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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holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
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Find a conformal map from semi-disc onto unit disc

This comes straight from Conway's Complex Analysis, VII.4, exercise 4. Find an analytic function $f$ which maps $G:=$ {${z: |z| < 1, Re(z) > 0}$} onto $B(0; 1)$ in a one-one fashion. ...
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38 views

Conformal mapping of the domain bounded by a line segment and a circular arc

I am trying to construct a conformal map from the region $R$ which is the set of points in the complex plane bounded by the segment connecting $i$ and $1$ and the part of the unit circle in the first ...
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A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
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Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
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Conformal group in two dimensions

In Conformal field theory, physicist says, the conformal group in two dimensions is infinite dimensional, so the associated with the infinity of generators and infinity conserved charges provided. Is ...
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21 views

Analytical form of conformal mapping of simple closed curve

I want to create a mapping to map a simple closed curve to a unit disk, in analytic forms. The curve is simple enough, with several segments: each with an analytical form, but no general analytical ...
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43 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
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The proof of the Area Theorem for Conformal Maps

The Area Theorem: Suppose $f(z)$ is one-to-one and analytic on the punctured unit disk, and is given by $f(z) = 1/z + \sum_0^\infty a_nz^n$ Then $\sum_0^\infty n|a_n|^2 \le 1$ I'm reading the ...
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How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$ X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3 $$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
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30 views

Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
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Conformal map two slits to circles

I am trying to find a conformal mapping that maps a double slitted plane onto a plane with two circles. The two slits are both located along the real axis with similar lengths. For a single slit ...
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47 views

Schwarz-Christoffel mapping onto infinite L-shaped region

I'm trying to map the upper half plane onto the infinite L-shaped region $$ \Omega = \{z = x+iy; \ x > 0, \ y > 0, \ \min(x,y) < 1 \} $$ My first try is a Schwarz-Christoffel function $$ F(w) ...
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19 views

Find a conformal mapping from lens to first quadradrant

Consider the disks of radius 1 centered at 0 and 1 in the complex plane. Their intersection forms a lens shape. I want a complex function which is a conformal map from this lens to the first quadrant. ...
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29 views

Injective polynomial on the unit disc

Let $P(z)=\sum_{k=0}^{n}{a_kz^k}$ be polynomial that is injective in the open unit disc. Show that $|a_n|\le |a_1|/n$. I know that if $P$ is injective function than $P$ is conformal map and therefore ...
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Does $T$ map a circle to a circle?

I have something to ask about the following map on in $\mathbb{C}$ $$ T \quad : \quad \longmapsto \frac{-2}{\bar{z}+i} -i $$ The map is well-defined whenever $z \neq i$. I have shown that it maps the ...
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18 views

Extremal length of a compound object

Lets say we want to find the extremal length of the family of curves $\Gamma$, which we say is $\mathcal{L}(\Gamma)$. $\Gamma$ moves through two adjacent Riemann manifolds, which have different ...
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A conformal image of a Brownian motion is a time changed Brownian motion

I have read a paper which has stated the following: A conformal image of a Brownian motion is a time changed Brownian motion. The paper cites R. Durret, Brownian motion and martingales in ...
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arctan maps the unit disk onto a band around the imaginary axis

Let $D\subseteq\mathbb{C}$ be the unit disk; that is, $D=\{z\in\mathbb{C}:\ |z|<1\}$. Let $B\subseteq \mathbb{C}$ be some band around the imaginary axis: $B=\{z\in\mathbb{C}:\ ...
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17 views

Conformal Mapping Question Relating Solution of Laplace Equation on Different Domains

Above is my question. The issue that I'm having is that I find that the given mapping, call it $f$, maps $C_1$ to itself, and $C_2$ maps to a circle of radius $5$ and centre $-2$. As much as this in ...
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65 views

Turing machine for the language a^nb ^2nc^3n

How can we give a Turing Machines that accept following language. $$a^nb^{2n}c^{3n}$$ I am allowed to use also pseudo-code descriptions (i.e. high level descriptions of movements of r/w head):
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22 views

Complex Variables Conformal Mapping in Complex Plane of harmonic Functions

Consider the harmonic function $u(x,y) = 1 - y + x/(x^2+y^2)$ on the upper half plane $y > 0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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Map a half sliced unit disk to upper half plane

"half sliced unit disk" Can somebody tell me how to map this conformally to the upper half plane? I think the symmetry principle should be applied here but stuck on that for hours. Pardon my hasty ...
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68 views

Conformal map entire domain to a strip with specific branchcuts

I am looking for a conformal mapping function that maps the entire z-plane to an infinite strip. (e.g. T=f(z) & -b < Real(T) > b ) I hope to find a function that cuts open to original domain ...
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130 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
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35 views

Zero-distortion map projection

Is it possible to take a limit of map projections (from a sphere to a plane) with ever-smaller distortion factors to get some kind of dendritic limit projection that has zero distortion everywhere? My ...
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96 views

Conformal equivalence of resistance

Link to the question in the physics portal. I'm currently working on a system that uses a logarithmic and a Schwarz-Christoffel transformation to calculate the resistance of a specific area. With ...
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1answer
57 views

Find a conformal map from the exterior of the closed unit disk to the unit disk

Question: Find a conformal map from the exterior of the closed unit disk to the unit disk. Also, prove that it is indeed a conformal map (bijective and holomorphic along with its inverse). I missed ...
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25 views

Conformal Mapping with homeomorphic extension

Suppose that $$D=\{z:0<x<a,0<y<b\}$$ and that $$D'=\{w:0<u<c,0,v<d\}$$ Then there is a conformal mapping $f$ of $D$ onto $D'$ whose homeomorphic extension $\tilde{f}$ to ...
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Explicitly reconstruct conformal mapping from conformal factor

Consider two smooth, compact surfaces $\mathcal{S_i} \subset \mathbb{R}^3$, with Riemannian metrics $g_i$, $i=1,2$ and a conformal mapping $f:S_1\rightarrow S_2$. Suppose we know the conformal ...
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Conformal Mapping Between Two Domains (log)

Does anyone have a recommendation as how to go about solving this problem? I want a conformal from G to H where $$ G = \{ z \in \Bbb C \ | \ |z|<1, |z+i|>\sqrt{2} \}, S = \{ z \in \Bbb C \ | \ ...
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48 views

Prove that a pseudo-hyperbolic ball is a Euclidean ball. Find the radius and center of the Euclidean ball.

We have that the pseudo-hyperbolic metric in the open unit disk $\mathbb D$ is defined by $$ \rho(z,w) = |\phi_w(z)|, \qquad \phi_w(z) = \frac{w - z}{1 - \overline w z}$$ where $z,w \in \mathbb D.$ ...
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128 views

Conformal map from exterior of unit circle to upper half plane

I'm trying to find a conformal map from the space $\Omega = \mathbb{H}\setminus\{z : |z-\frac{i}{2}|\leq\frac{1}{2}\}$ to the upper half plane. I think I'm most of the way there, but I wanted to check ...
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38 views

Schwarz-Christoffel transformation understanding

I've been reading this explanation (with pic and formula) about the Schwarz-Christoffel mapping. I'm not really used to this sort of argument. My question is why are all terms constant in $(21.3)$ ...
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35 views

Does this conformal map from a rectangle exist?

It is well known (by Schwarz-Christoffel) that if $k \in (0,1)$, then the Jacobi elliptic function $\mathrm{sn}(\cdot,k)$ provides a biholomorphic map from the rectangle $(-K(k),K(k)) \times ...
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31 views

The fixed points of analytic self-maps of $\mathbb{D}$

So far, I have assumed that $z_1$ is a fixed point of an analytic self map of $\mathbb{D}$. Then, I summoned the conformal self map of $\mathbb{D}$, $\phi$ to take $z_1\to 0$. It follows from Schwarz ...
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conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
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43 views

Conformal Mappings- Fluid Dynamics

a)Show that the transformation z=F(Z) where F(Z)=Z+$a^2\over Z$, a is real positive constant, z=x+iy and Z=X+iY, maps the exterior of the circle |Z|=a to the exterior of the plate Z=0, -2a b)Write ...
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56 views

Conformal mapping from exterior of semi disk onto exterior of unit disk

Can you construct a conformal mapping from exterior of upper semi-disk onto exterior of unit disk and fixes infinity?
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100 views

Conformal maps from the upper half-plane to the unit disc has the form

Prove that the conformal maps from the upper half-plane $\mathbb{H}$ to the unit disc $\mathbb{D}$ has the form $$e^{i\theta}\dfrac{z-\beta}{z-\overline{\beta}},\quad\theta \in \mathbb{R} \text { and ...
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38 views

Conformal Mapping and Relating Solutions (of Laplace) of Domains (via the Mapping)

Find a conformal equivalence between the following domains: the strip $ S = \{ z \in \Bbb C \ | \ 0 < \Bbb Im(z) < 1 \} $ and the quadrant $ Q = \{z \in \Bbb C \ | \ \Bbb Re(z) > 0, \Bbb ...
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Conformal map iff holomorphic

It seems like if $U$ is an open subset of the complex plane, $\mathbb{C}$, then a function $$f: U \rightarrow \mathbb{C}$$ is conformal if and only if it is holomorphic and its derivative is ...
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Conformal maps from the left half plane to a sector

I'm trying to do a conformal mapping from the left half plane to a sector symmetrical around -X axis. would it be $Z=W^m$ while $m= (\frac{2}{\pi})(\pi-\theta)$. theta is the angle of the sector ...
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Mapping circles via inversion in the complex plane

Consider two unshaded circles $C_r$ and $C_s$ with radii $r>s$ that touch at the origin of the complex plane. The shaded circles $C_1,C_2...C_7$ (labeled in counterclockwise direction sequentially) ...
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Why are Mobius map conformal at infinity?

Why are Mobius map conformal at infinity? I think I'm missing a subtlety! So we know an analytic function, $f$, is conformal at $z$ iff $f'(z) \neq 0$ But we see that the derivative of Mobius ...
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88 views

Conformal mapping on unit disk

Let $\mathbb{C}_{-}^r:=\mathbb{C}\setminus(]-\infty,-\frac{1}{r}])$, $~\mathbb{C}_{-}:=\mathbb{C}\setminus(]-\infty,0])$, $~\mathbb{D}:= \{z\in \mathbb{C}\mid |z|<1\}$ and $\mathbb{G}=\{z\in ...
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83 views

Conformal map from unit disc $D\setminus\lbrace0\rbrace$ to $\mathbb{C}\setminus\bar D$

Question: If $D$ is the unit disc, find a conformal equivalence from $D\setminus\lbrace0\rbrace$ to $\mathbb{C}\setminus \bar D$ My Attempt: I have no idea how to start this problem... I don't think ...
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163 views

What is the conformal equivalence from the half-plane to the unit disc?

Problem: Find a conformal equivalence from the half-plane {z : Re(z) > 1} to the unit disc D. My Attempt: Just a FYI that I am completely new to conformal mapping. Okay, so I know that $$h(z) = ...
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82 views

A holomorphic function is conformal

I am trying to show that if a function $f = u+iv$ is holomorphic with $\partial_z f(z)$ always non zero, then $f$ is a conformal mapping, i.e. it preserves angles between smooth curves. If $f$ is ...
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55 views

Mirror point with respect to Riemann circle (Möbius transformation)

The problem is "Find a Möbius transformation $w(z)$ that maps the area $Re( z) > 0, |z-1| > 1$ to the strip $0<Re(w)<2$." I realize there are many ways to skin a cat, but what I wanted ...