Questions tagged [conformal-geometry]
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 the Mercator projection from cartography, and the Möbius transformations of the Riemann sphere.
1,192 questions
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$w = \frac{2z + 3}{z - 4}$ maps the circle $x^2 + y^2 - 4x = 0$ onto the straight line $4u + 3 = 0$
Our question is:
Show that the transformation $w = \frac{2z + 3}{z - 4}$ maps the circle $x^2 + y^2 - 4x = 0$ onto the straight line $4u + 3 = 0$.
So, here's the solution:
We have,
$$
w = \frac{2z + 3}...
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Conformal map and the distance [closed]
I'm studying the paper 'Frank R L, König T, Tang H. Classification of solutions of an equation related to a conformal log Sobolev inequality. Advances in Mathematics, 2020.' and having trouble in ...
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Conformal mapping to the unit disc
Consider the region
$$
\Omega = \left\{z:\Re z > 0,\ \Im z \le 0\}\cup\{z:\Im z \in \left(0,1\right)\right\}
$$
I want to find an explicit conformal mapping from $\Omega$ to the unit disc.
I tried
...
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I need help resolving egregious typos in this complex analysis theorem statement
First of all, I tried to go online to find a similar theorem statement but couldn't find anything.
I underlined the points that I'm confused about.
The wording in the first underlined part is weird. ...
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How to derive the geometrical relationship between the Laplace equation and a rectangle with anisotropic properties?
My question is to ask how the geometrical relationships given for Equations 11-13 below were derived.
The question concerns transforming a rectangle with anisotropic properties to an isotropic ...
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Divergence Issues in Series Expansion of Conformal Transformation of n-gon polygon with negative angle.
Greeting everyone.
I am having a numerical transformation of given polygon to the surface of an unit cirle (following the procedure by this article "A Conformal Mapping Method to Predict Low-...
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Help finding the the mapping form the z-plane to the w-plane of the unit circle.
Here is the question:
A mapping from the $z$-plane to the $w$-plane, where $w = u + iv$, is given by:
$$w = \frac{z - 1}{z + i}$$
The unit circle in the $z$ plane maps to which of the following in ...
- 11
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Finding a conformal mapping
I am trying to find a conformal map between the regions:
$$\Omega_1 = \{z \,:\, |z| < 2,\, |z - 1| > 1\}, \quad \Omega_2 = \{z\,:\,\text{Im}(z) > 0 ,\, |z - i| > 1\}.$$
Specifically, I ...
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Complex Analytic 1-1 functions
Statement : If $f$ is analytic at $z_{0}$ and $f^{\prime}\left(z_{0}\right) \neq 0$, then there is an open disk $D$ centered at $z_{0}$ such that $f$ is one-to-one on $D$.
The first couple of ...
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Discrete conformal mapping between two genus zero closed surfaces
Given a discrete genus zero closed surface with triangle mesh, is it possible to find a discrete conformal mapping to a sphere?
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Schwarz-Christoffel transformation of rectangle
I have a rectangle of known aspect ratio $\frac{L_y}{L_x}$. While not necessary, assume that the rectangle is in the first quadrant of the XY-plane with the points given as:
$z_1 = (0, L_y)$, $z_2 = (...
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Complex plane conformal map $g\circ f(A) = B$
Let
$$A:=\{ z \in \Bbb C: |z+1|<2,\ \Re(z)<-1,\ \Im(z) >0\}, \quad f(z) = \frac{3+z}{1-z}$$
Find a conformal map $g$ such that $g\circ f(A) = B$ where
$$B = \{w\in \Bbb C : \Im(w) < \pi \}$...
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Uniformization and Constructive Analytic Continuation of Taylor-Maclaurin Series
Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
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Milne-Thomson Circle Theorem for Multiple Circles in Potential Flow
I'm currently studying potential flow in fluid dynamics and have been utilizing the Milne-Thomson Circle Theorem to analyze the flow around a single circular cylinder. This theorem has been ...
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elaboration proof for double of conformally flat manifold
I'm reading a paper and there is a proof that the double of a compact locally conformally flat Riemannian manifold with totally geodesic boundary again carries a locally conformally flat structure. ...
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I can't cancel out some terms in the Conformal invariance of 2D Laplace equation
I started with the equation
$$(\partial ^2_x + \partial _y ^2 ) f=0$$
Then I changed to $(r,w)$ co ordinates using a Conformal transform :
$$\partial _x= u \partial _r - v \partial w$$
$$\partial _y= ...
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Find a conformal map from $\{0 < \Im z < 1\}$ to $\{\max\{|z-1|,|z+1|\}< \sqrt{2}\}$
Find a conformal/biholomorphic map from $\{z\in \mathbb{C}: 0 < \Im z < 1\}$ to $\{z\in\mathbb{C}:\max\{|z-1|,|z+1|\}< \sqrt{2}\}$ (explicitly).
Such a map obviously exist, due to Riemann ...
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Solving PDE on domain bounded by two intersecting curves (parabolas) and conformal mapping
I am trying to solve a PDE similar to the heat equation:
$$
\frac{\partial T}{\partial t} + (\vec{u} \cdot \nabla)T = D \Delta T
$$
(where $\vec{u}$ is a known advecting vector field) with some ...
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Is the complex tangent Injective?
So if we consider the Analytic function $f(z)=\tan(z)$ when $z\in \{z\in\mathbb{C}\text{ : } -\pi/2<Re(z)<\pi/2 \}$ is it injective ? Its derivative is non zero however this only guarantees ...
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A problem in the proof of mapping formual (H →P) in stein Complex Analysis chapter8
We know that $F$ is a conformal map of $\mathbb{H}$ to $P$ ($P$ is a polygon)
Let$$
h_{k}(z)=(F(z)-a_{k})^{ \frac{1}{\alpha_{k}}}
$$
Then
$$
\frac{F^{''}(z)}{F^{'}(z)}=-\beta_{k} \frac{h^{'}_{k}(z)}{...
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Green's function of the conformal Laplacian
I am reading T. Parker, S. Rosenberg, "Invariants of conformal Laplacians", J. Differential Geom. 25(2): 199-222 (1987). I would like to understand how Green function changes if the metric ...
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Conformal coordinates - Isothermal coordinates
Given a smooth and closed surface $S$ parameterized by coordinates $u,v$ such that its metric can be written in the form $ ds^2 = \phi(u,v)^2(du^2 + dv^2)$ implies that the coordinates $u,v$ are ...
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Isometries as a subgroup of conformal diffeomorphisms
I have a question that, if true, does not seem trivial to prove, but if false, I am unable to find a counterexample. The question is as follows:
Given a 3-dimensional Riemannian manifold $M$, is the ...
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Conformal map of two intersecting circles
While studying for my Complex Analysis course, I came across the following exercise:
You are given the following region
$$\Omega = \left\{ z \in \mathbb{C} \mid |z - i| < \sqrt{2} \text{ or } |z + ...
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Doubt in Mobius geometry
I am currently studying Mobius geometry. I found a group in Mobius geometry called Mobius group which contains Mobius transformations. I have the following doubt.
Dose this group contain ...
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Area of image of holomorphic function on the disk [duplicate]
I'm working on an old qualifying exam problem, which asks me to show that if $f:\mathbb{D}\rightarrow \mathbb{C}$ is holomorphic and injective and $f'(0)=1$, then the area of $f(\mathbb{D})$ is at ...
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Comparing areas between conformal metrics in $\mathbb{R}^2$
I would like to ask for a reference on the following subject:
Let $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ be a radial positive function, i.e., $$f(x,y)=\lambda(x^2+y^2)$$ for some $C^\infty$ ...
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Torus equation in conformal geometric algebra
How would you define a torus using conformal geometric algebra?
Since CGA has circles as a primitive, It seems to me that we should be able to able to define a torus as a circle C rotated around a ...
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Integral representation of conformal map on the upper half plane
It is a fact that there is a one to one correspondence between the space $M(k)$ of finite, signed Borel measures on $\mathbb{S}^1$ with total mass equal to $2$ and total variation equal to some $2 \...
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Stein and Shakarchi Complex Analysis Chapter 8, Proof of Lemma 1.3
I am confused about the proof to Lemma 1.3 in Stein and Shakarchi's Complex Analysis Chapter 8 (Conformal Mappings).
Here is a screenshot of the Lemma and its proof:
My question is is the following: ...
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Image of upper half-plane under sin(z) is not the half strip (Stein and Shakarchi Complex Analysis, Chapter 8)
In section 1.2 of Chapter 8 Conformal Mappings of Stein and Shakarchi's Complex Analysis, it is said that the map $f(z) = \sin(z)$ takes the upper half-plane conformally onto the half-strip $\{w = x + ...
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Schwarz-Pick Theorem Exercise
I am trying to solve the following problem, which I believe should be solved using the Schwarz-Pick Theorem.
Let $\mathbb{D}$ be the open unit disk and let $f\colon \mathbb{D}\xrightarrow{} \{z\in \...
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Dirichlet problem with conformal maps
The problem I am struggling to solve is from Brown and Churchill's Complex Variables and Applications and is intended to be solved using conformal maps:
Derive an expression for steady state ...
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Conformal mapping concerning a branch cut
Map the region $\Omega = \{z=re^{i \theta} \in \mathbb{C}: r >0, \frac{\pi}{4} < \theta < \frac{7\pi}{4} \} \setminus [-1,0]$ onto $\mathbb{H}$.
Thus, the $\Omega$ is the complex plane with ...
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conformal maps of S2 to compact topologically equivalent region in R3?
How do I conformally, compactly map the unit sphere in $R^3$?
I know that if I have a compact, simply connected region of the plane, $R^2$, I can create a conformal map onto a new region in the same ...
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Definition of an inner product.
I was solving some problems on isometries on the upper half plane and I came up with the following post: Excercise in isometries of the half upper plane here the answer states the following inner ...
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Conformal map between a domain minus an arc and the unit circle
I was trying to understand the paper computability of Brolin-Lyubich measure and, in the last counter-example of the paper (page 25), they say about a conformal map between the set $(\mathbb{C}\...
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Prove that there is no conformal mapping between $\mathbb{D}\setminus\{0,1/2\}$ and $\mathbb{D}\setminus\{0,1/4\}$
I have to prove that there is no conformal mapping between $\mathbb{D}\setminus\{0,1/2\}$ and $\mathbb{D}\setminus\{0,1/4\}$. I honestly have no idea. Intuitively, I guess that the problem has to do ...
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Conformal map a quarter disc into a horizontal strip
I'm interested in mapping the quarter disk $\{z = x+iy \in \mathbb{C}: 0 < x < 1, 0 <y<1 \}$, i.e. the intersection of $\mathbb{D}$ with the first quadrant to the horizontal strip $\{z=x+...
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What is C minus infinitely many disjoint "Jordan Balls" conformally equivalent to?
So I recently came across a Theorem due to Koebe proved in 1918 that states that an M connected domain $\mathbb{D}$ in $\mathbb{C}$, that is a a domain such that $\partial \mathbb{D}=\bigcup_{i=1}^{k}\...
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Show that the conformally complete Schwarzschild spacetime is asymptotically flat at null infinity
I am trying to show that the conformal factor used to conformally complete the
Schwarzschild spacetime renders it asymptotically flat at null infinity (according to the mathematical definition given ...
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Surjectiity of a conformal map
I'm stuck on this exercise:
Let $\left\{ z \in \mathbb{C} : z = x + iy, \; y > 0 \right\}$
Check whether the application $\phi : {\mathbb H}→ {\mathbb C}$ where ${\mathbb C}$
given
$\phi(z) = z^4 - ...
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Construct an explicit biholomorphism between two domains
This problem is from my homework: Construct an explicit biholomorphism between $D_1=\mathbb{C}-\{-x\pm\sqrt{-1}\pi\mid x\ge1\}$ and $D_2=\{x+\sqrt{-1}y\mid -\infty<x<\infty, -\pi<y<\pi\}$.
...
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How to Find a Conformal Mapping from a Crescent-Shaped Region to an Annulus?
I am trying to find a conformal mapping from
$\{𝑧:∣𝑧∣<1\}\cap\{𝑧:|z−\frac{1}{2}∣>\frac{1}{2}\}$ onto an annulus.
This question is the very last problem from an old complex analysis exam.
I ...
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Determining a conformal mapping to the unit disk
Let
$$\Omega = \left\{z \in \mathbb{C} : |z - 1| > 1, |z - 2| < 2, \Im(z) > 0\right\}.$$
We want to determine a conformal map which maps $\Omega$ to the unit disk. We solved it by composing $...
2
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0
answers
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Using conformal mapping to solve Laplace equation
Given the circles $$C_1 : x^2 + y^2 = 1,\quad C_2 : 5x^2 - 4x + 5y^2 = 0$$ let $D$ be the finite region between $C_1$ and $C_2$. Using the conformal mapping $$w = \frac{z-2}{2z-1}$$ solve the problem $...
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Let $D=\{z\in \Bbb C; 0<\Im z<\pi, z\not\in [0,\pi i/4]\cup [3\pi i/4,\pi i]\}$. How to find a conformal map from $D$ to the upper half plane?
Let $D=\{z\in \Bbb C; 0<\Im z<\pi, z\not\in [0,\pi i/4]\cup [3\pi i/4,\pi i]\}$. Here $\Im z$ is the imaginary part of $z$. How to find a conformal map from $D$ to the upper half plane?
My ...
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Combining the two conformal mapping equations
The mapping function ${Z_{\rm{1}}}$, which transforms the outside region of a unit circle (in the $\zeta$-plane) onto the outside region of the unit circle with two unequal radial aligned cracks (in ...
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1
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Conformal map between $D-[\alpha,1)$ and $D-[0,1)$
Let $\alpha$ be real, $0 \leq \alpha < 1$. Let $U_{\alpha}$ be the open set obtained from the unit disc by deleting the segment $[\alpha, 1)$
Find an isomorphism of $U_{\alpha}$ with the unit disc ...
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answer
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Freitag and Busam Complex analysis conformal map condition theorem
Theorem I.5.15 of the book (Freitag&Busam complex analysis) says: A map $f:D\to D'$, where $D,D'$ open in $\mathbb{C}$, is locally conformal if and only if it is analytic and its derivative is ...
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