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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22 views
Mapping of a Lens-shaped region by a Möbius Transformation
Consider the 'lens' described by $\{z:|z-i|<\sqrt{2}\ \text{and}\ |z+i|<\sqrt{2} \}$ . We want to map this to the upper right quadrant using a Möbius transformation.
The two circles meet at ...
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2answers
53 views
Constructing a conformal map from $\mathbb{D}$ to a cut plane
Source: Oxford Exam $A2 \ 1999$
We want to construct a conformal map $F$ from the unit disc $\mathbb{D}=\{z:|z|<1\}$ to $\mathbb{C} \setminus S$ where $S$ is the half-line $\{x+i:x \in (-\infty,0] ...
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12 views
Conformal mapping in neural networks [closed]
Sir,
When we map our complex input range using conformal mapping, what is the advantage of doing so?
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1answer
24 views
Conformal Map from Vertical Strip to Unit Disc
I haven't found a similar question on here, though I suspect the question may be rather well-covered.
I want to find a conformal map from the vertical strip $\{z:-1<Re(z)<1\}$ onto the unit ...
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1answer
33 views
Maximum Modulus theorem applied on mapping
Question: For $|z_0|<R$, I want to show that the mapping $$T(z)=\frac {R(z-z_0)} {R^2-\bar{z_0}z}$$ takes the open disc of radius $R$ $1-1$ and onto the unit disc and $z_0\rightarrow 0$.
Hint: ...
4
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2answers
56 views
Looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex
I am looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when $n=2$. ...
3
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0answers
26 views
Optimization of Möbius transformation
Say I have a family of points $(w_i, z_i)$ for $i=1,2,...,n$, and I wish to find $a,b,c,d$ such that $\sum_i \left|\frac{a z_i -b}{c z_i - d} - w_i \right|^2 $ is minimized. I realize there are things ...
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1answer
53 views
Show that there is no surjective smooth function $S^1 \to S^1\times S^1\times S^1$
This is not homework, but a sample test question. The question is:
Show that there is no surjective smooth function $$S^1 \to S^1 \times S^1 \times S^1.$$
Now I can see that, for example ...
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1answer
24 views
image of a circle under conformal trasformation
Consider a circle:
$C_R=\{w=(x,y): |w|^2=x^2+y^2=R^2\}$
Prove that $A(C_R)$ remains a circle if $A$ is either a conformal or an anticonformal matrix.
My attempt:
I defined the complex number $z:=x ...
5
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2answers
88 views
Hopf's theorem on CMC surfaces
I got stuck reading the proof of the following theorem:
Theorem (Heinz Hopf) Let $X: S^2\to \mathbb R^3$ be a constant mean curvature immersion. Then $X(S^2)$ is a round sphere.
Proof: Let ...
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2answers
40 views
Is there a procedural way of finding a Möbius transformation given prescribed conditions?
Is there a procedural way of finding a Möbius transformation given prescribed conditions?
For example, I've been asked to find a Möbius tranformation which fixes $\mathcal{C}_2$, maps ...
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1answer
31 views
Riemann mapping theorem and an inequality inducing conformality
From an old complex analysis prelim:
Suppose that $\Omega\not= \mathbb{C}$ is a simply connected region, $a$ and $b$ are distinct points on $\Omega$, $f$ is a conformal map of $\Omega$ on ...
3
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1answer
62 views
A question on the uniformization theorem
Wikipedia reads, on the uniformization theorem:
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: ...
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0answers
24 views
formula for inverse multidimensional stereographic projection
i'm in need of formula for inverse multidimensional stereographic projection with variant radius of the sphere. Sadly the only ones i'm able to find have either fixed number of dimensions or don't ...
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1answer
63 views
Parallel transport for a conformally equivalent metric
Suppose $M$ is a smooth manifold equipped with a Riemannian metric $g$. Given a curve $c$, let $P_c$ denote parallel transport along $c$. Now suppose you consider a new metric $g'=fg$ where $f$ is a ...
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1answer
88 views
Why does the non-euclidean distance between the lines $x=0$, $x=1$ approach $0$ as $y \to \infty$?
Why does the non-euclidean distance between the lines $x=0$, $x=1$ approach $0$ as $y \to \infty$?
Please see http://books.google.ca/books?isbn=0387290524 on pg 191 for more information.
My ...
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66 views
Conformal transformation from 2 circles to 2 line segments
thank you very much for creating this website and allow people to ask questions here. This is my question, I want to know if it is possible to find a conformal transformation that maps the inside of ...
2
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2answers
273 views
Möbius Transformation help
Hey guys I need help on these 2 questions that I am having trouble on.
1) Show that the Möbius transformation $z \rightarrow \frac{2}{1-z}$ sends the unit circle and the line $x = 1$ to the lines $x ...
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1answer
27 views
Conformal map projecting a line to a sine wave
I'm looking for an analytic complex function that will map a straight line on to a sine wave. Are there any known examples?
To be more specific, let
$f(x+iy) = u(x,y) + iv(x,y)$.
I want to find a ...
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0answers
53 views
Conformal map between $D$ and $\{z : |z| < 1 \, \text{and}\, Im(z) > i/\sqrt{2}\}$
Find a conformal (biholomorphic) map between the unit disc $D = \{z : |z|<1\}$ and the domain $U = \left\{z \in \mathbb{C} : |z| < 1 \mbox{ and } Im(z) > \frac{1}{\sqrt{2}}\right\}$
The ...
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1answer
48 views
the reciprocal of an exterior conformal mapping
I would like to ask a question about an exterior conformal mapping. Could you please help me?
Let $L$ be a Jordan curve in $\mathbb{C}.$ Let $G$ be the exterior of $L.$ Then, there exists a conformal ...
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1answer
39 views
nth derivative of an exterior conformal mapping in complex analysis
I would like to ask a question about an exterior conformal mapping. Could you please help me?
Let $L$ be a Jordan curve in $\mathbb{C}.$ Let $G$ be the exterior of $L.$ Then, there exists a conformal ...
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1answer
71 views
Characterization of one-to-one conformal mapping from unit disc onto a square
I'm trying to solve the following exercise from Rudin's Real & Complex Analysis, chapter 14 exercise 22.
Suppose $f$ is a one-to-one conformal mapping of $U$ onto a square with center at $0$, ...
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1answer
48 views
Angle preserving transformation
I've been working on a problem where I need to know the angle between the tangent vectors of two curves at their intersection point in a flat torus...
Then I thought: Consider two geodesics ...
6
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1answer
134 views
Maximize absolute value of complex logarithm
I'm trying to solve exercise 9 in chapter 14 of Real & Complex Analysis of Walter Rudin:
Suppose $g \in H(U), |\Re(g)|<1$ in $U$, and $g(0)=0$. Prove that
...
4
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1answer
117 views
Green's function for the Yamabe problem
I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8.
Theorem 2.8 (Existence of the Green Function).
Suppose $M$ is a ...
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1answer
88 views
Show that $f(z)=z^2+z$ is a conformal mapping and preserves angle
How do i show that this function preserves angle? There is no function given along which this is mapped. I know that its derivative is $f'(z)=2z+1$ so it is conformal for all points except 1 and $z_0$ ...
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1answer
306 views
Finding a conformal map of slit disk on to unit disk
I am trying to find a conformal map of slit unit disk (slit in negative real axis)
i.e $${{z: |z|<1, z\notin (-1,0]}}$$ on to the unit disk that takes $\sqrt2 -1 $ to $0$.
This is what I think,
...
5
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2answers
131 views
Find a one-to-one conformal mapping from $D(0;1) \setminus D(1;1)$ onto $D(0;1)$
I'm working on the conformal mappings of complex analysis. I can find conformal mappings from&onto simple domains but cannot find one from the abnormal domains.
Can any one give me a hint? Is ...
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1answer
114 views
If $f,g: U \rightarrow \Omega$ are holomorphic, $f(0)=g(0)$ and $f$ is 1-1&onto, then $f$ has larger image of a disk than that of $g$.
I'm working on the RCA of rudin but having a difficulty in the following problem:
Suppose $f$ and $g$ are holomorphic mappings of $U$(the unit circle centered at 0) into $\Omega$, $f$is one to one ...
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1answer
161 views
Conformal mapping into a unit disc
$T$ is the upper half of the unit disc $U$. What is the conformal mapping $f$ of $T$ onto $U$ that transforms $\{-1,0,1\}$ to $\{-1,-i,1\}$?
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1answer
157 views
linear fractional transformation with two fixed point on the unit circle
Let $f$ be a linear fractional transformation of the unit disc in itself, fixing points 1 and -1. Can i conclude that $f$ fixes the real axis?
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2answers
189 views
Conformal mapping of two circles into a line and a circle
I have the following conformal mapping:
I need to find $\lambda = f(\zeta)$ and its reverse. Zeros on the figure are given, the axis are oriented as usual. The resulting distance between the line ...
2
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1answer
56 views
linear fractional transformations fixing a line
I want to find all linear fractional transformations that fix the points 1 and -1. In particular i'd like to give this set a group structure and see if it is some familiar group or not. I wrote ...
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1answer
43 views
When is a function conformal at a pole?
I was wondering when a function was conformal at a pole? In class, when learning about Möbius transformations we put down a definition saying $f$ is conformal at a pole $z$ if $1/f$ is conformal at ...
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99 views
A homeomorphism from a semidisk to the unit disk
In Bak and Newman's Complex Analysis Chapter 14, Problem 12, the reader is asked to find a conformal mapping from the upper semidisk (with norm 1) $S$ to the unit disk $U$. Then, they ask to show that ...
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1answer
97 views
Harmonic functions and conformal mappings
I would like to get some insight into the practicalities of applying conformal mapping techniques for the numerical solution of PDEs. Up until now I had the impression that conformal mapping ...
2
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1answer
77 views
Conformal equivalence of tori
In $(x,y,z)$-space, take the circle in the $(x,z)$-plane of radius $1$ centered at $(R,0,0)$, where $R>1$, and revolve it about the $z$-axis, getting a torus embedded in that $3$-dimensional ...
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0answers
252 views
A conformal map from a horizontal half-strip to $H$
I have seen many examples of mapping the vertical half-strip, ie $-\pi/2 \lt x < \pi/2$, $y \gt 0$ to $H$(the upper half-plane) in $\mathbb{C}$ using the transformation $f = \sin z$. Would the ...
3
votes
2answers
326 views
Finding a conformal map from unit disk to half-plane
I'm trying to find a conformal map $f$ from the open unit disk to the set $\mathbb{C}-[-1/4,-\infty)$ (I think this means the half-plane Re$(w)>-1/4$ with the properties $f(0)=0$ and $f'(0)>0$. ...
0
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2answers
94 views
Map 2D points inside a closed curve to unit disk
Hy everyone,
I have a set of 2D cartesian points (x,y coordinates) lying inside an arbitrary closed contour , something like this: arbitrary_closed_contour
by 'arbitrary' I mean that the closed ...
3
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1answer
89 views
conformally equivalent flat tori
The interiors of any two rectangles are conformally equivalent, by the Riemann mapping theorem. Suppose with each rectangle, we glue opposite sides together, and the metric on the quotient space, ...
3
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1answer
222 views
Find Möbius transformation that send Re(z)=Im(z) to a circle and the real axis to itself
Problem 3.3.7d in Complex Variables, 2nd edition, by Stephen D. Fisher.
Find a linear fractional transformation $T$ that maps the real axis onto itself and the line $y=x$ onto the circle ...
2
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1answer
333 views
Mapping circles using Möbius transformations.
I need some help with the following problem from Ahlfors' Complex Analysis.
Problem: Find a single Möbius transformation $\phi$ (that is, a map of the form $\phi(z) = \dfrac{az + b}{cz + d}$, where ...
1
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1answer
147 views
I need some help solving a Dirichlet problem using a conformal map
I'm struggling here, trying to understand how to do this, and after 4 hours of reading, i still can't get around the concept and how to use it.
Basically, i have this problem:
A={(x,y) / x≥0, 0≤y≤pi
...
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1answer
230 views
Conformal map in 3D
I'm looking for a method to transform a three dimensional geometry. This geometry has a rotational symmetry, so the $r$- and $z$-coordinates are all the same over $\phi$. I want to transform this ...
1
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0answers
56 views
Pullback of conformal killing field via conformal map
This is my first time posting on this forum, so to start with, it's good to meet you all and thanks in advance for the help!
My question is as follows. Suppose I have two semi-riemannian manifolds of ...
3
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3answers
163 views
Boundary of product manifolds such as $S^2 \times \mathbb R$
Simple question but I am confused.
What is the boundary of $S^2\times\mathbb{R}$? Is it just $S^2$?
What would be the general way to evaluate the boundary of a product manifold?
Thanks for the ...
4
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2answers
154 views
Moving to a conformal metric
Given a generic 2-dimensional metric
$$
ds^2=E(x,y)dx^2+2F(x,y)dxdy+G(x,y)dy^2
$$
what is the change of coordinates that move it into the conformal form
$$
...
2
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2answers
199 views
circles and linear fractional transformations
I'm realizing how little (in some respects) I know about circles. Here's something that emerged out of something I was fiddling with. My question is whether this is
"well known" in the way that ...
