \rightarrow S$ a holomorphic function so that $f(0)= 0$ and $f'(0) = 1$. Find $f''(0)$. It seems like I need to use Cauchy's ...
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24 views

inversion of the circle $t \mapsto (3 + is) + e^{it} $ around the unit circle.

We know that inversion interchanges lines and circles, but it's very hard to The inversion map about the unit circle is just $\displaystyle z \mapsto \frac{1}{\overline{z}}$. As a Möbius ...
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46 views

Riemann Mapping Theorem, the concept of a Riemann mapping

If I construct a composition of mappings that map the upper half of the unit disk conformally to the entire unit disk, then this mapping is a Riemann mapping, by the Riemann Mapping Theorem, since ...
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31 views

Uniqueness of conformal mappings with different normalizations: three boundary points, or an interior point

Some stuff I've seen in lecture but am still a little shaky on: 1) To determine my mapping explicitly, it suffices to know where 3 distinct points on my pre-image object, say, the unit circle, gets ...
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71 views

Automorphisms of the upper half plane

STATEMENT: Suppose $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ are two pairs of three distinct points on the real axis with$$x_1<x_2<x_3 \;\;\;\;\text{and} \;\;\;\;\;y_1<y_2<y_3$$ Prove that ...
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Conformal Map for Circle to Circle

I am trying to find a conformal map that maps a circle in the $\zeta$ plane to a circle in the $z$ plane. As far as I know, a Mobius transformation is appropriate for this. These are the conditions ...
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31 views

Conformally mapping the unit disk to the upper-half plane

Within $\mathbb{C} \cup \{ \infty \}$, consider the unit-disk $\mathbb{D} = \{ z : |z|\leq 1 \}$ with three points labelled as $a$, $b$, $c$ on its boundary. I want to map $\mathbb{D}$ conformally ...
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154 views

conformal mapping, regions of the complex plane marked +/-, find the function f,

The picture shows what the function f: $\mathbb{C}\to\mathbb{C}\cup\infty$ does to the plane. The values 0 at 0, 1 at $\pm$1, and $\infty$ at $\pm i$ are specified. To elaborate on the picture: ...
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Mapping a region between two circles on a half plane,

I've tried this problem for awhile now but something strange is happening at the end of my solution. The question asks to map the region between $|z| =1$ and $|z-\frac{1}{2}| = \frac{1}{2}$ on a half ...
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47 views

Conformal mapping of part of an annulus

I have a question about conformal mapping. I am wanting to map annuli to some other simple domain (probably rectangular). I have an image of my problem below In image A. we see a standard annulus ...
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1answer
92 views

Conformal mapping circle onto square (and back)

I'm programming an implementation of the Peirce quincuncial map projection. The projection involves a stereographic projection of a hemisphere of the globe onto a circle (I've got that part), then ...
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1answer
25 views

Excercise (and consequence) for Explicit Riemann Mapping

Set $V:=D_1(e^{\frac{i\pi}{6}})\cap D_1(e^{-\frac{i\pi}{6}})$. Find the (unique) explicit conformal function $f:V\rightarrow \mathbb{C}$ such that $f(V)=\mathbb{D}$ with $f(1)=0$ and $f'(1)>0$. ...
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50 views

Excercise for Explicit Riemann Mapping

Set $U:=\{ z\in \mathbb{C}\mid \arg(z)\in(-\alpha,\alpha)\}$ with $\alpha\in(0,\pi)$.Give explicitly the conformal mapping $f:U\rightarrow \mathbb{D}$ such that $f(U)=\mathbb{D}$ with $f(1)=0$ and ...
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60 views

Conformality of a map

A conformal mapping is a map $f:U\to V$ with $U,V\subseteq\mathbb{C}$ such that the angles are locally preserved. This can be reformulated saying the jacobian matrix is everywhere a scalar multiple of ...
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Can the derivative of a conformal homeomorphism onto a bounded domain be unbounded?

Consider $f$ being a holomorphic homeomorphism of the closed unit disk into the complex plane (ie. $f$ must be a homeomorphism - which also imply bijectivity to its image - of the closed unit disk, it ...
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Analytic onto maps from D to D

We just characterized using the Schwarz Lemma the conformal self maps of the open unit disk. I am now trying to characterize the holomorphic onto maps from $\mathbb{D}$ onto $\mathbb{D}$. As a ...
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27 views

Möbius transformation $(-1,0,1)\to (i,-1,-i)$

Find Möbius transformation $S$ that maps points $(-1,0,1)$ to points $(i,-1,-i)$. And what's the image of real axis and upper half of imaginary axis $\{z\in\mathbb{C}| \operatorname{Im} z \geq 0 \} $ ...
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A conformal mapping from a sector to a strip

What is the simplest function that maps the sector $r < 1$, $0 < \theta < \pi$ conformally onto the strip $0 < u < \pi/2$, $v > 0$? Here, $r$, $\theta$, $u$, $v$ have their usual ...
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1answer
46 views

Conformal transformation of metric on $\mathbb{R}^n$

Let us define the following metric on $\mathbb{R}^n$: $$ g|_v(X, Y) := e^{-|v|^2} \langle X, Y\rangle,$$ where the brackets denote the standard scalar product. How does the resulting manifold look ...
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37 views

Flow of a Metric & Conformality

What is the flow of a metric mathematically? I want to be able to understand what it means to say that a metric preserves a conformal structure through the actual definition $\Theta_t^*g = ...
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31 views

Conformal mapping of nonsimply connected domains

The question asks: Map the complement of the arc $|z|=1$, $y\geq 0$ on the outside of the unit circle so that the points at $\infty$ correspond to each other. How would you construct such conformal ...
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1answer
44 views

Conformal classes and almost-complex structures

It is well-known that on closed oriented surfaces $S$, conformal classes of metrics on $S$ correspond bijectively to complex structures on $S$. My understanding is that this correspondance goes as ...
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Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature ...
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62 views

“Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be ...
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69 views

Conformal cobblestones

Open-ended question: It strikes me that the cobblestone pattern known as Bogen, which can be seen in many European cities, is a fairly accurate representation of the conformal mapping defined by the ...
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1answer
55 views

Conformal mapping from a tetrahedron to a spherical sector

I have been trying to find a way to do a conformal mapping from tetrahedron, specifically a trirectangular tetrahedron, to a spherical sector, but being an engineer with no training in this subject it ...